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<!doctype html>
<html lang="en-US">
<head>
<title>STATS 131 Refresher</title>
<meta name="description"
content="Author: Power I. Sunday, Desc: A STATS 131 Web App (refresher) for 100 Level Physical Sciences students University of Nigeria Nsukka Nigeria." />
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<link rel='stylesheet' as="font" href="https://fonts.googleapis.com/css?family=PT+Sans&display=swap" />
<link rel="preload" as="style" href="dist/css/main.min.css" onload="this.onload=null;this.rel='stylesheet';" />
<!-- sign in page style included here to prevent FOUC (Flash Of Unstyled Content) -->
<style id='critical-css'>
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</style>
<link rel="manifest" href="manifest.json">
<meta name="theme-color" content="#1f54f3" />
<meta name="mobile-web-app-capable" content="yes" />
<meta name="apple-mobile-web-app-capable" content="yes" />
<meta name="application-name" content="Space Missions" />
<meta name="apple-mobile-web-app-title" content="Space Missions" />
<meta name="msapplication-navbutton-color" content="#1f54f3" />
<meta name="apple-mobile-web-app-status-bar-style" content="black-translucent" />
<meta name="msapplication-starturl" content="/index.html" />
<meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no" />
<link rel="icon" sizes="128x128" href="icons/icon-128x128.png" />
<link rel="apple-touch-icon" sizes="128x128" href="icons/icon-128x128.png" />
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<script src='dist/js/main.min.js' async></script>
</head>
<body>
<!-- Animated fixed top div - START -->
<header id="page-title-bar" class="title-bar" data-state="hidden">
<h3 id="heading-1" class="page-title" data-state='visible' role="heading" aria-level="1">Mean, Median and Mode</h3>
<h3 id="heading-2" class="page-title" data-state='hidden' role='heading' aria-level="1">Solve Questions</h4>
<h3 id="heading-3" class="page-title" data-state='hidden' role='heading' aria-level="1">Quartiles, Deciles and Percentiles</h4>
</header>
<!-- Animated fixed top div - START -->
<div id='left-elastic'></div>
<div id='right-elastic'></div>
<!-- Sign in page - START -->
<div id="sign-in-page" class="custom-scroll-bar prevent-swipe" data-state="visible" role="document" aria-label="document">
<div id="input-wrapper" class="row" role="form">
<label>
<input type="text" id="username" placeholder="Enter first name to sign in" spellcheck="false"
aria-label="username" aria-describedby="sign-in-status" />
</label>
<button id="sign-in">Sign in</button>
<label id="remember-me-label">
<input type="checkbox" id="remember-me" aria-label="remember me" />
Keep me signed in
</label>
<div id="sign-in-status" class="bad-input-feedback"></div>
</div>
<!-- Sign-in page footer - START -->
<div id="sign-in-page-footer" class="row" role="doc-footnote" aria-label='doc-footnote'>
<span>MEASURES OF CENTRAL TENDENCY</span>
<span>STAT131 Refresher for Upcoming Test</span><br />
<span>Written & Developed By</span>
<span>
<b><a href="https://m.facebook.com/power.sunday3?refid=8" target="_blank" id="my-name" rel="noreferrer">SundayPower Inemesit</a></b>
</span>
<span>Computer Science Dept. <b>¾</b></span>
<br />
<span>© 2018</span>
</div>
<!-- Sign-in page footer - END -->
</div>
<!-- Sign in page - END -->
<!-- BREAK BREAK BREAK BREAK BREAK BREAK -->
<main>
<!-- Home page - START -->
<section id="welcome-page" class="custom-scroll-bar prevent-swipe" data-role='page' role="document" aria-label='document' data-state="hidden">
<header>
<h1>MEASURES OF CENTRAL TENDENCY</h1>
<div class="fixed-top">
<h3 class="sub-heading">The Mean, Median and the Mode</h3>
<h4 class="rider"><i>(Grouped Data)</i></i></h4>
</div>
</header>
<!-- Intro text - START -->
<div id="intro-text-wrapper">
<div id="intro-text" class="row">
<p>
Once again, I welcome you, <span class="username"> Power</span>, to
this refresher 'course' in preparation for our upcoming STAT131 test. :)
</p>
<p>
Before we begin solving questions with the electronic machine I programmed, ;D,
let's go through the formulas or formulae for a reminder and for comprehension.
