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  <title>P2119R0: Feedback on P1708: Simple Statistical Functions</title>
<style data-fill-with="stylesheet">/******************************************************************************
 *                   Style sheet for the W3C specifications                   *
 *
 * Special classes handled by this style sheet include:
 *
 * Indices
 *   - .toc for the Table of Contents (<ol class="toc">)
 *     + <span class="secno"> for the section numbers
 *   - #toc for the Table of Contents (<nav id="toc">)
 *   - ul.index for Indices (<a href="#ref">term</a><span>, in §N.M</span>)
 *   - table.index for Index Tables (e.g. for properties or elements)
 *
 * Structural Markup
 *   - table.data for general data tables
 *     -> use 'scope' attribute, <colgroup>, <thead>, and <tbody> for best results !
 *     -> use <table class='complex data'> for extra-complex tables
 *     -> use <td class='long'> for paragraph-length cell content
 *     -> use <td class='pre'> when manual line breaks/indentation would help readability
 *   - dl.switch for switch statements
 *   - ol.algorithm for algorithms (helps to visualize nesting)
 *   - .figure and .caption (HTML4) and figure and figcaption (HTML5)
 *     -> .sidefigure for right-floated figures
 *   - ins/del
 *
 * Code
 *   - pre and code
 *
 * Special Sections
 *   - .note       for informative notes             (div, p, span, aside, details)
 *   - .example    for informative examples          (div, p, pre, span)
 *   - .issue      for issues                        (div, p, span)
 *   - .assertion  for assertions                    (div, p, span)
 *   - .advisement for loud normative statements     (div, p, strong)
 *   - .annoying-warning for spec obsoletion notices (div, aside, details)
 *
 * Definition Boxes
 *   - pre.def   for WebIDL definitions
 *   - table.def for tables that define other entities (e.g. CSS properties)
 *   - dl.def    for definition lists that define other entitles (e.g. HTML elements)
 *
 * Numbering
 *   - .secno for section numbers in .toc and headings (<span class='secno'>3.2</span>)
 *   - .marker for source-inserted example/figure/issue numbers (<span class='marker'>Issue 4</span>)
 *   - ::before styled for CSS-generated issue/example/figure numbers:
 *     -> Documents wishing to use this only need to add
 *        figcaption::before,
 *        .caption::before { content: "Figure "  counter(figure) " ";  }
 *        .example::before { content: "Example " counter(example) " "; }
 *        .issue::before   { content: "Issue "   counter(issue) " ";   }
 *
 * Header Stuff (ignore, just don't conflict with these classes)
 *   - .head for the header
 *   - .copyright for the copyright
 *
 * Miscellaneous
 *   - .overlarge for things that should be as wide as possible, even if
 *     that overflows the body text area. This can be used on an item or
 *     on its container, depending on the effect desired.
 *     Note that this styling basically doesn't help at all when printing,
 *     since A4 paper isn't much wider than the max-width here.
 *     It's better to design things to fit into a narrower measure if possible.
 *   - js-added ToC jump links (see fixup.js)
 *
 ******************************************************************************/

/******************************************************************************/
/*                                   Body                                     */
/******************************************************************************/

	body {
		counter-reset: example figure issue;

		/* Layout */
		max-width: 50em;               /* limit line length to 50em for readability   */
		margin: 0 auto;                /* center text within page                     */
		padding: 1.6em 1.5em 2em 50px; /* assume 16px font size for downlevel clients */
		padding: 1.6em 1.5em 2em calc(26px + 1.5em); /* leave space for status flag     */

		/* Typography */
		line-height: 1.5;
		font-family: sans-serif;
		widows: 2;
		orphans: 2;
		word-wrap: break-word;
		overflow-wrap: break-word;
		hyphens: auto;

		/* Colors */
		color: black;
		background: white top left fixed no-repeat;
		background-size: 25px auto;
	}


/******************************************************************************/
/*                         Front Matter & Navigation                          */
/******************************************************************************/

/** Header ********************************************************************/

	div.head { margin-bottom: 1em }
	div.head hr { border-style: solid; }

	div.head h1 {
		font-weight: bold;
		margin: 0 0 .1em;
		font-size: 220%;
	}

	div.head h2 { margin-bottom: 1.5em;}

/** W3C Logo ******************************************************************/

	.head .logo {
		float: right;
		margin: 0.4rem 0 0.2rem .4rem;
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	.head img[src*="logos/W3C"] {
		display: block;
		border: solid #1a5e9a;
		border-width: .65rem .7rem .6rem;
		border-radius: .4rem;
		background: #1a5e9a;
		color: white;
		font-weight: bold;
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	.head a:hover > img[src*="logos/W3C"],
	.head a:focus > img[src*="logos/W3C"] {
		opacity: .8;
	}

	.head a:active > img[src*="logos/W3C"] {
		background: #c00;
		border-color: #c00;
	}

	/* see also additional rules in Link Styling section */

/** Copyright *****************************************************************/

	p.copyright,
	p.copyright small { font-size: small }

/** Back to Top / ToC Toggle **************************************************/

	@media print {
		#toc-nav {
			display: none;
		}
	}
	@media not print {
		#toc-nav {
			position: fixed;
			z-index: 2;
			bottom: 0; left: 0;
			margin: 0;
			min-width: 1.33em;
			border-top-right-radius: 2rem;
			box-shadow: 0 0 2px;
			font-size: 1.5em;
			color: black;
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		#toc-nav > a {
			display: block;
			white-space: nowrap;

			height: 1.33em;
			padding: .1em 0.3em;
			margin: 0;

			background: white;
			box-shadow: 0 0 2px;
			border: none;
			border-top-right-radius: 1.33em;
			background: white;
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		#toc-nav > #toc-jump {
			padding-bottom: 2em;
			margin-bottom: -1.9em;
		}

		#toc-nav > a:hover,
		#toc-nav > a:focus {
			background: #f8f8f8;
		}
		#toc-nav > a:not(:hover):not(:focus) {
			color: #707070;
		}

		/* statusbar gets in the way on keyboard focus; remove once browsers fix */
		#toc-nav > a[href="#toc"]:not(:hover):focus:last-child {
			padding-bottom: 1.5rem;
		}

		#toc-nav:not(:hover) > a:not(:focus) > span + span {
			/* Ideally this uses :focus-within on #toc-nav */
			display: none;
		}
		#toc-nav > a > span + span {
			padding-right: 0.2em;
		}

		#toc-toggle-inline {
			vertical-align: 0.05em;
			font-size: 80%;
			color: gray;
			color: hsla(203,20%,40%,.7);
			border-style: none;
			background: transparent;
			position: relative;
		}
		#toc-toggle-inline:hover:not(:active),
		#toc-toggle-inline:focus:not(:active) {
			text-shadow: 1px 1px silver;
			top: -1px;
			left: -1px;
		}

		#toc-nav :active {
			color: #C00;
		}
	}

/** ToC Sidebar ***************************************************************/

	/* Floating sidebar */
	@media screen {
		body.toc-sidebar #toc {
			position: fixed;
			top: 0; bottom: 0;
			left: 0;
			width: 23.5em;
			max-width: 80%;
			max-width: calc(100% - 2em - 26px);
			overflow: auto;
			padding: 0 1em;
			padding-left: 42px;
			padding-left: calc(1em + 26px);
			background: inherit;
			background-color: #f7f8f9;
			z-index: 1;
			box-shadow: -.1em 0 .25em rgba(0,0,0,.1) inset;
		}
		body.toc-sidebar #toc h2 {
			margin-top: .8rem;
			font-variant: small-caps;
			font-variant: all-small-caps;
			text-transform: lowercase;
			font-weight: bold;
			color: gray;
			color: hsla(203,20%,40%,.7);
		}
		body.toc-sidebar #toc-jump:not(:focus) {
			width: 0;
			height: 0;
			padding: 0;
			position: absolute;
			overflow: hidden;
		}
	}
	/* Hide main scroller when only the ToC is visible anyway */
	@media screen and (max-width: 28em) {
		body.toc-sidebar {
			overflow: hidden;
		}
	}

	/* Sidebar with its own space */
	@media screen and (min-width: 78em) {
		body:not(.toc-inline) #toc {
			position: fixed;
			top: 0; bottom: 0;
			left: 0;
			width: 23.5em;
			overflow: auto;
			padding: 0 1em;
			padding-left: 42px;
			padding-left: calc(1em + 26px);
			background: inherit;
			background-color: #f7f8f9;
			z-index: 1;
			box-shadow: -.1em 0 .25em rgba(0,0,0,.1) inset;
		}
		body:not(.toc-inline) #toc h2 {
			margin-top: .8rem;
			font-variant: small-caps;
			font-variant: all-small-caps;
			text-transform: lowercase;
			font-weight: bold;
			color: gray;
			color: hsla(203,20%,40%,.7);
		}

		body:not(.toc-inline) {
			padding-left: 29em;
		}
		/* See also Overflow section at the bottom */

		body:not(.toc-inline) #toc-jump:not(:focus) {
			width: 0;
			height: 0;
			padding: 0;
			position: absolute;
			overflow: hidden;
		}
	}
	@media screen and (min-width: 90em) {
		body:not(.toc-inline) {
			margin: 0 4em;
		}
	}

/******************************************************************************/
/*                                Sectioning                                  */
/******************************************************************************/

/** Headings ******************************************************************/

	h1, h2, h3, h4, h5, h6, dt {
		page-break-after: avoid;
		page-break-inside: avoid;
		font: 100% sans-serif;   /* Reset all font styling to clear out UA styles */
		font-family: inherit;    /* Inherit the font family. */
		line-height: 1.2;        /* Keep wrapped headings compact */
		hyphens: manual;         /* Hyphenated headings look weird */
	}

	h2, h3, h4, h5, h6 {
		margin-top: 3rem;
	}

	h1, h2, h3 {
		color: #005A9C;
		background: transparent;
	}

	h1 { font-size: 170%; }
	h2 { font-size: 140%; }
	h3 { font-size: 120%; }
	h4 { font-weight: bold; }
	h5 { font-style: italic; }
	h6 { font-variant: small-caps; }
	dt { font-weight: bold; }

/** Subheadings ***************************************************************/

	h1 + h2,
	#subtitle {
		/* #subtitle is a subtitle in an H2 under the H1 */
		margin-top: 0;
	}
	h2 + h3,
	h3 + h4,
	h4 + h5,
	h5 + h6 {
		margin-top: 1.2em; /* = 1 x line-height */
	}

/** Section divider ***********************************************************/

	:not(.head) > hr {
		font-size: 1.5em;
		text-align: center;
		margin: 1em auto;
		height: auto;
		border: transparent solid 0;
		background: transparent;
	}
	:not(.head) > hr::before {
		content: "\2727\2003\2003\2727\2003\2003\2727";
	}

/******************************************************************************/
/*                            Paragraphs and Lists                            */
/******************************************************************************/

	p {
		margin: 1em 0;
	}

	dd > p:first-child,
	li > p:first-child {
		margin-top: 0;
	}

	ul, ol {
		margin-left: 0;
		padding-left: 2em;
	}

	li {
		margin: 0.25em 0 0.5em;
		padding: 0;
	}

	dl dd {
		margin: 0 0 .5em 2em;
	}

	.head dd + dd { /* compact for header */
		margin-top: -.5em;
	}

	/* Style for algorithms */
	ol.algorithm ol:not(.algorithm),
	.algorithm > ol ol:not(.algorithm) {
	 border-left: 0.5em solid #DEF;
	}

	/* Put nice boxes around each algorithm. */
	[data-algorithm]:not(.heading) {
	  padding: .5em;
	  border: thin solid #ddd; border-radius: .5em;
	  margin: .5em calc(-0.5em - 1px);
	}
	[data-algorithm]:not(.heading) > :first-child {
	  margin-top: 0;
	}
	[data-algorithm]:not(.heading) > :last-child {
	  margin-bottom: 0;
	}

	/* Style for switch/case <dl>s */
	dl.switch > dd > ol.only,
	dl.switch > dd > .only > ol {
	 margin-left: 0;
	}
	dl.switch > dd > ol.algorithm,
	dl.switch > dd > .algorithm > ol {
	 margin-left: -2em;
	}
	dl.switch {
	 padding-left: 2em;
	}
	dl.switch > dt {
	 text-indent: -1.5em;
	 margin-top: 1em;
	}
	dl.switch > dt + dt {
	 margin-top: 0;
	}
	dl.switch > dt::before {
	 content: '\21AA';
	 padding: 0 0.5em 0 0;
	 display: inline-block;
	 width: 1em;
	 text-align: right;
	 line-height: 0.5em;
	}