<br />So, let's get started. Click the next button or swipe if you're on mobile. ;)
</p>
</div>
</div>
<!-- Intro text - END -->
</section>
<!-- Home page - END -->
<!-- BREAK BREAK BREAK BREAK BREAK BREAK -->
<!-- PAGE 1 - Formulae Explications - START -->
<section id="mean-median-mode-page" class="custom-scroll-bar prevent-swipe" data-role='page' role="document" aria-labelledby='heading-1' data-state="hidden">
<header id="header1">
<h3 class="rider"><i>(Grouped Data)</i></i></h3>
</header>
<h3 class="padded whte-bg page-name"> Formulae Explications:</h3>
<!-- Quick links - START -->
<div class="quick-links padded">
<div class="row">
<i>Quick links:</i><br />
<ol>
<li><a href="#mean-formula-ref" class="padded">The mean</a></li>
<li><a href="#median-formula-ref" class="padded">The median</a></li>
<li><a href="#mode-formula-ref" class="padded">The mode</a></li>
</ol>
</div>
</div>
<!-- Quick links - END -->
<br /><br />
<!-- The mean container - START -->
<div id="mean">
<div class="row">
The formulae for finding the <b class="white-bg">mean</b>, the <b class="white-bg">median</b>,
and the <b id="mean-formula-ref" class="white-bg">mode</b> of any grouped data set are as given below:
</div>
<br /><br />
<div class="padded white-bg">
<div class="row">
<b>1.</b> The <b>mean</b> (coding method):
</div>
</div>
<!-- Wrapper for the 'mean' formula - START -->
<div class="padded">
<div class="row">
<div class="mean formula-wrapper custom-scroll-bar prevent-swipe">
<div class="formula-wrapper-first-child">
<div class="formula-wrapper-second-child">
<div class="div1">
<span class="align-vertical"> <b>X̄</b> = A +</span>
</div>
<div class="div2">
<!-- Mean formula fraction - START -->
<div class="div2-childDiv">
<span class="brace-open"> (</span>
<span class="fraction-wrapper">
<span class="numerator"> Σf<sub>i</sub>U<sub>i</sub></span>
<span class="denominator"> Σf<sub>i</sub></span>
</span>
<span class="brace-close"> )</span>
</div>
<!-- Mean formula fraction - END -->
</div>
<div class="div3">
<span class="align-vertical"> × C</span>
</div>
</div>
</div>
</div>
</div>
</div>
<!-- Wrapper for the 'mean' formula - END -->
<!-- Formula explanation for 'mean' - START -->
<div class="formula-explanation">
<div class="p">
<div id="assumed-mean-ref" class="row">
Where;
</div> <br />
<!-- Assumed mean - START-->
<div class="variable-names">
<div class="row">
<i><b>A</b></i> = <b>the assumed mean</b>
</div>
</div>
<div class="variable-definitions">
<div class="row">
(The assumed mean is the value of any class
mark chosen by you from the frequency table. Generally, it is advised
that you choose or take the median class mark as your assumed mean because
it makes finding <b>U<sub>i</sub></b> easier. For instance, if you
<a href="#solve-questions-page" id="take-example" rel="noreferrer">take an example</a>, you'll notice
how the sequence of values in the <b>U<sub>i</sub></b> column of the table,
starting from the middle, 0 upwards, descends towards <b><span
class="infinity">−∞</span></b>
(negatively), and
ascends downwards towards <b><span class="infinity" id="i-definition-ref">+∞</span></b>
(positively) with increments of 1 for each.
<br /><br />
<b><u>Note:</u></b> <b>X<sub>i</sub></b>, the i'th class mark, is calculated
by taking the average of the limits of the i'th class. As in this:<br />
<b>X<sub>4</sub></b>, i.e. the <b>4th</b> class mark, will be equal to;
<!-- the i'th class mark, Xi, formula - START -->
<div id="class-mark" class="variable-names variable-U">
<div class="div1">
<span class="align-vertical"> <b>X<sub>4</sub></b> =</span>
</div>
<div class="div2">
<!-- Formula fraction - START -->
<div class="div2-childDiv">
<span class="fraction-wrapper">
<span class="numerator"><b>UCL<sub>4</sub> + LCL<sub>4</sub></b>;</span>
<span class="denominator"><b>2</b></span>
</span>
</div>
<!-- Formula fraction - END -->
</div>
</div>
<!-- the i'th class mark, Xi, formula - END -->
where <b>UCL<sub>4</sub></b> is the <b>upper class limit</b> of the <b>4th</b> class, and
<b>LCL<sub>4</sub></b>
is the <b>lower class limit</b> of the same. I hope you understand, <span class="username">Power</span>?
Okay,
let's continue);
</div>
</div>
<!-- Assumed mean - END-->
<!-- i'th class frequency - START -->
<div class="variable-names">
<div class="row">
<i><b>f<sub>i</sub></b></i> = <b>i'th class frequency</b>
</div>
</div>
<div class="variable-definitions">
<div class="row">
(<b><u>Note:</u></b> 'i' is a variable and is as a placeholder. As in the <a
href="#i-definition-ref">above
definition</a>, the i'th class mark/frequency
in a table could be the <b>4</b>th class mark/frequency or <b>6</b>th class mark/frequency etc. in that
table.
Just know that wherever you see 'i', it is a variable; it changes, nothing more. For instance
<i><b>f<sub>4</sub></b></i>,
(<b>4</b> in place of 'i') is the frequency of the fourth class. Do you now comprende', <span
class="username">Power</span>?
Good. We're making progress. ;))
</div>
</div>
<!-- i'th class frequency - END -->
<!-- The coding method variable, 'U' - START -->
<div class="variable-names variable-U">
<div class="row">
<div class="div1">
<span class="align-vertical"> <b>U<sub>i</sub></b> =</span>
</div>
<div class="div2">
<!-- Coding method formula variable fraction, U - START -->
<div class="div2-childDiv">
<span class="fraction-wrapper">
<span class="numerator"><b>X<sub>i</sub> − A</b></span>
<span class="denominator"><b>C</b></span>
</span>
</div>
<!-- Coding method formula variable fraction, U - END -->
</div>
</div>
</div>
<!-- The coding method variable, 'U' - END -->
<div class="variable-definitions">
<div class="row">
(<b><i>X<sub>i</sub></i></b> is the i'th class mark, <b><i>A</i></b>
is the <a href="#assumed-mean-ref">assumed mean</a>, <b><i id="C">C</i></b> is the class size);
</div>
</div>
<!-- Class size - START -->
<div class="variable-names">
<div class="row">
<b><i>C</i></b> = <b>class size...</b>
</div>
</div>
<div class="variable-definitions">
<div class="row">
(Calculated as: The difference between the upper class limit and the
lower class limit of any class + 1. <br />
<b>Mathematically:<br /> (U<sub>limit</sub> − L<sub id="median-formula-ref">limit</sub>) + 1.</b>
<br />
Get the logic? Nice. Moving forward.
</div>
</div>
<!-- Class size - END -->
</div>
</div>
</div>
<!-- The 'mean' container - END -->
<br />
<!-- BREAK BREAK BREAK BREAK BREAK BREAK BREAK BREAK BREAK BREAK BREAK BREAK BREAK BREAK BREAK BREAK -->
<!-- The 'mediam' container - START -->
<div id="median">
<!-- Wrapper for 'median' formula - START -->
<div class="padded white-bg">
<div class="row">
<b>2.</b> The <b>median</b>:
</div>
</div>
<div class="padded">
<div class="row">
<div class="median formula-wrapper custom-scroll-bar prevent-swipe">
<div class="formula-wrapper-first-child">
<div class="formula-wrapper-second-child">
<div class="div1">
<span class="align-vertical"> <b>X̃</b> = L<sub>med</sub> +</span>
</div>
<div class="div2">
<!-- Formula fraction -->
<div class="div2-childDiv">
<span class="brace-open"> (</span>
<span class="fraction-wrapper">
<span class="numerator"> <sup>N</sup>/<sub>2</sub> − Σf<sub><med</sub></span>
<span class="denominator"> f<sub>med</sub></span>
</span>
<span class="brace-close"> )</span>
</div>
</div>
<div class="div3">
<span class="align-vertical"> × C</span>
</div>
</div>
</div>
</div>
</div>
</div>
<!-- Wrapper for 'median' formula - END -->
<!-- Formula explain for 'median' -START -->
<div class="formula-explanation">
<div class="p">
<div id="Lmed-ref" class="row">
Where;
</div><br />
<!-- Lower boundary of median class -->
<div class="variable-names">
<div class="row">
<i><b>L<sub>med</sub></b></i> = <b>lower class boundary of median class</b>
</div>
</div>
<div class="variable-definitions">
<div class="row">
(The lower class boundary of the median class is obtained by substracting 0.5 from the lower class limit
of the same. For instance, say we have median class limits of <b>55 - 67</b>, the lower class boundary
would then be <b>54.5</b>);
</div>
</div>
<!-- Summation of frequencies of all classes -->
<div class="variable-names">
<div class="row">
<i><b>N</b></i> = <b>Σf</b> <i>(read as summation-f or sigma-f)</i>
</div>
</div>
<div class="variable-definitions">
<div class="row">
(This is the same as the summation of the frequencies of all the classes);
</div>
</div>
<!-- Cummulative frequency below median class -->
<div class="variable-names">
<div class="row">
<i><b>Σf<sub><med</sub></b></i> = <b>cummulative frequency below median
class</b>
</div>
</div>
<div class="variable-definitions">
<div class="row">
(I like to call it "sigma-f below", i.e. summation of frequencies below median class);
</div>
</div>
<!-- Frequency of median class -->
<div class="variable-names">
<div class="row">
<b><i>f<sub>med</sub></i></b> = <b>frequency of median class</b>
</div>
</div>
<div class="variable-definitions">
<div class="row">
(This is obtained directly from the frequency distribution table);
</div>
</div>
<!-- Class size -->
<div class="variable-names">
<div class="row">
<b><i>C</i></b> = <b>class size</b>
</div>
</div>
<div class="variable-definitions">
<div class="row">
(Refer to <a href="#C" id="mode-formula-ref">class size</a> under 'The mean' above).