/** Terminology Markup ********************************************************/


/******************************************************************************/
/*                                 Inline Markup                              */
/******************************************************************************/

/** Terminology Markup ********************************************************/
	dfn   { /* Defining instance */
		font-weight: bolder;
	}
	a > i { /* Instance of term */
		font-style: normal;
	}
	dt dfn code, code.idl {
		font-size: medium;
	}
	dfn var {
		font-style: normal;
	}

/** Change Marking ************************************************************/

	del { color: red;  text-decoration: line-through; }
	ins { color: #080; text-decoration: underline;    }

/** Miscellaneous improvements to inline formatting ***************************/

	sup {
		vertical-align: super;
		font-size: 80%
	}

/******************************************************************************/
/*                                    Code                                    */
/******************************************************************************/

/** General monospace/pre rules ***********************************************/

	pre, code, samp {
		font-family: Menlo, Consolas, "DejaVu Sans Mono", Monaco, monospace;
		font-size: .9em;
		page-break-inside: avoid;
		hyphens: none;
		text-transform: none;
	}
	pre code,
	code code {
		font-size: 100%;
	}

	pre {
		margin-top: 1em;
		margin-bottom: 1em;
		overflow: auto;
	}

/** Inline Code fragments *****************************************************/

  /* Do something nice. */

/******************************************************************************/
/*                                    Links                                   */
/******************************************************************************/

/** General Hyperlinks ********************************************************/

	/* We hyperlink a lot, so make it less intrusive */
	a[href] {
		color: #034575;
		text-decoration: none;
		border-bottom: 1px solid #707070;
		/* Need a bit of extending for it to look okay */
		padding: 0 1px 0;
		margin: 0 -1px 0;
	}
	a:visited {
		border-bottom-color: #BBB;
	}

	/* Use distinguishing colors when user is interacting with the link */
	a[href]:focus,
	a[href]:hover {
		background: #f8f8f8;
		background: rgba(75%, 75%, 75%, .25);
		border-bottom-width: 3px;
		margin-bottom: -2px;
	}
	a[href]:active {
		color: #C00;
		border-color: #C00;
	}

	/* Backout above styling for W3C logo */
	.head .logo,
	.head .logo a {
		border: none;
		text-decoration: none;
		background: transparent;
	}

/******************************************************************************/
/*                                    Images                                  */
/******************************************************************************/

	img {
		border-style: none;
	}

	/* For autogen numbers, add
	   .caption::before, figcaption::before { content: "Figure " counter(figure) ". "; }
	*/

	figure, .figure, .sidefigure {
		page-break-inside: avoid;
		text-align: center;
		margin: 2.5em 0;
	}
	.figure img,    .sidefigure img,    figure img,
	.figure object, .sidefigure object, figure object {
		max-width: 100%;
		margin: auto;
	}
	.figure pre, .sidefigure pre, figure pre {
		text-align: left;
		display: table;
		margin: 1em auto;
	}
	.figure table, figure table {
		margin: auto;
	}
	@media screen and (min-width: 20em) {
		.sidefigure {
			float: right;
			width: 50%;
			margin: 0 0 0.5em 0.5em
		}
	}
	.caption, figcaption, caption {
		font-style: italic;
		font-size: 90%;
	}
	.caption::before, figcaption::before, figcaption > .marker {
		font-weight: bold;
	}
	.caption, figcaption {
		counter-increment: figure;
	}

	/* DL list is indented 2em, but figure inside it is not */
	dd > .figure, dd > figure { margin-left: -2em }

/******************************************************************************/
/*                             Colored Boxes                                  */
/******************************************************************************/

	.issue, .note, .example, .assertion, .advisement, blockquote {
		padding: .5em;
		border: .5em;
		border-left-style: solid;
		page-break-inside: avoid;
	}
	span.issue, span.note {
		padding: .1em .5em .15em;
		border-right-style: solid;
	}

	.issue,
	.note,
	.example,
	.advisement,
	.assertion,
	blockquote {
		margin: 1em auto;
	}
	.note  > p:first-child,
	.issue > p:first-child,
	blockquote > :first-child {
		margin-top: 0;
	}
	blockquote > :last-child {
		margin-bottom: 0;
	}

/** Blockquotes ***************************************************************/

	blockquote {
		border-color: silver;
	}

/** Open issue ****************************************************************/

	.issue {
		border-color: #E05252;
		background: #FBE9E9;
		counter-increment: issue;
		overflow: auto;
	}
	.issue::before, .issue > .marker {
		text-transform: uppercase;
		color: #AE1E1E;
		padding-right: 1em;
		text-transform: uppercase;
	}
	/* Add .issue::before { content: "Issue " counter(issue) " "; } for autogen numbers,
	   or use class="marker" to mark up the issue number in source. */

/** Example *******************************************************************/

	.example {
		border-color: #E0CB52;
		background: #FCFAEE;
		counter-increment: example;
		overflow: auto;
		clear: both;
	}
	.example::before, .example > .marker {
		text-transform: uppercase;
		color: #827017;
		min-width: 7.5em;
		display: block;
	}
	/* Add .example::before { content: "Example " counter(example) " "; } for autogen numbers,
	   or use class="marker" to mark up the example number in source. */

/** Non-normative Note ********************************************************/

	.note {
		border-color: #52E052;
		background: #E9FBE9;
		overflow: auto;
	}

	.note::before, .note > .marker,
	details.note > summary::before,
	details.note > summary > .marker {
		text-transform: uppercase;
		display: block;
		color: hsl(120, 70%, 30%);
	}
	/* Add .note::before { content: "Note"; } for autogen label,
	   or use class="marker" to mark up the label in source. */

	details.note > summary {
		display: block;
		color: hsl(120, 70%, 30%);
	}
	details.note[open] > summary {
		border-bottom: 1px silver solid;
	}

/** Assertion Box *************************************************************/
	/*  for assertions in algorithms */

	.assertion {
		border-color: #AAA;
		background: #EEE;
	}

/** Advisement Box ************************************************************/
	/*  for attention-grabbing normative statements */

	.advisement {
		border-color: orange;
		border-style: none solid;
		background: #FFEECC;
	}
	strong.advisement {
		display: block;
		text-align: center;
	}
	.advisement > .marker {
		color: #B35F00;
	}

/** Spec Obsoletion Notice ****************************************************/
	/* obnoxious obsoletion notice for older/abandoned specs. */

	details {
		display: block;
	}
	summary {
		font-weight: bolder;
	}

	.annoying-warning:not(details),
	details.annoying-warning:not([open]) > summary,
	details.annoying-warning[open] {
		background: #fdd;
		color: red;
		font-weight: bold;
		padding: .75em 1em;
		border: thick red;
		border-style: solid;
		border-radius: 1em;
	}
	.annoying-warning :last-child {
		margin-bottom: 0;
	}

@media not print {
	details.annoying-warning[open] {
		position: fixed;
		left: 1em;
		right: 1em;
		bottom: 1em;
		z-index: 1000;
	}
}

	details.annoying-warning:not([open]) > summary {
		text-align: center;
	}

/** Entity Definition Boxes ***************************************************/

	.def {
		padding: .5em 1em;
		background: #DEF;
		margin: 1.2em 0;
		border-left: 0.5em solid #8CCBF2;
	}

/******************************************************************************/
/*                                    Tables                                  */
/******************************************************************************/

	th, td {
		text-align: left;
		text-align: start;
	}

/** Property/Descriptor Definition Tables *************************************/

	table.def {
		/* inherits .def box styling, see above */
		width: 100%;
		border-spacing: 0;
	}

	table.def td,
	table.def th {
		padding: 0.5em;
		vertical-align: baseline;
		border-bottom: 1px solid #bbd7e9;
	}

	table.def > tbody > tr:last-child th,
	table.def > tbody > tr:last-child td {
		border-bottom: 0;
	}

	table.def th {
		font-style: italic;
		font-weight: normal;
		padding-left: 1em;
		width: 3em;
	}

	/* For when values are extra-complex and need formatting for readability */
	table td.pre {
		white-space: pre-wrap;
	}

	/* A footnote at the bottom of a def table */
	table.def           td.footnote {
		padding-top: 0.6em;
	}
	table.def           td.footnote::before {
		content: " ";
		display: block;
		height: 0.6em;
		width: 4em;
		border-top: thin solid;
	}

/** Data tables (and properly marked-up index tables) *************************/
	/*
		 <table class="data"> highlights structural relationships in a table
		 when correct markup is used (e.g. thead/tbody, th vs. td, scope attribute)

		 Use class="complex data" for particularly complicated tables --
		 (This will draw more lines: busier, but clearer.)

		 Use class="long" on table cells with paragraph-like contents
		 (This will adjust text alignment accordingly.)
		 Alternately use class="longlastcol" on tables, to have the last column assume "long".
	*/

	table {
		word-wrap: normal;
		overflow-wrap: normal;
		hyphens: manual;
	}

	table.data,
	table.index {
		margin: 1em auto;
		border-collapse: collapse;
		border: hidden;
		width: 100%;
	}
	table.data caption,
	table.index caption {
		max-width: 50em;
		margin: 0 auto 1em;
	}

	table.data td,  table.data th,
	table.index td, table.index th {
		padding: 0.5em 1em;
		border-width: 1px;
		border-color: silver;
		border-top-style: solid;
	}

	table.data thead td:empty {
		padding: 0;
		border: 0;
	}

	table.data  thead,
	table.index thead,
	table.data  tbody,
	table.index tbody {
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/*
Alternate table alignment rules

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Possible extra rowspan handling

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*/

/******************************************************************************/
/*                                  Indices                                   */
/******************************************************************************/


/** Table of Contents *********************************************************/

	.toc a {
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		/* Larger, more consistently-sized click target */
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		.toc > li li li .secno       { margin-left: -7rem; }
		.toc > li li li li .secno    { margin-left: -9rem; }
		.toc > li li li li li .secno { margin-left: -11rem; }

		/* Tighten up indentation in narrow ToCs */
		@media (max-width: 30em) {
			:not(li) > .toc              { margin-left:  4rem; }
			.toc .secno                  { margin-left: -4rem; }
			.toc > li li li              { margin-left:  1rem; }
			.toc > li li li .secno       { margin-left: -5rem; }
			.toc > li li li li .secno    { margin-left: -6rem; }
			.toc > li li li li li .secno { margin-left: -7rem; }
		}
	/* } */

	@supports (display:grid) and (display:contents) {
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		#toc {
			display: grid;
			align-content: start;
			grid-template-columns: auto 1fr;
			grid-column-gap: 1rem;
			column-gap: 1rem;
			grid-row-gap: .6rem;
			row-gap: .6rem;
		}
		#toc h2 {
			grid-column: 1 / -1;
			margin-bottom: 0;
		}
		#toc ol,
		#toc li,
		#toc a {
			display: contents;
			/* Switch <a> to subgrid when supported */
		}
		#toc span {
			margin: 0;
		}
		#toc > .toc > li > a > span {
			/* The spans of the top-level list,
			   comprising the first items of each top-level section. */
			margin-top: 1.1rem;
		}
		#toc#toc .secno { /* Ugh, need more specificity to override base.css */
			grid-column: 1;
			width: auto;
			margin-left: 0;
		}
		#toc .content {
			grid-column: 2;
			width: auto;
			margin-right: 1rem;
		}
		#toc .content:hover {
			background: rgba(75%, 75%, 75%, .25);
			border-bottom: 3px solid #054572;
			margin-bottom: -3px;
		}
		#toc li li li .content {
			margin-left: 1rem;
		}
		#toc li li li li .content {
			margin-left: 2rem;
		}
	}


/** Index *********************************************************************/

	/* Index Lists: Layout */
	ul.index       { margin-left: 0; columns: 15em; text-indent: 1em hanging; }
	ul.index li    { margin-left: 0; list-style: none; break-inside: avoid; }
	ul.index li li { margin-left: 1em }
	ul.index dl    { margin-top: 0; }
	ul.index dt    { margin: .2em 0 .2em 20px;}
	ul.index dd    { margin: .2em 0 .2em 40px;}
	/* Index Lists: Typography */
	ul.index ul,
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			white-space: nowrap;
			color: transparent; }
		ul.index li a:hover + span,
		ul.index li a:focus + span {
			color: #707070;
		}
	}

/** Index Tables *****************************************************/
	/* See also the data table styling section, which this effectively subclasses */

	table.index {
		font-size: small;
		border-collapse: collapse;
		border-spacing: 0;
		text-align: left;
		margin: 1em 0;
	}

	table.index td,
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		padding: 0.4em;
	}

	table.index tr:hover td:not([rowspan]),
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		background: #f7f8f9;
	}

	/* The link in the first column in the property table (formerly a TD) */
	table.index th:first-child a {
		font-weight: bold;
	}

/******************************************************************************/
/*                                    Print                                   */
/******************************************************************************/

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		/* Pages have their own margins. */
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			margin: 0;
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		/* Serif for print. */
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			font-family: serif;
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	@page {
		margin: 1.5cm 1.1cm;
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/******************************************************************************/
/*                                    Legacy                                  */
/******************************************************************************/

	/* This rule is inherited from past style sheets. No idea what it's for. */
	.hide { display: none }



/******************************************************************************/
/*                             Overflow Control                               */
/******************************************************************************/

	.figure .caption, .sidefigure .caption, figcaption {
		/* in case figure is overlarge, limit caption to 50em */
		max-width: 50rem;
		margin-left: auto;
		margin-right: auto;
	}
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		/* Magic to create good table positioning:
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		   Extra space (after ToC + content) is empty on the right.