</div>
</div>
</div>
</div>
<!-- Formula explain for 'median' - END -->
</div>
<!-- THe 'median' container - END -->
<br />
<!-- Break Break Break Break Break Break Break Break Break Break Break Break Break Break Break -->
<!-- THe 'modal' container - START -->
<div id="mode">
<!-- Wrapper for 'mode' formula - START -->
<div class="padded white-bg">
<div class="row">
<b>3.</b> The <b>mode</b>:
</div>
</div>
<div class="padded">
<div class="row">
<div class="mode formula-wrapper custom-scroll-bar prevent-swipe">
<div class="formula-wrapper-first-child">
<div class="formula-wrapper-second-child">
<div class="div1">
<span class="align-vertical"> <b>X̂</b> = L<sub>mod</sub> +</span>
</div>
<div class="div2">
<!-- Formula fraction -->
<div class="div2-childDiv">
<span class="brace-open"> (</span>
<span class="fraction-wrapper">
<span class="numerator"> D<sub>1</sub></span>
<span class="denominator"> D<sub>1</sub> + D<sub>2</sub></span>
</span>
<span class="brace-close"> )</span>
</div>
</div>
<div class="div3">
<span class="align-vertical"> × C</span>
</div>
</div>
</div>
</div>
</div>
</div>
<!-- Wrapper for 'mode' formula - END -->
<!-- Formula explanation for 'mode' - START -->
<div class="formula-explanation">
<div class="p">
<div class="row">
Where;
</div><br />
<!-- Lower boundary of modal class -->
<div class="variable-names">
<div class="row">
<i><b>L<sub>mod</sub></b></i> = <b>lower class boundary of modal class</b>
</div>
</div>
<div class="variable-definitions">
<div class="row">
(The lower class boundary of the modal class is obtained by substracting 0.5 from the lower class limit
of
the same. See <a href="#Lmed-ref">L<sub>med</sub></a>);
</div>
</div>
<!-- Frequency of next lower class -->
<div class="variable-names">
<div class="row">
<i><b>D<sub>1</sub></b></i> = <b>frequency difference between modal class and
next lower class</b>
</div>
</div>
<div class="variable-definitions">
<div class="row">
(This is obtained directly from the table. It is the difference between the frequencies of the modal
class
and the class just before it, the modal class.);
</div>
</div>
<!-- Frequency of next higher class -->
<div class="variable-names">
<div class="row">
<i><b>D<sub>2</sub></b></i> = <b>frequency difference between modal class and
next higher class</b>
</div>
</div>
<div class="variable-definitions">
<div class="row">
(This is also obtained directly from the table. It is the difference between the frequencies of the
modal
class and the class just after the modal class.);
</div>
</div>
<!-- Class size -->
<div class="variable-names">
<div class="row"> <b><i>C</i></b> = <b>class size</b></div>
</div>
<div class="variable-definitions">
<div class="row">
(Refer to <a href="#C">class size</a> under 'The mean'</a> above).
</div>
</div>
</div>
</div>
<!-- Formula explanation for 'mode' - END -->
</div>
<!-- The 'modal' container - END -->
<br />
<div class="row">
<p class="padded p">
Okay now, <span class="username">Power</span>, having refreshed our minds on these formulas, let's move on to
solve any questions you'll be providing. ;)
</p>
</div>
</section>
<!-- PAGE 1 - Formulae Explications - END -->
<!-- BREAK BREAK BREAK BREAK BREAK BREAK BREAK -->
<!-- PAGE 2 - Solving Questions - START -->
<section id="solve-questions-page" class="custom-scroll-bar prevent-swipe" data-role='page' role="document" aria-labelledby='heading-2' data-state="hidden">
<header id="header2">
<h3 class="rider"><i>(Grouped Data)</i></i></h3>
</header>
<h3 class="padded page-name">The Distribution Computer:</h3>
<!-- Input for calculations -->
<div class="row">
<div id="container" class="padded" role="form" aria-label='form'>
<label>
<input id="interval" type="text" value="" aria-label="class intervals" aria-describedby="result"
placeholder="Input limits of only 1st class e.g. 45 50" />
</label>
<label>
<input id="frequencies" type="text" value="" aria-label="frequencies" aria-describedby="result"
placeholder="Input respective frequencies of all classes e.g. 3 2 5 6 4..." />
</label>
<button id="compute">Compute</button>
</div>
<div id="result" class="prevent-swipe custom-scroll-bar">
In the input boxes above, for the first, input the class limits of
only the first class. You don't have to input the limits for all the classes. And for the
second, input the frequencies of all the classes respectively. Don't forget
to separate the values you input ('23' is not same as
'2 3', you know). <br /> <i>Note: For full width of table when result is displayed, rotate screen.</i>;)
</div>
</div>
<!-- Container for input question solutions - START -->
<div class="row">
<div id="solutions-wrapper">
<b>Let's solve:</b>
<!-- Wrapper for 'mean' solution - START -->
<div id="mean-solution">
<div class="variable-definitions">
Solving for the <b>mean</b>, from the values you provided, and from the table above, we have that: <br />
<b>A</b> = <span class="A"> A</span>; <br />
<b>Σf<sub>i</sub>U<sub>i</sub></b> = <span class="EfU">Σf<sub>i</sub>U<sub>i</sub></span>;
<br />
<b>Σf<sub>i</sub></b> = <span class="Ef">Σf<sub>i</sub></span>; and<br />
<b>C</b> = <span class="C">C</span>. <br /><br />
Therefore, using the <b>mean</b> formula:
<!-- The mean formula - START -->
<div class="formula-wrapper custom-scroll-bar prevent-swipe">
<div class="formula-wrapper-first-child">
<div class="formula-wrapper-second-child">
<div class="div1">
<span class="align-vertical"><b>X̄</b> = A +</span>
</div>
<div class="div2">
<!-- Formula fraction -->
<div class="div2-childDiv">
<span class="brace-open"> (</span>
<span class="fraction-wrapper">
<span class="numerator">Σf<sub>i</sub>U<sub>i</sub></span>
<span class="denominator">Σf<sub>i</sub></span>
</span>
<span class="brace-close"> )</span>
</div>
</div>
<div class="div3">
<span class="align-vertical"> × C</span>
</div>
</div>
</div>
</div>
<!-- The mean formula - END -->
We have,
<!-- Formula values inputted - START -->
<div class="mean formula-wrapper custom-scroll-bar prevent-swipe">
<div class="formula-wrapper-first-child">
<div class="formula-wrapper-second-child">
<div class="div1">
<span class="align-vertical"> <b class="slant">X̄</b> = <span class="A">A</span> +</span>
</div>
<div class="div2">
<!-- Formula fraction -->
<div class="div2-childDiv">
<span class="brace-open"> (</span>
<span class="fraction-wrapper">
<span class="EfU numerator"> Σf<sub>i</sub>U<sub>i</sub></span>
<span class="Ef denominator"> Σf<sub>i</sub></span>
</span>
<span class="brace-close"> )</span>
</div>
</div>
<div class="div3">
<span class="align-vertical fraction-wrapper"> × <span class="C"> C</span></span>
</div>
</div>
</div>
</div>
<!-- Formula values inputted - END -->
<!-- Solving further - START -->
<div class="mean formula-wrapper custom-scroll-bar prevent-swipe">
<div class="formula-wrapper-first-child">
<div class="formula-wrapper-second-child">
<div class="div1">
<span class="align-vertical">
<b class="slant">X̄</b> =
<span class="A"> A</span> + <b class="unslant"> (</b>
<span class="mean-fraction"> EfU/Ef</span> × <span class="C"> C</span><b class="unslant">
)
</b>
</span>
</div>
</div>
</div>
</div>
<!-- Solving further - END -->
The <b>mean</b>, therefore, of the frequency distribution is <b>≈ <span id="computed-mean"> MEAN
</span></b>.