		   1. When table < content column, centers table in column.
		   2. When content < table < available, left-aligns.
		   3. When table > available, fills available + scroll bar.
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		grid-template-columns: minmax(0, 50em);
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		}
	}

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			 *
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			background-repeat: no-repeat;
			*/
		}
	}
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 <body class="h-entry" dir="auto">
  <div class="head">
   <p data-fill-with="logo"></p>
   <h1 class="p-name no-ref" id="title">P2119R0<br>Feedback on P1708: Simple Statistical Functions</h1>
   <h2 class="no-num no-toc no-ref heading settled" id="subtitle"><span class="content">Published Proposal, <time class="dt-updated" datetime="2020-02-13">2020-02-13</time></span></h2>
   <div data-fill-with="spec-metadata">
    <dl>
     <dt>Author:
     <dd>
      <dd class="editor p-author h-card vcard"><a class="p-name fn u-email email" href="mailto:jolanta.opara@mobica.com">Jolanta Opara</a> (<a class="p-org org" href="http://mobica.com">Mobica Limited sp.z o.o. Oddział w Polsce</a>)
     <dt>Audience:
     <dd>SG19
     <dt>Project:
     <dd>ISO/IEC JTC1/SC22/WG21 14882: Programming Language — C++
    </dl>
   </div>
   <div data-fill-with="warning"></div>
   <hr title="Separator for header">
  </div>
  <div class="p-summary" data-fill-with="abstract">
   <h2 class="no-num no-toc no-ref heading settled" id="abstract"><span class="content">Abstract</span></h2>
   <p>The paper contains comments about <a data-link-type="biblio" href="#biblio-p1708r0">[P1708R0]</a>, <a data-link-type="biblio" href="#biblio-p1708r1">[P1708R1]</a> and <a data-link-type="biblio" href="#biblio-p1708r2">[P1708R2]</a>. It also provides a comparison of the analogical functionalities provided in python, R, SAS and Matlab and some comments from a users' perspective.</p>
  </div>
  <nav data-fill-with="table-of-contents" id="toc">
   <h2 class="no-num no-toc no-ref" id="contents">Table of Contents</h2>
   <ol class="toc" role="directory">
    <li><a href="#magicparlabel-53"><span class="secno">1</span> <span class="content">Motivation</span></a>
    <li><a href="#magicparlabel-56"><span class="secno">2</span> <span class="content">Users' perspective</span></a>
    <li><a href="#magicparlabel-260"><span class="secno">3</span> <span class="content"> Common expectations for all statistics</span></a>
    <li>
     <a href="#magicparlabel-76"><span class="secno">4</span> <span class="content">Free-standing statistical functions</span></a>
     <ol class="toc">
      <li>
       <a href="#magicparlabel-78"><span class="secno">4.1</span> <span class="content">Mean</span></a>
       <ol class="toc">
        <li><a href="#magicparlabel-79"><span class="secno">4.1.1</span> <span class="content">Definition</span></a>
        <li><a href="#magicparlabel-81"><span class="secno">4.1.2</span> <span class="content">Users' perspective</span></a>
        <li><a href="#magicparlabel-83"><span class="secno">4.1.3</span> <span class="content"> Interfaces in other languages</span></a>
       </ol>
      <li>
       <a href="#magicparlabel-262"><span class="secno">4.2</span> <span class="content"> Median</span></a>
       <ol class="toc">
        <li><a href="#magicparlabel-263"><span class="secno">4.2.1</span> <span class="content">Definition</span></a>
        <li><a href="#magicparlabel-265"><span class="secno">4.2.2</span> <span class="content">Users' perspective</span></a>
        <li><a href="#magicparlabel-267"><span class="secno">4.2.3</span> <span class="content"> Interfaces in other languages</span></a>
       </ol>
      <li>
       <a href="#magicparlabel-458"><span class="secno">4.3</span> <span class="content">Quantiles</span></a>
       <ol class="toc">
        <li><a href="#magicparlabel-460"><span class="secno">4.3.1</span> <span class="content">Definition</span></a>
        <li><a href="#magicparlabel-460①"><span class="secno">4.3.2</span> <span class="content">Users' perspective</span></a>
        <li><a href="#magicparlabel-462"><span class="secno">4.3.3</span> <span class="content"> Interfaces in other languages</span></a>
       </ol>
      <li>
       <a href="#magicparlabel-658"><span class="secno">4.4</span> <span class="content">Mode</span></a>
       <ol class="toc">
        <li><a href="#magicparlabel-659"><span class="secno">4.4.1</span> <span class="content">Definition</span></a>
        <li><a href="#magicparlabel-661"><span class="secno">4.4.2</span> <span class="content">Users' perspective</span></a>
        <li><a href="#magicparlabel-663"><span class="secno">4.4.3</span> <span class="content"> Interfaces in other languages</span></a>
       </ol>
      <li>
       <a href="#magicparlabel-848"><span class="secno">4.5</span> <span class="content"> Standard deviation</span></a>
       <ol class="toc">
        <li><a href="#magicparlabel-849"><span class="secno">4.5.1</span> <span class="content">Definition </span></a>
        <li><a href="#magicparlabel-851"><span class="secno">4.5.2</span> <span class="content"> Users' perspective</span></a>
        <li><a href="#magicparlabel-853"><span class="secno">4.5.3</span> <span class="content"> Interfaces in other languages</span></a>
       </ol>
      <li>
       <a href="#magicparlabel-1048"><span class="secno">4.6</span> <span class="content">Variance</span></a>
       <ol class="toc">
        <li><a href="#magicparlabel-1050"><span class="secno">4.6.1</span> <span class="content"> Users' perspective</span></a>
        <li><a href="#magicparlabel-1052"><span class="secno">4.6.2</span> <span class="content"> Interfaces in other languages</span></a>
       </ol>
     </ol>
    <li>
     <a href="#magicparlabel-1249"><span class="secno">5</span> <span class="content">Grouped statistics</span></a>
     <ol class="toc">
      <li><a href="#magicparlabel-1251"><span class="secno">5.1</span> <span class="content"> Expectations from the users' perspective</span></a>
      <li><a href="#magicparlabel-1257"><span class="secno">5.2</span> <span class="content"> Users' perspective</span></a>
     </ol>
    <li><a href="#magicparlabel-1264"><span class="secno">6</span> <span class="content">Some use-cases</span></a>
    <li><a href="#magicparlabel-1258"><span class="secno">7</span> <span class="content"> Summary</span></a>
    <li><a href="#magicparlabel-1261"><span class="secno">8</span> <span class="content">Document history</span></a>
    <li><a href="#magicparlabel-1263"><span class="secno">9</span> <span class="content">Acknowledgements</span></a>
    <li>
     <a href="#references"><span class="secno"></span> <span class="content">References</span></a>
     <ol class="toc">
      <li><a href="#normative"><span class="secno"></span> <span class="content">Normative References</span></a>
      <li><a href="#informative"><span class="secno"></span> <span class="content">Informative References</span></a>
     </ol>
   </ol>
  </nav>
  <main>
   <p></p>
   <h2 class="section heading settled" data-level="1" id="magicparlabel-53"><span class="secno">1. </span><span class="content">Motivation</span><a class="self-link" href="#magicparlabel-53"></a></h2>
   <div class="standard" id="magicparlabel-55">This paper reviews <a data-link-type="biblio" href="#biblio-p1708r0">[P1708R0]</a>, <a data-link-type="biblio" href="#biblio-p1708r1">[P1708R1]</a> and <a data-link-type="biblio" href="#biblio-p1708r2">[P1708R2]</a>: Simple Statistical Functions and summarizes the discussions about the functionality. All the reviews are mentioned because the approach towards the functionality changes between the reviews.</div>
   <div class="standard" id="magicparlabel-55①"> The idea of the paper is to provide an overview of users' expectations towards the statistical functions and a comparison of analogical features available in other programming languages. The paper does not propose any syntax. Subsections about users' perspective contain opinions only.</div>
   <h2 class="section heading settled" data-level="2" id="magicparlabel-56"><span class="secno">2. </span><span class="content">Users' perspective</span><a class="self-link" href="#magicparlabel-56"></a></h2>
   <div class="standard" id="magicparlabel-57">Simple statistical functions are available not only in python, as it was mentioned in <a data-link-type="biblio" href="#biblio-p1708r0">[P1708R0]</a>, but also in R, S, SAS, Matlab and many other languages used by statisticians. </div>
   <div class="standard" id="magicparlabel-58">Potential users expect both free-standing simple statistical functions (for simlicity) and some way to calculate them all in one pass, or as fewer passes as possible, over a container (for speed). Additionally, they expect user-friendly, intuitive and extendable interface for both use-cases. Preferably, it should be easy to use for an inexperienced user and easy to customize for more sophisticated use-cases. </div>
   <div class="standard" id="magicparlabel-59">Typically, users need to calculate one statistic with the default parameters. It seems excessive to require an additional object in these cases. The requirement that the collection is sorted is unintuitive. Hence, it is unexpected as the default option and it may be error-prone. Morover, the necessity to sort the collection before the calculation of statistics may lead to computational overhead.</div>
   <h2 class="section heading settled" data-level="3" id="magicparlabel-260"><span class="secno">3. </span><span class="content"> Common expectations for all statistics</span><a class="self-link" href="#magicparlabel-260"></a></h2>
   <ul class="itemize" id="magicparlabel-61">
    <li class="itemize_item">both iterator-based and range-based versions
    <li class="itemize_item">free-standing functions that are simple in use
    <li class="itemize_item">possibility to compute multiple statistics of the same sample in a comutationally effective and intuitive way, consistent with the interface of free-standing functions
    <li class="itemize_item">possibility to choose internal representation/return type for the calculation (e.g. the mean of integers can be fractional, so should computations be made on ints, floats or doubles?)
    <li class="itemize_item">customization of NANs and missing data: yes/no, functor, list of values (?). This is a common issue in many statistical domains.
    <li class="itemize_item">transformation possibility 
   </ul>
   <div class="standard" id="magicparlabel-67">The last two points seem to be achievable with ranges, for example:</div>
   <div class="float-listings">
<pre class="listings highlight"><c- n>range</c-> <c- o>|</c-> <c- n>std</c-><c- o>::</c-><c- n>views</c-><c- o>::</c-><c- n>filter</c-><c- p>(</c-><c- n>not_nan_in_key</c-><c- p>)</c-> <c- o>|</c-> <c- n>std</c-><c- o>::</c-><c- n>views</c-><c- o>::</c-><c- n>transform</c-><c- p>(</c-><c- n>value_only</c-><c- p>)</c-> </pre>
   </div>
   <div class="standard" id="magicparlabel-67①">or depending on the user needs</div>
   <div class="float-listings">
<pre class="listings highlight"><c- n>range</c-> <c- o>|</c-> <c- n>std</c-><c- o>::</c-><c- n>views</c-><c- o>::</c-><c- n>trasform</c-><c- p>(</c-><c- n>key_only</c-><c- p>)</c-> <c- o>|</c-> <c- n>std</c-><c- o>::</c-><c- n>views</c-><c- o>::</c-><c- n>filter</c-><c- p>(</c-><c- n>not_nan</c-><c- p>)</c-></pre>
   </div>
   <h2 class="section heading settled" data-level="4" id="magicparlabel-76"><span class="secno">4. </span><span class="content">Free-standing statistical functions</span><a class="self-link" href="#magicparlabel-76"></a></h2>
   <div class="standard" id="magicparlabel-77">Some statistical functions have been proposed in [<a data-link-type="biblio" href="#biblio-n1668">[N1668]</a>: A Proposal to add Mathematical Functions for Statistics to the C++ Standard Library, but the proposal did not cover the topics described in the P1708 paper. The N1668 proposes functions to calculate the statistical properties of common probability distributions (e.g. quantiles, densities of the normal distribution with chosen parameters) whereas P1708 proposes functions to calculate statistical properties of a sample.</div>