</div>
</div>
<!-- Wrapper for 'mean' solution - END -->
<!-- Wrapper for 'median' solution - START -->
<div id="median-solution">
<div class="variable-definitions">
In the same vein, solving for the <b>median</b>, we have: <br />
<b>L<sub>med</sub></b> = <span class="Lmed"> L<sub>med</sub></span>; <br />
<b>N</b> = <span class="N"> N</span>; <br />
<b>Σf<sub><med</sub></b> = <span class="EfBelow"> Σf<sub><med</sub></span>; <br />
<b>f<sub>med</sub></b> = <span class="fmed"> f<sub>med</sub></span>; <br />
<b>C</b> = <span class="C"> C</span>. <br /><br />
Therefore, using the <b>median</b> formula:
<!-- The mean formula wrapper - START -->
<div class="median formula-wrapper custom-scroll-bar prevent-swipe">
<div class="formula-wrapper-first-child">
<div class="formula-wrapper-second-child">
<div class="div1">
<span class="align-vertical"> <b class="slant">X̃</b> = L<sub>med</sub> +</span>
</div>
<div class="div2">
<!-- Formula fraction -->
<div class="div2-childDiv">
<span class="brace-open"> (</span>
<span class="fraction-wrapper">
<span class="numerator"> <sup>N</sup>/<sub>2</sub> − Σf<sub><med</sub></span>
<span class="denominator"> f<sub>med</sub></span>
</span>
<span class="brace-close"> )</span>
</div>
</div>
<div class="div3">
<span class="align-vertical"> × C</span>
</div>
</div>
</div>
</div>
<!-- The mean formula wrapper - END -->
We also have that,
<!-- Formula values inputted - START -->
<div class="median formula-wrapper custom-scroll-bar prevent-swipe">
<div class="formula-wrapper-first-child">
<div class="formula-wrapper-second-child">
<div class="div1">
<span class="align-vertical"> <b class="slant">X̃</b> = <span class="Lmed">
L<sub>med</sub></span> +</span>
</div>
<div class="div2">
<!-- Formula fraction -->
<div class="div2-childDiv">
<span class="brace-open"> (</span>
<span class="fraction-wrapper">
<span class="numerator"> <sup class="N">N</sup>/<sub>2</sub> − <span class="EfBelow">
Σf<sub><med</sub></span></span>
<span class="fmed denominator"> f<sub>med</sub></span>
</span>
<span class="brace-close"> )</span>
</div>
</div>
<div class="div3">
<span class="align-vertical"> × <span class="C"> C</span></span>
</div>
</div>
</div>
</div>
<!-- Formula values inputted - END -->
<!-- Solving further - START -->
<div class="mean formula-wrapper custom-scroll-bar prevent-swipe">
<div class="formula-wrapper-first-child">
<div class="formula-wrapper-second-child">
<div class="div1">
<span class="align-vertical">
<b class="slant">X̃</b> =
<span class="Lmed"> Lmed</span> +
<b class="unslant"> (</b>
<span class="median-fraction"> N/2</span> × <span class="C"> C</span>
<b class="unslant"> )</b>
</span>
</div>
</div>
</div>
</div>
<!-- Solving further - END -->
Hence, the <b>median</b> of the distribution is <b>≈ <span id="computed-median"> MEDIAN</span></b>.
</div>
</div>
<!-- Wrapper for 'median' solution - END -->
<!-- Wrapper for 'mode' solution - START -->
<div id="mode-solution">
<div class="variable-definitions">
Finally, solving for the <b>mode</b>, we have that: <br />
<b>L<sub>mod</sub></b> = <span class="Lmod"> L<sub>mod</sub></span>; <br />
<b>D<sub>1</sub></b> = <span class="D1"> D<sub>1</sub></span>; <br />
<b>D<sub>2</sub></b> = <span class="D2"> D<sub>2</sub></span>; and<br />
<b>C</b> = <span class="C"> C</span>. <br /><br />
Therefore, using the <b>mode</b> formula:
<!-- The modal formula wrapper - START -->
<div class="mode formula-wrapper custom-scroll-bar prevent-swipe">
<div class="formula-wrapper-first-child">
<div class="formula-wrapper-second-child">
<div class="div1">
<span class="align-vertical"> <b class="slant">X̂</b> = L<sub>mod</sub> +</span>
</div>
<div class="div2">
<!-- Formula fraction -->
<div class="div2-childDiv">
<span class="brace-open"> (</span>
<span class="fraction-wrapper">
<span class="numerator"> D<sub>1</sub></span>
<span class="denominator"> D<sub>1</sub> + D<sub>2</sub></span>
</span>
<span class="brace-close"> )</span>
</div>
</div>
<div class="div3">
<span class="align-vertical"> × C</span>
</div>
</div>
</div>
</div>
<!-- The modal formula wrapper - END -->
We have,
<!-- Formula values inputted - START -->
<div class="mode formula-wrapper custom-scroll-bar prevent-swipe">
<div class="formula-wrapper-first-child">
<div class="formula-wrapper-second-child">
<div class="div1">
<span class="align-vertical"> <b class="slant">X̂</b> = <span class="Lmod"> L<sub>mod</sub>
</span> +</span>
</div>
<div class="div2">
<!-- Formula fraction -->
<div class="div2-childDiv">
<span class="brace-open"> (</span>
<span class="fraction-wrapper">
<span class="D1 numerator"> D<sub>1</sub></span>
<span class="denominator"><span class="D1"> D<sub>1</sub></span> + <span class="D2">
D<sub>2</sub></span></span>
</span>
<span class="brace-close"> )</span>
</div>
</div>
<div class="div3">
<span class="align-vertical"> × <span class="C"> C</span></span>
</div>
</div>
</div>
</div>
<!-- Formula values inputted - END -->
<!-- Solving further - START -->
<div class="mean formula-wrapper custom-scroll-bar prevent-swipe">
<div class="formula-wrapper-first-child">
<div class="formula-wrapper-second-child">
<div class="div1">
<span class="align-vertical">
<b class="slant">X̂</b> =
<span class="Lmod"> Lmod</span> +</span> <b class="unslant"> (</b>
<span class="mode-fraction"> D1/D2</span> × <span class="C"> C</span><b class="unslant"> )
</b>
</span>
</div>
</div>
</div>
</div>
<!-- Solving further - END -->
Consequentially, the <b>mode</b> of the distribution is <b>≈ <span id="computed-mode">
MODE</span></b>.