   <h3 class="subsection heading settled" data-level="4.1" id="magicparlabel-78"><span class="secno">4.1. </span><span class="content">Mean</span><a class="self-link" href="#magicparlabel-78"></a></h3>
   <h4 class="subsubsection heading settled" data-level="4.1.1" id="magicparlabel-79"><span class="secno">4.1.1. </span><span class="content">Definition</span><a class="self-link" href="#magicparlabel-79"></a></h4>
   <div class="standard" id="magicparlabel-80">The arithmetic mean of the sample. The definition is given in <a data-link-type="biblio" href="#biblio-p1708r2">[P1708R2]</a></div>
   <h4 class="subsubsection heading settled" data-level="4.1.2" id="magicparlabel-81"><span class="secno">4.1.2. </span><span class="content">Users' perspective</span><a class="self-link" href="#magicparlabel-81"></a></h4>
   <div class="standard" id="magicparlabel-82">The most commonly used mean function is an arithmetic mean defined in P1708. However, it could be extended to compute the trimmed arithmetic mean, which is robust against outliers. It would be useful to have it supported under the same or different function name.</div>
   <h4 class="subsubsection heading settled" data-level="4.1.3" id="magicparlabel-83"><span class="secno">4.1.3. </span><span class="content"> Interfaces in other languages</span><a class="self-link" href="#magicparlabel-83"></a></h4>
   <div class="standard" id="magicparlabel-84">
    <table>
     <tbody>
      <tr>
       <td align="center" style="width: 4%;" valign="top">
        <div class="plain_layout" id="magicparlabel-131">Language</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-134">Python</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-137">R</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-140">SAS</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-143">Matlab</div>
      <tr>
       <td align="center" style="width: 4%;" valign="top">
        <div class="plain_layout" id="magicparlabel-146">Syntax</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="float-listings">
<pre class="listings highlight"><c- n>mean</c-><c- p>([</c-><c- n>data</c-><c- o>-</c-><c- n>set</c-><c- p>])</c-></pre>
        </div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="float-listings">
<pre class="listings highlight"><c- n>mean</c-><c- p>([</c-><c- n>x</c-><c- p>,</c-> <c- n>trim</c-> <c- o>=</c-> <c- mi>0</c-><c- p>,</c-> <c- n>na</c-><c- p>.</c-><c- n>rm</c-> <c- o>=</c-> <c- n>FALSE</c-><c- p>,...</c-> <c- p>)</c-></pre>
        </div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="float-listings">
<pre class="listings highlight"><c- n>MEAN</c-><c- p>(</c-><c- n>argument</c-><c- o>-</c-><c- mi>1</c-> <c- o>&lt;</c-><c- p>,</c-> <c- p>...</c-> <c- n>argument</c-><c- o>-</c-><c- n>n</c-><c- o>></c-><c- p>)</c-></pre>
        </div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="float-listings">
<pre class="listings highlight"><c- n>M</c-> <c- o>=</c-> <c- n>mean</c-><c- p>(</c-><c- n>A</c-><c- p>)</c-> 
<c- n>M</c-> <c- o>=</c-> <c- n>mean</c-><c- p>(</c-><c- n>A</c-><c- p>,</c->'<c- n>all</c->'<c- p>)</c-> 
<c- n>M</c-> <c- o>=</c-> <c- n>mean</c-><c- p>(</c-><c- n>A</c-><c- p>,</c-><c- n>dim</c-><c- p>)</c-> 
<c- n>M</c-> <c- o>=</c-> <c- n>mean</c-><c- p>(</c-><c- n>A</c-><c- p>,</c-><c- n>vecdim</c-><c- p>)</c->
<c- n>M</c-> <c- o>=</c-> <c- n>mean</c-><c- p>(</c-><c- n>___</c-><c- p>,</c-><c- n>outtype</c-><c- p>)</c-> 
<c- n>M</c-> <c- o>=</c-> <c- n>mean</c-><c- p>(</c-><c- n>___</c-><c- p>,</c-><c- n>nanflag</c-><c- p>)</c-></pre>
        </div>
      <tr>
       <td align="center" style="width: 4%;" valign="top">
        <div class="plain_layout" id="magicparlabel-182">Parameters</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-185">[data-set]: List or tuple of a set of numbers.</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-188">
         x: An R object. 
         <div class="plain_layout" id="magicparlabel-188①">trim: the fraction (0 to 0.5) of observations to be trimmed from each end of x before the mean is computed.</div>
         <div class="plain_layout" id="magicparlabel-188②"> na.rm: a logical value indicating whether NA values should be stripped before the computation proceeds.</div>
         <div class="plain_layout" id="magicparlabel-188③"> …: further arguments passed to or from other methods.</div>
        </div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-191">argument: specifies a numeric constant, variable, or expression.</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-194">A: matrix or vector</div>
        <div class="plain_layout" id="magicparlabel-195">all: computes the mean over all elements of A.</div>
        <div class="plain_layout" id="magicparlabel-196">dim: returns the mean along dimension dim</div>
        <div class="plain_layout" id="magicparlabel-197">vecdim: computes the mean based on the dimensions specified in the vector vecdim</div>
        <div class="plain_layout" id="magicparlabel-198">outtype: returns the mean with a specified data type, using any of the input arguments in the previous syntaxes. outtype can be <a class="property" data-link-type="propdesc">default</a>, <a class="property" data-link-type="propdesc">double</a>, or <a class="property" data-link-type="propdesc">native</a>.</div>
        <div class="plain_layout" id="magicparlabel-199">nanflag: specifies whether to include or omit NaN values from the calculation for any of the previous syntaxes. mean(A,<a class="property" data-link-type="propdesc">includenan</a>) includes all NaN values in the calculation while mean(A,<a class="property" data-link-type="propdesc">omitnan</a>) ignores them.</div>
      <tr>
       <td align="center" style="width: 4%;" valign="top">
        <div class="plain_layout" id="magicparlabel-202">Returns</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-205">Sample arithmetic mean of the provided data-set.</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-208">Sample arithmetic mean of the (optionally: non-missing) provided data-set.</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-211">Sample arithmetic mean of the non-missing values provided data-set.</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-214">The mean of the elements of A along the first array dimension whose size does not equal 1.</div>
      <tr>
       <td align="center" style="width: 4%;" valign="top">
        <div class="plain_layout" id="magicparlabel-217">Restrictions</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-220">Numeric values must be passed as parameter.</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-223">Numeric/logical vectors and date, date-time and time interval objects. Complex vectors are allowed for trim = 0, only.</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-226">At least one non-missing argument is required.</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-229">specified in <a class="property" data-link-type="propdesc">details</a></div>
      <tr>
       <td align="center" style="width: 4%;" valign="top">
        <div class="plain_layout" id="magicparlabel-232">Error handling</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-235">Exceptions TypeError</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-238">If x is not logical (coerced to numeric), numeric (including integer) or complex, NA_real_ is returned, with a warning.</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-241">The function returns a missing value.</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-244">returns a missing value</div>
      <tr>
       <td align="center" style="width: 4%;" valign="top">
        <div class="plain_layout" id="magicparlabel-247">Details</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-250"><a href="https://docs.python.org/3/library/statistics.html#statistics.mean">python_doc_mean</a> <a href="https://www.geeksforgeeks.org/python-statistics-mean-function/">geeksforgeeks_mean</a></div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-253"><a href="https://documentation.sas.com/?docsetId=lefunctionsref&amp;docsetTarget=p0vl2htnkjkvejn1lsx2004n6dqp.htm&amp;docsetVersion=9.4&amp;locale=en">r_doc_mean</a></div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-256">Matlab_doc<a href="https://documentation.sas.com/?docsetId=lefunctionsref&amp;docsetTarget=p0vl2htnkjkvejn1lsx2004n6dqp.htm&amp;docsetVersion=9.4&amp;locale=en">SAS_doc_mean</a></div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-259"><a href="https://www.mathworks.com/help/matlab/ref/mean.html">Matlab_doc_mean</a></div>
    </table>
   </div>
   <div class="standard" id="magicparlabel-260①">All the languages listed above, except python, provide a way to exclude missing value. It is worth to mention that all the languages have a NaN “type” for missing data. The interfaces that do not have a trim parameter provide separate functions for a trimmed mean.</div>
   <h3 class="subsection heading settled" data-level="4.2" id="magicparlabel-262"><span class="secno">4.2. </span><span class="content"> Median</span><a class="self-link" href="#magicparlabel-262"></a></h3>
   <h4 class="subsubsection heading settled" data-level="4.2.1" id="magicparlabel-263"><span class="secno">4.2.1. </span><span class="content">Definition</span><a class="self-link" href="#magicparlabel-263"></a></h4>
   <div class="standard" id="magicparlabel-264">Median is a quantile with probability 0.5. The definition of a quantile will be provided in section 4.3. One of the definitions of median is described in <a data-link-type="biblio" href="#biblio-p1708r1">[P1708R1]</a>.</div>
   <h4 class="subsubsection heading settled" data-level="4.2.2" id="magicparlabel-265"><span class="secno">4.2.2. </span><span class="content">Users' perspective</span><a class="self-link" href="#magicparlabel-265"></a></h4>
   <div class="standard" id="magicparlabel-266">It is a quantile so one can either treat it this way, with all the customization points available for quantiles, or reduce the use-cases for std::median to the definition from <a data-link-type="biblio" href="#biblio-p1708r1">[P1708R1]</a> and types that support the required mathematical mathematical operations. </div>
   <div class="standard" id="magicparlabel-266①"> In other programming languages median returns a single value. For details about python, R, Matlab and SAS, see the comparison in the following subsection. Moreover, in <a data-link-type="biblio" href="#biblio-unur_2016">[UNUR_2016]</a> A. Sinan Unur described median calculation results from perl, Stata, R, Octave, WolframAlfa and Boost library and they all returned a single number, so it would be counter-intuitive to return a tuple in the same case in the C++ standard library as proposed in <a data-link-type="biblio" href="#biblio-p1708r2">[P1708R2]</a>.</div>
   <h4 class="subsubsection heading settled" data-level="4.2.3" id="magicparlabel-267"><span class="secno">4.2.3. </span><span class="content"> Interfaces in other languages</span><a class="self-link" href="#magicparlabel-267"></a></h4>
   <div class="standard" id="magicparlabel-268">
    <table>
     <tbody>
      <tr>
       <td align="center" valign="top">
        <div class="plain_layout" id="magicparlabel-315">Language</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-318">Python</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-321">R</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-324">SAS</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-327">Matlab</div>
      <tr>
       <td align="center" valign="top">
        <div class="plain_layout" id="magicparlabel-330">Syntax</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="float-listings">
<pre class="listings highlight"><c- n>statistics</c-><c- p>.</c-><c- n>median</c-><c- p>(</c-><c- n>data</c-><c- p>)</c->