</div>
<!-- Wrapper for 'mode' solution - END -->
<span class="padded">
Alright, <span class="username"> Power</span>. Having understood all these, let's discuss a little bit
further.
Tap the 'next' button or swipe left. ;)
</span>
</div>
</div>
</div>
</section>
<!-- BREAK BREAK BREAK BREAK BREAK BREAK -->
<!-- PAGE 3 - Quartiles, Percentiles and Deciles -->
<section id="further-discussion-page" class="custom-scroll-bar prevent-swipe" data-role='page' role="document" aria-labelledby='heading-3' data-state="hidden">
<header id="header3">
<h3 class="rider"><i>(Grouped Data)</i></i></h3>
</header>
<h3 class="padded page-name">Further Discussion:</h3>
<!-- Quick links - START -->
<div class="quick-links white-bg">
<div class="row">
<i>Quick links:</i><br />
<ol type="1">
<li>
<a href="#the-quartiles-ref"> The quartiles</a>
<ol type="A">
<li><a href="#quartile-formula-ref" class="padded"> The quartile formula</a></li>
<li>
<a href="#semi-interquartile-range-ref" class="padded"> The semi-interquartile-range
</a>
</li>
</ol>
</li>
<li>
<a href="#the-deciles-ref"> The deciles</a>
<ol type="A">
<li><a href="#decile-formula-ref" class="padded"> The decile formula</a></li>
</ol>
</li>
<li>
<a href="#the-percentiles-ref"> The percentiles</a>
<ol type="A">
<li><a href="#percentile-formula-ref" class="padded"> The percentile formula</a></li>
</ol>
</li>
</ol>
</div>
</div>
<!-- Quick links - END -->
<br />
<div class="row">
<div class="padded">
Alright. Now, <span class="username"> Power</span>? I'm sure you must have heard of these words that end with
<b><i>"iles"</i></b> (hmm...what about we call them <b><i>"the iles"</i></b>? Cool, right? ;)). Okay. They are
the
<b class="white-bg">quartiles</b>, the <b class="white-bg">deciles</b>,
and the <b class="white-bg">percentiles</b>.<br />
When you meet them, do not let them scare you. They are easy to handle so long you can get the logic behind
handling them well enough.<br />
So, let's talk about them, about <b><i id="the-quartiles-ref">"the iles".</i></b>
</div>
</div>
<!-- Wrapper for 'quartiles' formula - START -->
<div id="quartiles">
<h4>
<div class="white-bg padded">
<div class="row">
<b>1.</b> The quartiles:
</div>
</div>
</h4>
<div class="row">
The <b>quartiles</b> consist of four parts, (just like from the word, <i>'quarter'</i> which implies a fourth
part of something), obviously
the first part, <b><i>Q1</i></b>, the second, <b><i>Q2</i></b>, the third, <b><i>Q3</i></b> and the fourth,
<b><i>Q4</i></b>. <br />
Now, we are going to consider only two of the parts here and discard the other two. I'll tell you why. Let's
continue. <br />
The two parts we'll consider are <b><i>Q1</i></b> and <b><i>Q3</i></b>. <br /><br />
From the elaboration of quartiles and quarter above, we can say that <b><i>Q1</i></b>
is the 1'th part of the quartile (called as first-quartile) and <b><i>Q3</i></b> is
the 3'th part of the quartile (called as third-quartile). This means that <b><i>Q1</i></b>
= ¼ of <b><i>Q4</i></b> and <b><i>Q3</i></b> = ¾ of <b><i>Q4</i></b>. Notice I
said, <b>"...of Q4"</b>, because Q4 is itself the quartile. And also <b><i>Q2</i></b>, the
2'th part, would be <sup>2</sup><i class="slant">/</i><sub>4</sub> of <b><i>Q4</i></b>
which is equal to ½ of <b><i>Q4</i></b> which is also the same as the <b><i>median</i></b>,
because when we divide a dataset by 2 we get <a href="#median-formula-ref" id="link-to-median">the median</a>
which we discussed earlier. So, <b><i>Q2</i></b> is the median (half of the quartile) and <b><i>Q4</i></b>
is the quartile itself, the full package. Now, you see why we had to discard them both. Good. So,
let's move forward.<br /> <br />
Again, from the above illustration, we see how the quartiles are related to each other and the median.