<p><c- n>numpy</c-><c- p>.</c-><c- n>median</c-><c- p>(</c-><c- n>a</c-><c- p>,</c-> <c- n>axis</c-><c- o>=</c-><c- n>None</c-><c- p>,</c-> <c- n>out</c-><c- o>=</c-><c- n>None</c-><c- p>,</c-> <c- n>overwrite_input</c-><c- o>=</c-><c- n>False</c-><c- p>,</c-> <c- n>keepdims</c-><c- o>=</c-><c- n>False</c-><c- p>)</c-></p></pre>
        </div>
        <p></p>
       <td align="left" style="width: 15%;" valign="top">
        <div class="float-listings">
<pre class="listings highlight"><c- n>median</c-><c- p>(</c-><c- n>x</c-><c- p>,</c-> <c- n>na</c-><c- p>.</c-><c- n>rm</c-> <c- o>=</c-> <c- n>FALSE</c-><c- p>,...)</c-></pre>
        </div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="float-listings">
<pre class="listings highlight"><c- n>MEDIAN</c-><c- p>(</c-><c- n>value1</c-> <c- o>&lt;</c-><c- p>,</c-> <c- n>value2</c-><c- p>,</c-> <c- p>...</c-><c- o>></c-><c- p>)</c-></pre>
        </div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="float-listings">
<pre class="listings highlight"><c- n>M</c-> <c- o>=</c-> <c- n>median</c-><c- p>(</c-><c- n>A</c-><c- p>)</c-> 
<c- n>M</c-> <c- o>=</c-> <c- n>median</c-><c- p>(</c-><c- n>A</c-><c- p>,</c->'<c- n>all</c->'<c- p>)</c-> 
<c- n>M</c-> <c- o>=</c-> <c- n>median</c-><c- p>(</c-><c- n>A</c-><c- p>,</c-><c- n>dim</c-><c- p>)</c-> 
<c- n>M</c-> <c- o>=</c-> <c- n>median</c-><c- p>(</c-><c- n>A</c-><c- p>,</c-><c- n>vecdim</c-><c- p>)</c-> 
<c- n>M</c-> <c- o>=</c-> <c- n>median</c-><c- p>(</c-><c- n>___</c-><c- p>,</c-><c- n>nanflag</c-><c- p>)</c-></pre>
        </div>
      <tr>
       <td align="center" valign="top">
        <div class="plain_layout" id="magicparlabel-367">Parameters</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-370">a: array_like object</div>
        <div class="plain_layout" id="magicparlabel-371">axis: {int, sequence of int, None}, optional Axis or axes along which the medians are computed. </div>
        <div class="plain_layout" id="magicparlabel-372">out: ndarray, optional Alternative output array in which to place the result.</div>
        <div class="plain_layout" id="magicparlabel-373">overwrite_input: bool, optional If True, then allow use of memory of input array a for calculations. </div>
        <div class="plain_layout" id="magicparlabel-374">keepdims: bool, optional If this is set to True, the axes which are reduced are left in the result as dimensions with size one.</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-377">x: an object for which a method has been defined, or a numeric vector containing the values whose median is to be computed.</div>
        <div class="plain_layout" id="magicparlabel-378">na.rm: a logical value indicating whether NA values should be stripped before the computation proceeds.</div>
        <div class="plain_layout" id="magicparlabel-379">...: potentially further arguments for methods; not used in the default method.</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-382">value: is a numeric constant, variable, or expression</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-385">A: matrix or vector</div>
        <div class="plain_layout" id="magicparlabel-386">all: computes the mean over all elements of A.</div>
        <div class="plain_layout" id="magicparlabel-387">dim: returns the mean along dimension dim</div>
        <div class="plain_layout" id="magicparlabel-388">vecdim: computes the mean based on the dimensions specified in the vector vecdim</div>
        <div class="plain_layout" id="magicparlabel-389">outtype: returns the mean with a specified data type, using any of the input arguments in the previous syntaxes. outtype can be <a class="property" data-link-type="propdesc">default</a>, <a class="property" data-link-type="propdesc">double</a>, or <a class="property" data-link-type="propdesc">native</a>.</div>
        <div class="plain_layout" id="magicparlabel-390">nanflag: specifies whether to include or omit NaN values from the calculation for any of the previous syntaxes. function(A,<a class="property" data-link-type="propdesc">includenan</a>) includes all NaN values in the calculation while function(A,<a class="property" data-link-type="propdesc">omitnan</a>) ignores them.</div>
      <tr>
       <td align="center" valign="top">
        <div class="plain_layout" id="magicparlabel-393">Returns</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-396">statistics:</div>
        <div class="plain_layout" id="magicparlabel-397">statistics.median return the median (middle value) of numeric data, using the common “mean of middle two” method</div>
        <div class="plain_layout" id="magicparlabel-398">numpy:</div>
        <div class="plain_layout" id="magicparlabel-399">ndarray A new array holding the result. If the input contains integers or floats smaller than float64, then the output data-type is np.float64. Otherwise, the data-type of the output is the same as that of the input. If out is specified, that array is returned instead..</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-402">The default method returns a length-one object of the same type as x, except when x is logical or integer of even length, when the result will be double.</div>
        <div class="plain_layout" id="magicparlabel-403">If there are no values or if na.rm = FALSE and there are NA values the result is NA of the same type as x.</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-406">the median of non-missing values</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-409">For ordinal categorical arrays, MATLAB interprets the median of an even number of elements as follows:</div>
        <div class="plain_layout" id="magicparlabel-410">If the number of categories between the middle two values is ... Then the median is ... zero (values are from consecutive categories) larger of the two middle values an odd number value from category occurring midway between the two middle values an even number value from larger of the two categories occurring midway between the two middle values</div>
      <tr>
       <td align="center" valign="top">
        <div class="plain_layout" id="magicparlabel-413">Restrictions</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-416">statistics.median: data can be a sequence or iterable.</div>
        <div class="plain_layout" id="magicparlabel-417">The input array will be modified by the call to median. </div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-420">See: details</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-423">See: details</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-426">See: details</div>
      <tr>
       <td align="center" valign="top">
        <div class="plain_layout" id="magicparlabel-429">Error handling</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-432">If data is empty, StatisticsError is raised</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-435">the result is NA of the same type as x</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-438">If all arguments have missing values, the result is a missing value.</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-441">returns NaN</div>
      <tr>
       <td align="center" valign="top">
        <div class="plain_layout" id="magicparlabel-444">Details</div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-447"><a href="https://docs.python.org/3/library/statistics.html#statistics.median">python_doc_median</a></div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-450"><a href="https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/median">R_doc_median</a></div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-453"><a href="https://documentation.sas.com/?docsetId=lefunctionsref&amp;docsetTarget=p10gg32gykwv40n1mfbqkbhlbr2q.htm&amp;docsetVersion=9.4&amp;locale=en">SAS_doc_median</a></div>
       <td align="left" style="width: 15%;" valign="top">
        <div class="plain_layout" id="magicparlabel-456"><a href="https://www.mathworks.com/help/matlab/ref/median.html">Matlab_doc_median</a></div>
    </table>
   </div>
   <div class="standard" id="magicparlabel-457">All the languages listed above, except python, provide a way to exclude missing values. It is worth to mention that all the languages have a NaN “type” for missing data. Also, all the interfaces return one value as a result.</div>
   <h3 class="subsection heading settled" data-level="4.3" id="magicparlabel-458"><span class="secno">4.3. </span><span class="content">Quantiles</span><a class="self-link" href="#magicparlabel-458"></a></h3>
   <div class="standard" id="magicparlabel-459">The only quantile mentioned in P1708 is median, but I think that quantile function is worth considering as an input to discussion about a median of non-numeric types. </div>
   <h4 class="subsubsection heading settled" data-level="4.3.1" id="magicparlabel-460"><span class="secno">4.3.1. </span><span class="content">Definition</span><a class="self-link" href="#magicparlabel-460"></a></h4>
   <div class="standard" id="magicparlabel-461">
    Quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. See: <a data-link-type="biblio" href="#biblio-hyndmanfan_1996">[HYNDMAN+FAN_1996]</a> 
    <h4 class="subsubsection heading settled" data-level="4.3.2" id="magicparlabel-460①"><span class="secno">4.3.2. </span><span class="content">Users' perspective</span><a class="self-link" href="#magicparlabel-460①"></a></h4>
    <div class="standard" id="magicparlabel-459①">In <a data-link-type="biblio" href="#biblio-hyndmanfan_1996">[HYNDMAN+FAN_1996]</a> Rob J. Hyndman and Yanan Fan discussed nine algorithms of quantile calculation. They are summarized in <a href="https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/quantile">R_doc_quantile</a> in Type section. Types 1 and 3 can be applied to non-numeric types as they do not require interpolation and so they are adequate for discontiunous sample quantiles.</div>
    <h4 class="subsubsection heading settled" data-level="4.3.3" id="magicparlabel-462"><span class="secno">4.3.3. </span><span class="content"> Interfaces in other languages</span><a class="self-link" href="#magicparlabel-462"></a></h4>
    <div class="standard" id="magicparlabel-463">
     <table>
      <tbody>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-510">Language</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-513">Python</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-516">R</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-519">SAS</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-522">Matlab</div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-525">Syntax</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="float-listings">
<pre class="listings highlight"><c- n>numpy</c-><c- p>.</c-><c- n>quantile</c-><c- p>(</c-><c- n>a</c-><c- p>,</c-> <c- n>q</c-><c- p>,</c-> <c- n>axis</c-><c- o>=</c-><c- n>None</c-><c- p>,</c-> <c- n>out</c-><c- o>=</c-><c- n>None</c-><c- p>,</c-> <c- n>overwrite_input</c-><c- o>=</c-><c- n>False</c-><c- p>,</c-> <c- n>interpolation</c-><c- o>=</c->'<c- n>linear</c->'<c- p>,</c-> <c- n>keepdims</c-><c- o>=</c-><c- n>False</c-><c- p>)</c-></pre>
         </div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="float-listings">
<pre class="listings highlight"><c- n>quantile</c-><c- p>(</c-><c- n>x</c-><c- p>,</c-> <c- n>probs</c-> <c- o>=</c-> <c- n>seq</c-><c- p>(</c-><c- mi>0</c-><c- p>,</c-> <c- mi>1</c-><c- p>,</c-> <c- mf>0.25</c-><c- p>),</c-> <c- n>na</c-><c- p>.</c-><c- n>rm</c-> <c- o>=</c-> <c- n>FALSE</c-><c- p>,</c-> <c- n>names</c-> <c- o>=</c-> <c- n>TRUE</c-><c- p>,</c-> <c- n>type</c-> <c- o>=</c-> <c- mi>7</c-><c- p>,</c-> <c- p>...</c-> <c- p>)</c-></pre>
         </div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="float-listings">
<pre class="listings highlight"><c- n>QNTL</c-> <c- p>(</c-><c- n>q</c-><c- p>,</c-> <c- n>x</c-><c- p>,</c-> <c- o>&lt;</c-><c- p>,</c-> <c- n>probs</c-><c- o>></c-> <c- o>&lt;</c-><c- p>,</c-> <c- n>method</c-><c- o>></c-> <c- p>)</c-></pre>
         </div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="float-listings">
<pre class="listings highlight"><c- n>Y</c-> <c- o>=</c-> <c- n>quantile</c-><c- p>(</c-><c- n>X</c-><c- p>,</c-><c- n>p</c-><c- p>)</c-> 
<c- n>Y</c-> <c- o>=</c-> <c- n>quantile</c-><c- p>(</c-><c- n>X</c-><c- p>,</c-><c- n>N</c-><c- p>)</c-> 
<c- n>Y</c-> <c- o>=</c-> <c- n>quantile</c-><c- p>(</c-><c- n>___</c-><c- p>,</c->'<c- n>all</c->'<c- p>)</c-> 
<c- n>Y</c-> <c- o>=</c-> <c- n>quantile</c-><c- p>(</c-><c- n>___</c-><c- p>,</c-><c- n>dim</c-><c- p>)</c-> 
<c- n>Y</c-> <c- o>=</c-> <c- n>quantile</c-><c- p>(</c-><c- n>___</c-><c- p>,</c-><c- n>vecdim</c-><c- p>)</c-> 
<c- n>Y</c-> <c- o>=</c-> <c- n>quantile</c-><c- p>(</c-><c- n>___</c-><c- p>,</c->'<c- n>Method</c->'<c- p>,</c-><c- n>method</c-><c- p>)</c-></pre>
         </div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-561">Parameters</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-564">a: array_like input array or object that can be converted to an array. </div>
         <div class="plain_layout" id="magicparlabel-565">q: array_like of float Quantile or sequence of quantiles to compute, which must be between 0 and 1 inclusive. </div>
         <div class="plain_layout" id="magicparlabel-566">axis{int, tuple of int, None}: Axis or axes along which the quantiles are computed.</div>
         <div class="plain_layout" id="magicparlabel-567">out: ndarray Alternative output array in which to place the result. </div>
         <div class="plain_layout" id="magicparlabel-568">overwrite_inputbool: If True, then allow the input array a to be modified by intermediate calculations, to save memory. </div>