<br /><br />
Take this logic and never forget it, because you'll use the same when solving for <span
class="white-bg">deciles</span>
and <span class="white-bg">percentiles</span>: <br />
If you take a look at the formula for finding <a href="#median-formula-ref" id="quartiles-median-ref">the
median</a>
of a grouped data, you'll notice there's a variable <b><i>N</i></b> involved in the calculation. This variable
is the 'mother'
quartile, I mean <b><i>Q4</i></b>; same with <b><i>D10</i></b> for the <b class="white-bg"> deciles</b> and
<b><i>P100</i></b>
for the <b class="white-bg">percentiles</b>. Also, notice the variable, <b><i>N</i></b>, is divided by 2 which
is the same as saying ½ of
<b><i>N</i></b> which also implies ½ of <b><i>Q4</i></b> which is, as well, equal to <b><i>Q2</i></b>,
the median.
<span class="username"> Power</span>? I hope you are grasping? <br />
Now, using the same logic, we can compute the values of <b><i>Q1</i></b> and <b><i>Q3</i></b> by substituting
'<b><i><sup class="unslant">1</sup><i class="slant">/</i><sub class="unslant">2</sub></i></b>' in the formula
for '¼'
and '¾' respectively since <b><i>Q4</i></b> = <b><i>N</i></b>.<br /><br />
<b id="quartile-formula-ref"> <b class="white-bg">1.A</b></b>For <b><i>Q1</i></b>, the formula would look
like this:
</div>
<!-- Quartile formula wrapper - START -->
<div class="row">
<div class="median formula-wrapper custom-scroll-bar prevent-swipe">
<div class="formula-wrapper-first-child">
<div class="formula-wrapper-second-child">
<div class="div1">
<span class="align-vertical"> <b>Q1</b> = L<sub>Q1</sub> +</span>
</div>
<div class="div2">
<!-- Formula fraction -->
<div class="div2-childDiv">
<span class="brace-open"> (</span>
<span class="fraction-wrapper">
<span class="numerator"> <b class="unslant"> [</b><span class="unslant">¼</span> ×
N<b class="unslant">]</b> − Σf<sub><Q1</sub></span>
<span class="denominator"> f<sub>Q1</sub></span>
</span>
<span class="brace-close"> )</span>
</div>
</div>
<div class="div3">
<span class="align-vertical"> × C</span>
</div>
</div>
</div>
</div>
</div>
<!-- Quartile formula wrapper - END -->
<!-- Formula explanation for 'first-quartile' - START -->
<div class="formula-explanation">
<div class="p">
<div class="row">Where;</div><br />
<!-- Lower boundary of median class -->
<div class="variable-names">
<div class="row">
<i><b>L<sub>Q1</sub></b></i> = <b>lower class boundary of the first-quartile
class</b>
</div>
</div>
<div class="variable-definitions">
<div class="row">
(Having known how to calculate for lower class boundaries, the quartile class in this case is the class
whose frequency has the result<i class="slant">/</i>value of <b class="unslant">[</b><span
class="unslant">¼</span> × N<b class="unslant">]</b> embedded in it. I.e. the value<i
class="slant">/</i>result of <b class="unslant">[</b><span class="unslant">¼</span> × N<b
class="unslant">]</b> is in the range of values of the frequency of the class);
</div>
</div>
<!-- Summation of frequencies of all classes -->
<div class="variable-names">
<div class="row">
<b><i>N</i> = Q4 = <i>Σf</i></b> <i>(read as summation-f or sigma-f)</i>
</div>
</div>
<div class="variable-definitions">
<div class="row">
(This is the same as the summation of the frequencies of all the classes);
</div>
</div>
<!-- Cummulative frequency below median class -->
<div class="variable-names">
<div class="row">
<i><b>Σf<sub><Q1</sub></b></i> = <b>cummulative frequency below
<b><i>Q1</i></b> class</b>
</div>
</div>
<div class="variable-definitions">
<div class="row">
(I.e. the summation of the frequencies below the frequecy of the quartile class, <b><i>Q1 (the first
quartile class)</i></b>, in this case);
</div>
</div>
<!-- Frequency of median class -->
<div class="variable-names">
<div class="row"> <b><i>f<sub>Q1</sub></i></b> = <b>frequency of <b><i>Q1</i></b> class</b></div>
</div>
<div class="variable-definitions">
<div class="row">
(This is obtained directly from the frequency distribution table having gotten the value of <b
class="unslant">[</b><span class="unslant">¼</span> × N<b class="unslant">]</b>);
</div>
</div>
<!-- Class size -->
<div class="variable-names">
<div class="row"> <b><i>C</i></b> = <b>class size</b></div>
</div>