         <div class="plain_layout" id="magicparlabel-569">interpolation{‘linear’, ‘lower’, ‘higher’, ‘midpoint’, ‘nearest’}: This optional parameter specifies the interpolation method to use when the desired quantile lies between two data points i &lt; j: </div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-572">x: an object for which a method has been defined, or a numeric vector containing the values whose median is to be computed.</div>
         <div class="plain_layout" id="magicparlabel-573">probs: numeric vector of probabilities with values in [ 0 , 1 ] . </div>
         <div class="plain_layout" id="magicparlabel-574">na.rm: a logical value indicating whether NA values should be stripped before the computation proceeds.</div>
         <div class="plain_layout" id="magicparlabel-575">type: number - one of 9 types described in <a data-link-type="biblio" href="#biblio-hyndmanfan_1996">[HYNDMAN+FAN_1996]</a></div>
         <div class="plain_layout" id="magicparlabel-576">...: potentially further arguments for methods; not used in the default method</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-579">q: specifies a matrix to contain the quantiles of the x matrix.</div>
         <div class="plain_layout" id="magicparlabel-580">x: specifies an numerical matrix of data. The QNTL subroutine computes quantiles for each column of the matrix.</div>
         <div class="plain_layout" id="magicparlabel-581">probs: specifies a numeric vector of probabilities used to compute the quantiles. </div>
         <div class="plain_layout" id="magicparlabel-582">method: specifies the method (one of the 9 types described in <a data-link-type="biblio" href="#biblio-hyndmanfan_1996">[HYNDMAN+FAN_1996]</a>) </div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-585">x: matrix or vector</div>
         <div class="plain_layout" id="magicparlabel-586">p: probability or probabilities p in the interval [0,1]</div>
         <div class="plain_layout" id="magicparlabel-587">N: number N of evenly spaced cumulative probabilities (1/(N + 1), 2/(N + 1), ..., N/(N + 1)) for integer N>1</div>
         <div class="plain_layout" id="magicparlabel-588">all: computes the quantiles over all elements of A.</div>
         <div class="plain_layout" id="magicparlabel-589">dim: returns the quantiles along dimension dim</div>
         <div class="plain_layout" id="magicparlabel-590">vecdim: computes the quantiles based on the dimensions specified in the vector vecdim</div>
         <div class="plain_layout" id="magicparlabel-591">method: approximate quantiles based on the value of method, using any of the input argument combinations in the previous syntaxes</div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-594">Returns</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-597">scalar or ndarray </div>
         <div class="plain_layout" id="magicparlabel-598">If q is a single quantile and axis=None, then the result is a scalar. If multiple quantiles are given, first axis of the result corresponds to the quantiles. </div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-601">scalar or matrix</div>
         <div class="plain_layout" id="magicparlabel-602">estimates of underlying distribution quantiles based on one or two order statistics from the supplied elements in x at probabilities in probs.</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-605">scalar or matrix</div>
         <div class="plain_layout" id="magicparlabel-606">The QNTL subroutine computes sample quantiles for data.</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-609">scalar or matrix</div>
         <div class="plain_layout" id="magicparlabel-610">quantile treats NaNs as missing values and removes them.</div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-613">Restrictions</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-616">See: details</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-619">See: details</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-622">See: details</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-625">See: details</div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-628">Error handling</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-631">not specified</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-634">the result is NA of the same type as x</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-637">If all arguments have missing values, the result is a missing value.</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-640">treats NaNs as missing values and removes them</div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-643">Details</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-646"><a href="https://numpy.org/devdocs/reference/generated/numpy.nanquantile.html#numpy.nanquantile">python_doc_quantile</a></div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-649"><a href="https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/quantile">R_doc_quantile</a></div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-652"><a href="https://documentation.sas.com/?docsetId=lefunctionsref&amp;docsetTarget=n0uhywbqfucg6qn18woziy41flqp.htm&amp;docsetVersion=9.4&amp;locale=en">SAS_doc_quantile</a></div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-655"><a href="https://www.mathworks.com/help/stats/quantile.html">Matlab_doc_quantile</a></div>
     </table>
    </div>
    <div class="standard" id="magicparlabel-656">Every syntax described above provides a way to customize how the quantiles are estimated. It can be configured by choosing an approximation method (Python, Matlab), or algorithm from <a data-link-type="biblio" href="#biblio-hyndmanfan_1996">[HYNDMAN+FAN_1996]</a> (SAS, R).</div>
    <h3 class="subsection heading settled" data-level="4.4" id="magicparlabel-658"><span class="secno">4.4. </span><span class="content">Mode</span><a class="self-link" href="#magicparlabel-658"></a></h3>
    <h4 class="subsubsection heading settled" data-level="4.4.1" id="magicparlabel-659"><span class="secno">4.4.1. </span><span class="content">Definition</span><a class="self-link" href="#magicparlabel-659"></a></h4>
    <div class="standard" id="magicparlabel-660">The definition is provided in <a data-link-type="biblio" href="#biblio-p1708r2">[P1708R2]</a></div>
    <h4 class="subsubsection heading settled" data-level="4.4.2" id="magicparlabel-661"><span class="secno">4.4.2. </span><span class="content">Users' perspective</span><a class="self-link" href="#magicparlabel-661"></a></h4>
    <div class="standard" id="magicparlabel-662">Despite a possibly linear complexity the function may have huge space requirements for example when a container is big and contains unique numbers. It would be good to have some policy that allows returning a single value in case of multiple modes.</div>
    <h4 class="subsubsection heading settled" data-level="4.4.3" id="magicparlabel-663"><span class="secno">4.4.3. </span><span class="content"> Interfaces in other languages</span><a class="self-link" href="#magicparlabel-663"></a></h4>
    <div class="standard" id="magicparlabel-664">
     <table>
      <tbody>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-711">Language</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-714">Python</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-717">R</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-720">SAS</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-723">Matlab</div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-726">Syntax</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="float-listings">
<pre class="listings highlight"><c- n>statistics</c-><c- p>.</c-><c- n>mode</c-><c- p>(</c-><c- n>data</c-><c- p>)</c->
<c- n>scipy</c-><c- p>.</c-><c- n>stats</c-><c- p>.</c-><c- n>mode</c-><c- p>(</c-><c- n>a</c-><c- p>,</c-> <c- n>axis</c-><c- o>=</c-><c- mi>0</c-><c- p>,</c-> <c- n>nan_policy</c-><c- o>=</c->'<c- n>propagate</c->'<c- p>)</c-></pre>
         </div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="float-listings">
<pre class="listings highlight"><c- n>Mode</c-><c- p>(</c-><c- n>x</c-><c- p>,</c-> <c- n>na</c-><c- p>.</c-><c- n>rm</c-> <c- o>=</c-> <c- n>FALSE</c-><c- p>)</c-> </pre>
         </div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-744">not provided</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="float-listings">
<pre class="listings highlight"><c- n>M</c-> <c- o>=</c-> <c- n>mode</c-><c- p>(</c-><c- n>A</c-><c- p>)</c-> 
<c- n>M</c-> <c- o>=</c-> <c- n>mode</c-><c- p>(</c-><c- n>A</c-><c- p>,</c->'<c- n>all</c->'<c- p>)</c-> 
<c- n>M</c-> <c- o>=</c-> <c- n>mode</c-><c- p>(</c-><c- n>A</c-><c- p>,</c-><c- n>dim</c-><c- p>)</c-> 
<c- n>M</c-> <c- o>=</c-> <c- n>mode</c-><c- p>(</c-><c- n>A</c-><c- p>,</c-><c- n>vecdim</c-><c- p>)</c-> 
<c- p>[</c-><c- n>M</c-><c- p>,</c-><c- n>F</c-><c- p>]</c-> <c- o>=</c-> <c- n>mode</c-><c- p>(</c-><c- n>___</c-><c- p>)</c-> 
<c- p>[</c-><c- n>M</c-><c- p>,</c-><c- n>F</c-><c- p>,</c-><c- n>C</c-><c- p>]</c-> <c- o>=</c-> <c- n>mode</c-><c- p>(</c-><c- n>___</c-><c- p>)</c-></pre>
         </div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-759">Parameters</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-762">data: discrete or nominal data</div>
         <div class="plain_layout" id="magicparlabel-763">a: array_like n-dimensional array of which to find mode(s).</div>
         <div class="plain_layout" id="magicparlabel-764">axis: int or None, Axis along which to operate. None means full matrix</div>
         <div class="plain_layout" id="magicparlabel-765">nan_policy: {‘propagate’, ‘raise’, ‘omit’} Nan handling</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-768">x: an object for which a method has been defined, or a numeric vector containing the values </div>
         <div class="plain_layout" id="magicparlabel-769">na.rm: a logical value indicating whether NA values should be stripped before the computation proceeds.</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-772">—</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-775">A: matrix or vector</div>
         <div class="plain_layout" id="magicparlabel-776">all: computes the mean over all elements of A.</div>
         <div class="plain_layout" id="magicparlabel-777">dim: returns the mean along dimension dim</div>
         <div class="plain_layout" id="magicparlabel-778">vecdim: computes the mean based on the dimensions specified in the vector vecdim</div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-781">Returns</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-784">statistics:</div>
         <div class="plain_layout" id="magicparlabel-2785">Return the single most common data point from discrete or nominal data. If there are multiple modes with the same frequency, returns the first one encountered in the data</div>
         <div class="plain_layout" id="magicparlabel-786">scipy:</div>
         <div class="plain_layout" id="magicparlabel-787">If there is more than one such value, only the smallest is returned.</div>
         <div class="plain_layout" id="magicparlabel-788">Returns mode: ndarray Array of modal values and count: ndarray Array of counts for each mode.</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-791">Returns the most frequent value. If there are more than one, all of them will be returned in a vector.</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-794">—</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-797">M = mode(A) returns the sample mode of A, which is the most frequently occurring value in A. When there are multiple values occurring equally frequently, mode returns the smallest of those values. For complex inputs, the smallest value is the first value in a sorted list.</div>
         <div class="plain_layout" id="magicparlabel-798">[M,F] = mode(___) also returns a frequency array F</div>
         <div class="plain_layout" id="magicparlabel-799">[M,F,C] = mode(___) also returns a cell array C that is a sorted vector of all values that have the same frequency as the corresponding element of M.</div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-802">Restrictions</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-805">statistics:</div>
         <div class="plain_layout" id="magicparlabel-806">assumes discrete data and returns a single value</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-809">See: details</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-812">—</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-815">The mode function is most useful with discrete or coarsely rounded data. Also, the mode function is not suitable for finding peaks in distributions having multiple modes.</div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-818">Error handling</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-821">statistics:</div>
         <div class="plain_layout" id="magicparlabel-822">StatisticsError is raised when more than one mode was found.</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-825">NaNs can be ignored</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-828">—</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-831">See: details</div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-834">Details</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-837"><a href="https://docs.python.org/3/library/statistics.html#statistics.mode">python_statistics_mode</a></div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-840"><a href="https://www.rdocumentation.org/packages/DescTools/versions/0.99.32/topics/Mode">R_doc_mode</a></div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-843">—</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-846"><a href="https://www.mathworks.com/help/matlab/ref/mode.html">Matlab_doc_mode</a></div>
     </table>
    </div>
    <div class="standard" id="magicparlabel-847">Not each of the above interfaces provides a mode function. Python always returns a single value (however, there is a separate function that returns multiple values), Matlab can provide both single (lowest) mode or all of them and R always returns all of the modes. Both python and R provide a way to handle missing values (NaNs and NAs).</div>
    <h3 class="subsection heading settled" data-level="4.5" id="magicparlabel-848"><span class="secno">4.5. </span><span class="content"> Standard deviation</span><a class="self-link" href="#magicparlabel-848"></a></h3>
    <h4 class="subsubsection heading settled" data-level="4.5.1" id="magicparlabel-849"><span class="secno">4.5.1. </span><span class="content">Definition </span><a class="self-link" href="#magicparlabel-849"></a></h4>
    <div class="standard" id="magicparlabel-850">Definition of population and sample standard deviation is provided in <a data-link-type="biblio" href="#biblio-p1708r2">[P1708R2]</a> in the equations (2) and (3). However, there is a typo in equation (2).</div>
    <h4 class="subsubsection heading settled" data-level="4.5.2" id="magicparlabel-851"><span class="secno">4.5.2. </span><span class="content"> Users' perspective</span><a class="self-link" href="#magicparlabel-851"></a></h4>
    <div class="standard" id="magicparlabel-852">To compute standard deviation, a type must support subtration, multiplication, division by a scalar and square root. This reduces the number of potentially supported types. Both the population standard deviation and the sample standard deviation are commonly used.</div>
    <h4 class="subsubsection heading settled" data-level="4.5.3" id="magicparlabel-853"><span class="secno">4.5.3. </span><span class="content"> Interfaces in other languages</span><a class="self-link" href="#magicparlabel-853"></a></h4>
    <div class="standard" id="magicparlabel-854">
     <table>
      <tbody>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-901">Language</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-904">Python</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-907">R</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-910">SAS</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-913">Matlab</div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-916">Syntax</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="float-listings">
<pre class="listings highlight"><c- n>statistics</c-><c- p>.</c-><c- n>pstdev</c-><c- p>(</c-><c- n>data</c-><c- p>,</c-> <c- n>m</c-><c- o>=</c-><c- n>None</c-><c- p>)</c->
<c- n>statistics</c-><c- p>.</c-><c- n>stdev</c-><c- p>(</c-><c- n>data</c-><c- p>,</c-> <c- n>m</c-><c- o>=</c-><c- n>None</c-><c- p>)</c->
<c- n>numpy</c-><c- p>.</c-><c- n>std</c-><c- p>(</c-><c- n>a</c-><c- p>,</c-> <c- n>axis</c-><c- o>=</c-><c- n>None</c-><c- p>,</c-> <c- n>dtype</c-><c- o>=</c-><c- n>None</c-><c- p>,</c-> <c- n>out</c-><c- o>=</c-><c- n>None</c-><c- p>,</c-> <c- n>ddof</c-><c- o>=</c-><c- mi>0</c-><c- p>,</c-> <c- n>keepdims</c-><c- o>=&lt;</c-><c- n>no</c-> <c- n>value</c-><c- o>></c-><c- p>)</c-></pre>
         </div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="float-listings">
<pre class="listings highlight"><c- n>sd</c-><c- p>(</c-><c- n>x</c-><c- p>,</c-> <c- n>na</c-><c- p>.</c-><c- n>rm</c-> <c- o>=</c-> <c- n>FALSE</c-><c- p>)</c-></pre>
         </div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="float-listings">
<pre class="listings highlight"><c- n>STD</c-><c- p>(</c-> <c- n>x</c-> <c- p>)</c-></pre>
         </div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="float-listings">
<pre class="listings highlight"><c- n>S</c-> <c- o>=</c-> <c- n>std</c-><c- p>(</c-><c- n>A</c-><c- p>)</c-> 
<c- n>S</c-> <c- o>=</c-> <c- n>std</c-><c- p>(</c-><c- n>A</c-><c- p>,</c-><c- n>w</c-><c- p>)</c-> 
<c- n>S</c-> <c- o>=</c-> <c- n>std</c-><c- p>(</c-><c- n>A</c-><c- p>,</c-><c- n>w</c-><c- p>,</c->'<c- n>all</c->'<c- p>)</c-> 
<c- n>S</c-> <c- o>=</c-> <c- n>std</c-><c- p>(</c-><c- n>A</c-><c- p>,</c-><c- n>w</c-><c- p>,</c-><c- n>dim</c-><c- p>)</c-> 
<c- n>S</c-> <c- o>=</c-> <c- n>std</c-><c- p>(</c-><c- n>A</c-><c- p>,</c-><c- n>w</c-><c- p>,</c-><c- n>vecdim</c-><c- p>)</c-> 
<c- n>S</c-> <c- o>=</c-> <c- n>std</c-><c- p>(</c-><c- n>___</c-><c- p>,</c-><c- n>nanflag</c-><c- p>)</c-></pre>
         </div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-954">Parameters</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-957">data: an iterable of at least two real-valued numbers.</div>
         <div class="plain_layout" id="magicparlabel-958">m: the mean of data (optional parameter)</div>
         <div class="plain_layout" id="magicparlabel-959">a: array_like Calculate the standard deviation of these values.</div>
         <div class="plain_layout" id="magicparlabel-960">axis: None or int or tuple of ints, optional Axis or axes along which the standard deviation is computed. </div>
         <div class="plain_layout" id="magicparlabel-961">d: typedtype, optional Type to use in computing the standard deviation. </div>
         <div class="plain_layout" id="magicparlabel-962">out: ndarray, optional Alternative output array in which to place the result. </div>
         <div class="plain_layout" id="magicparlabel-963">ddof: int, optional Means Delta Degrees of Freedom. </div>
         <div class="plain_layout" id="magicparlabel-964">keepdims: bool, optional If this is set to True, the axes which are reduced are left in the result as dimensions with size one. </div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-967">x: a numeric vector or an R object but not a factor coercible to numeric by as.double(x).</div>
         <div class="plain_layout" id="magicparlabel-968">na.rm: a logical value indicating whether NA values should be stripped before the computation proceeds.</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-971">x: a numerical matrix or vector</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-974">A: matrix or vector</div>
         <div class="plain_layout" id="magicparlabel-2975">w: specifies a weighting scheme for any of the previous syntaxes. When w = 0 (default), S is normalized by N-1. When w = 1, S is normalized by the number of observations, N. w also can be a weight vector containing nonnegative elements. In this case, the length of w must equal the length of the dimension over which std is operating.</div>
         <div class="plain_layout" id="magicparlabel-976">all: computes the mean over all elements of A.</div>
         <div class="plain_layout" id="magicparlabel-977">dim: returns the mean along dimension dim</div>
         <div class="plain_layout" id="magicparlabel-978">vecdim: computes the mean based on the dimensions specified in the vector vecdim</div>
         <div class="plain_layout" id="magicparlabel-979">nanflag: specifies whether to include or omit NaN values from the calculation for any of the previous syntaxes </div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-982">Returns</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-985">statistics:</div>
         <div class="plain_layout" id="magicparlabel-986">pstdev returns the population standard deviation</div>
         <div class="plain_layout" id="magicparlabel-987">stdev returns the sample standard deviation</div>
         <div class="plain_layout" id="magicparlabel-988">numpy:</div>
         <div class="plain_layout" id="magicparlabel-989">returns population or sample standard deviation depanding on parameters</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-992">the sample standard deviation of the values in x</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-995">the sample standard deviation of data</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-998">the sample or population standarddeviation of the data depending on the parameters</div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-1001">Restrictions</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1004">statistics: least two real-valued numbers</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1007">vectors of numerics or type that are converible to double</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1010">numeric data</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1013">numeric data</div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-1016">Error handling</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1019">statistics: Raises StatisticsError if data is empty.</div>
         <div class="plain_layout" id="magicparlabel-1020">numpy: If the sub-class’ method does not implement keepdims any exceptions will be raised.</div>
         <div class="plain_layout" id="magicparlabel-1021">numpy provides a separate function that handles nan (nanstdev)</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1024">The standard deviation of a length-one or zero-length vector is NA.</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1027">not specified</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1030">returns NaN</div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-1033">Details</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1036"><a href="https://docs.python.org/3/library/statistics.html">python_doc_statistics</a></div>
         <div class="plain_layout" id="magicparlabel-1037"><a href="https://numpy.org/doc/1.18/reference/generated/numpy.std.html">python_doc_stdev</a></div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1040"><a href="https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/sd">R_doc_sd</a></div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1043"><a href="http://support.sas.com/documentation/cdl/en/imlug/64248/HTML/default/viewer.htm#imlug_langref_sect287.html">SAS_doc_sdev</a></div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1046"><a href="https://www.mathworks.com/help/matlab/ref/std.html?s_tid=doc_ta">Matlab_doc_stdev</a></div>
     </table>
    </div>
    <div class="standard" id="magicparlabel-1047">Python and Matlab support sample and population standard deviation, while R and SAS calculate sample standard deviation only. Python, R, Matlab support excluding NaN values either as a parameter or as a separate function.</div>
    <h3 class="subsection heading settled" data-level="4.6" id="magicparlabel-1048"><span class="secno">4.6. </span><span class="content">Variance</span><a class="self-link" href="#magicparlabel-1048"></a></h3>
    <div class="standard" id="magicparlabel-1049">Definition of variance is provided in <a data-link-type="biblio" href="#biblio-p1708r2">[P1708R2]</a>.</div>
    <h4 class="subsubsection heading settled" data-level="4.6.1" id="magicparlabel-1050"><span class="secno">4.6.1. </span><span class="content"> Users' perspective</span><a class="self-link" href="#magicparlabel-1050"></a></h4>
    <div class="standard" id="magicparlabel-1051">Variance is the squared standard deviation. 
Due to this relationship they are expected to have very similar interfaces.</div>
    <h4 class="subsubsection heading settled" data-level="4.6.2" id="magicparlabel-1052"><span class="secno">4.6.2. </span><span class="content"> Interfaces in other languages</span><a class="self-link" href="#magicparlabel-1052"></a></h4>
    <div class="standard" id="magicparlabel-1053">
     <table>
      <tbody>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-1100">Language</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1103">Python</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1106">R</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1109">SAS</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1112">Matlab</div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-1115">Syntax</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="float-listings">
<pre class="listings highlight"><c- n>statistics</c-><c- p>.</c-><c- n>pvariance</c-><c- p>(</c-><c- n>data</c-><c- p>,</c-> <c- n>m</c-><c- o>=</c-><c- n>None</c-><c- p>)</c->
<c- n>statistics</c-><c- p>.</c-><c- n>variance</c-><c- p>(</c-><c- n>data</c-><c- p>,</c-> <c- n>m</c-><c- o>=</c-><c- n>None</c-><c- p>)</c->
<c- n>numpy</c-><c- p>.</c-><c- n>var</c-><c- p>(</c-><c- n>a</c-><c- p>,</c-> <c- n>axis</c-><c- o>=</c-><c- n>None</c-><c- p>,</c-> <c- n>dtype</c-><c- o>=</c-><c- n>None</c-><c- p>,</c-> <c- n>out</c-><c- o>=</c-><c- n>None</c-><c- p>,</c-> <c- n>ddof</c-><c- o>=</c-><c- mi>0</c-><c- p>,</c-> <c- n>keepdims</c-><c- o>=&lt;</c-><c- n>no</c-> <c- n>value</c-><c- o>></c-><c- p>)</c-></pre>
         </div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="float-listings">
<pre class="listings highlight"><c- n>var</c-><c- p>(</c-><c- n>x</c-><c- p>,</c-> <c- n>y</c-> <c- o>=</c-> NULL<c- p>,</c-> <c- n>na</c-><c- p>.</c-><c- n>rm</c-> <c- o>=</c-> <c- n>FALSE</c-><c- p>,</c-> <c- n>use</c-><c- p>)</c-></pre>
         </div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="float-listings">
<pre class="listings highlight"><c- n>VAR</c-><c- p>(</c-> <c- n>x</c-> <c- p>)</c-></pre>
         </div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="float-listings">
<pre class="listings highlight"><c- n>V</c-> <c- o>=</c-> <c- n>var</c-><c- p>(</c-><c- n>A</c-><c- p>)</c-> 
<c- n>V</c-> <c- o>=</c-> <c- n>var</c-><c- p>(</c-><c- n>A</c-><c- p>,</c-><c- n>w</c-><c- p>)</c-> 
<c- n>V</c-> <c- o>=</c-> <c- n>var</c-><c- p>(</c-><c- n>A</c-><c- p>,</c-><c- n>w</c-><c- p>,</c->'<c- n>all</c->'<c- p>)</c-> 
<c- n>V</c-> <c- o>=</c-> <c- n>var</c-><c- p>(</c-><c- n>A</c-><c- p>,</c-><c- n>w</c-><c- p>,</c-><c- n>dim</c-><c- p>)</c-> 
<c- n>V</c-> <c- o>=</c-> <c- n>var</c-><c- p>(</c-><c- n>A</c-><c- p>,</c-><c- n>w</c-><c- p>,</c-><c- n>vecdim</c-><c- p>)</c-> 
<c- n>V</c-> <c- o>=</c-> <c- n>var</c-><c- p>(</c-><c- n>___</c-><c- p>,</c-><c- n>nanflag</c-><c- p>)</c-></pre>
         </div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-1153">Parameters</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1156">data: an iterable of at least two real-valued numbers.</div>
         <div class="plain_layout" id="magicparlabel-1157">m: the mean of data (optional parameter)</div>
         <div class="plain_layout" id="magicparlabel-1158">a: array_like Calculate the standard deviation of these values.</div>
         <div class="plain_layout" id="magicparlabel-1159">axis: None or int or tuple of ints, optional Axis or axes along which the standard deviation is computed. </div>
         <div class="plain_layout" id="magicparlabel-1160">d: typedtype, optional Type to use in computing the standard deviation. </div>
         <div class="plain_layout" id="magicparlabel-1161">out: ndarray, optional Alternative output array in which to place the result. </div>
         <div class="plain_layout" id="magicparlabel-1162">ddof: int, optional Means Delta Degrees of Freedom. </div>
         <div class="plain_layout" id="magicparlabel-1163">keepdims: bool, optional If this is set to True, the axes which are reduced are left in the result as dimensions with size one. </div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1166">x: a numeric vector or an R object but not a factor coercible to numeric by as.double(x).</div>
         <div class="plain_layout" id="magicparlabel-1167">y: NULL (default) or a vector, matrix or data frame with compatible dimensions to x.</div>
         <div class="plain_layout" id="magicparlabel-1168">na.rm: a logical value indicating whether NA values should be stripped before the computation proceeds.</div>
         <div class="plain_layout" id="magicparlabel-1169">use: ignored</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1172">x: a numerical matrix or vector</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1175">A: matrix or vector</div>
         <div class="plain_layout" id="magicparlabel-21176">w: specifies a weighting scheme for any of the previous syntaxes. When w = 0 (default), S is normalized by N-1. When w = 1, S is normalized by the number of observations, N. w also can be a weight vector containing nonnegative elements. In this case, the length of w must equal the length of the dimension over which std is operating.</div>
         <div class="plain_layout" id="magicparlabel-1177">all: computes the mean over all elements of A.</div>
         <div class="plain_layout" id="magicparlabel-1178">dim: returns the mean along dimension dim</div>
         <div class="plain_layout" id="magicparlabel-1179">vecdim: computes the mean based on the dimensions specified in the vector vecdim</div>
         <div class="plain_layout" id="magicparlabel-1180">nanflag: specifies whether to include or omit NaN values from the calculation for any of the previous syntaxes </div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-1183">Returns</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1186">statistics:</div>
         <div class="plain_layout" id="magicparlabel-1187">pvariance returns the population variance</div>
         <div class="plain_layout" id="magicparlabel-1188">variance returns the sample variance</div>
         <div class="plain_layout" id="magicparlabel-1189">numpy:</div>
         <div class="plain_layout" id="magicparlabel-1190">returns population or sample variance depanding on parameters</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1193">the sample variance of the values in x</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1196">The VAR function computes the sample variance of the columns of this matrix.</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1199">the sample or population variance of the data depending on the parameters</div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-1202">Restrictions</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1205">statistics: least two real-valued numbers</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1208">vectors of numerics or type that are converible to double</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1211">numeric data</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1214">numeric data</div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-1217">Error handling</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1220">statistics: Raises StatisticsError if data is empty.</div>
         <div class="plain_layout" id="magicparlabel-1221">numpy: If the sub-class’ method does not implement keepdims any exceptions will be raised.</div>
         <div class="plain_layout" id="magicparlabel-1222">numpy provides a separate function that handles nan (nanstdev)</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1225">The variance of a length-one or zero-length vector is NA.</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1228">not specified</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1231">returns NaN</div>
       <tr>
        <td align="center" valign="top">
         <div class="plain_layout" id="magicparlabel-1234">Details</div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1237"><a href="https://docs.python.org/3/library/statistics.html">python_doc_statistics</a></div>
         <div class="plain_layout" id="magicparlabel-1238"><a href="https://numpy.org/doc/1.18/reference/generated/numpy.var.html">python_doc_var</a></div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1241"><a href="https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/var">R_doc_var</a></div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1244"><a href="http://support.sas.com/documentation/cdl/en/imlug/64248/HTML/default/viewer.htm#imlug_langref_sect323.htm">SAS_doc_var</a></div>
        <td align="left" style="width: 15%;" valign="top">
         <div class="plain_layout" id="magicparlabel-1247"><a href="https://www.mathworks.com/help/matlab/ref/var.html?s_tid=doc_ta">Matlab_doc_var</a></div>
     </table>
    </div>
    <div class="standard" id="magicparlabel-1248"> Matlab, Python and SAS provide similar interfaces for variance and standard deviation. Python, R, Matlab support NaN excluding either as a parameter or as a separate function.</div>
    <h2 class="section heading settled" data-level="5" id="magicparlabel-1249"><span class="secno">5. </span><span class="content">Grouped statistics</span><a class="self-link" href="#magicparlabel-1249"></a></h2>
    <div class="standard" id="magicparlabel-1250">Grouped statistics are expected to provide a way to calculate a group of statistics in one pass (or as fewer passes as possible) over a container.</div>
    <h3 class="subsection heading settled" data-level="5.1" id="magicparlabel-1251"><span class="secno">5.1. </span><span class="content"> Expectations from the users' perspective</span><a class="self-link" href="#magicparlabel-1251"></a></h3>
    <div class="standard" id="magicparlabel-1252">The interface for computing a group of statistics is expected to fulfil common requirements stated in section 3 and the following requirements:</div>
    <ul class="itemize" id="magicparlabel-1253">
     <li class="itemize_item">extendable 
     <li class="itemize_item">a way to pass parameters to every statistic
     <li class="itemize_item">information about required statistics handled in types 
     <li class="itemize_item">consistent with stand-alone statistical functions
    </ul>
    <h3 class="subsection heading settled" data-level="5.2" id="magicparlabel-1257"><span class="secno">5.2. </span><span class="content"> Users' perspective</span><a class="self-link" href="#magicparlabel-1257"></a></h3>
    <div class="standard" id="magicparlabel-1260"> Mean, variance, skewness and kurtosis are commonly described as (raw, central or standardized) moments of order 1, 2, 3, and 4. Consequently, it might be reasonable to implement a function for computing moments of a chosen order. There are diverging opinions whether these moments should be available for group computation alongside statistics of different origin, such as quantiles. Proponents of common treatment stress user-friendliness and the statistical notions of location (mean, median) and scale (variance, inter quantile range) parameters. Opponents highlight implementation-based difficulties and point out conceptual differences in the way these statistics are constructed. </div>
    <h2 class="section_ heading settled" data-level="6" id="magicparlabel-1264"><span class="secno">6. </span><span class="content">Some use-cases</span><a class="self-link" href="#magicparlabel-1264"></a></h2>
    <ol>
     <li class="itemize_item">A user wants to calculate a mean of a vector of doubles.
     <li class="itemize_item">A user wants to calculate a mean of a list of a custom numeric type that support addition as well as division by integer.
     <li class="itemize_item">A user wants to calculate a mean of a vector of floats, where missing data are marked as values below -90.
     <li class="itemize_item"> A user wants to calculate a mean of a vector of floats using a double as a return value and internal representation during calculations (for precision). 
     <li class="itemize_item">A user wants to calculate mean, variance and standard deviation of a sample in a computationally effective way.
     <li class="itemize_item">A user wants to calculate mean, variance and standard deviation of a sample in a computationally effective way, but her biggest concern is numerical stability of the calculation.
     <li class="itemize_item">A user wants to calculate a median of values stored in a map if keys are in a certian range.
     <li class="itemize_item">A user wants to calculate quantiles 0.25, 0.5 and 0.75 of a container of floats excluding NANs that are in the data. 
     <li class="itemize_item">A user wants to calculate percentils of a sample (i.e. quantiles with order 0.01, 0.02, ..., 0.99).
     <li class="itemize_item">A user wants to calculate percentils of a sorted sample.
    </ol>
    <h2 class="section heading settled" data-level="7" id="magicparlabel-1258"><span class="secno">7. </span><span class="content"> Summary</span><a class="self-link" href="#magicparlabel-1258"></a></h2>
    <div class="standard" id="magicparlabel-1259">In my opinion, the solution proposed in <a data-link-type="biblio" href="#biblio-p1708r2">[P1708R2]</a> needs a major revision to fulfill the expectations stated in the previous sections. </div>
    <h2 class="section_ heading settled" data-level="8" id="magicparlabel-1261"><span class="secno">8. </span><span class="content">Document history</span><a class="self-link" href="#magicparlabel-1261"></a></h2>
    <ul class="itemize" id="magicparlabel-1262">
     <li class="itemize_item"><b>R0</b>, 2020-02-13: Initial version. 
    </ul>
    <h2 class="section_ heading settled" data-level="9" id="magicparlabel-1263"><span class="secno">9. </span><span class="content">Acknowledgements</span><a class="self-link" href="#magicparlabel-1263"></a></h2>
    <div class="standard" id="magicparlabel-1264①">I would like to express my gratitude to everyone who provided comments on the use cases and acknowleadge discussions in SG6, SG14, SG18 and SG19.</div>
    <div class="standard" id="magicparlabel-1265">Jolanta Opara’s work was supported by Mobica Limited sp. z o.o. Oddział w Polsce.</div>
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  <h2 class="no-num no-ref heading settled" id="references"><span class="content">References</span><a class="self-link" href="#references"></a></h2>
  <h3 class="no-num no-ref heading settled" id="normative"><span class="content">Normative References</span><a class="self-link" href="#normative"></a></h3>
  <dl>
   <dt id="biblio-p1708r0">[P1708R0]
   <dd>Richard Dosselmann, Michael Wong. <a href="https://wg21.link/p1708r0">Simple Statistics functions</a>. 17 June 2019. URL: <a href="https://wg21.link/p1708r0">https://wg21.link/p1708r0</a>
   <dt id="biblio-p1708r1">[P1708R1]
   <dd>Michael Wong. <a href="https://wg21.link/p1708r1">Simple Statistical Functions</a>. 10 October 2019. URL: <a href="https://wg21.link/p1708r1">https://wg21.link/p1708r1</a>
   <dt id="biblio-p1708r2">[P1708R2]
   <dd>Michael Wong, Micheal Chiu, Richard Dosselmann, Eric Niebler, Phillip Ratzlof, Vincent Reverdy. <a href="https://wg21.link/p1708r2">Simple Statistical Functions</a>. 10 January 2020. URL: <a href="https://wg21.link/p1708r2">https://wg21.link/p1708r2</a>
  </dl>
  <h3 class="no-num no-ref heading settled" id="informative"><span class="content">Informative References</span><a class="self-link" href="#informative"></a></h3>
  <dl>
   <dt id="biblio-hyndmanfan_1996">[HYNDMAN+FAN_1996]
   <dd>Rob J. Hyndman; Yanan Fan. <a href="http://www.amherst.edu/media/view/129116/original/Sample+Quantiles.pdf">Sample Quantiles in Statistical Packages</a>. The American Statistician, Vol. 50, No. 4 (Nov., 1996), pp. 361-365. URL: <a href="http://www.amherst.edu/media/view/129116/original/Sample+Quantiles.pdf">http://www.amherst.edu/media/view/129116/original/Sample+Quantiles.pdf</a>
   <dt id="biblio-n1668">[N1668]
   <dd>Paul A Bristow. <a href="https://wg21.link/n1668">A Proposal to add Mathematical Functions for Statistics to the C++ Standard Library</a>. 11 August 2004. URL: <a href="https://wg21.link/n1668">https://wg21.link/n1668</a>
   <dt id="biblio-unur_2016">[UNUR_2016]
   <dd>A. Sinan Unur. <a href="http://www.nu42.com/2016/12/descriptive-stats-with-cpp-boost.html">Descriptive Stats with C++ and Boost</a>. URL: <a href="http://www.nu42.com/2016/12/descriptive-stats-with-cpp-boost.html">http://www.nu42.com/2016/12/descriptive-stats-with-cpp-boost.html</a>
  </dl>