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variance.html
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variance.html
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<!DOCTYPE html>
<html data-require="math math-format word-problems stat">
<head>
<meta charset="UTF-8" />
<title>Variance</title>
<script src="../khan-exercise.js"></script>
<style>
#answer_area .short input[type=text] {
width: 40px;
}
</style>
</head>
<body>
<div class="exercise">
<div class="vars">
<var id="DATA_POINTS">randRange( 4, 6 )</var>
<var id="POPULATION">randRange( 20, 50 )</var>
<var id="TGT_MEAN">animalAvgLifespan( 1 )</var>
<var id="TGT_STDDEV">animalStddevLifespan( 1 )</var>
<var id="DATA">$.map( randGaussian( TGT_MEAN, TGT_STDDEV, DATA_POINTS ), function( lifespan ) {
lifespan = lifespan < 1 ? 1 : round( lifespan );
return randRange( 1, lifespan );
} )</var>
<var id="MEAN">roundTo( 1, mean( DATA ) )</var>
<var id="SQR_DEV">$.map( DATA, function( x ) { return roundTo( 2, ( x - MEAN ) * ( x - MEAN ) ); })</var>
<var id="VARIANCE">roundTo( 2, sum( SQR_DEV ) / ( DATA_POINTS - 1 ) )</var>
<var id="VARIANCE_POP">roundTo( 2, sum( SQR_DEV ) / DATA_POINTS )</var>
</div> <!-- vars -->
<div class="problems">
<div id="population">
<div class="problem" data-else>
<p>You have found the following ages (in years) of all <var>plural( DATA_POINTS, animal( 1 ) )</var> at your local zoo:</p>
<p><code>\qquad<var>DATA.join( ",\\enspace " )</var></code></p>
</div>
<p class="question">
What is the average age of the <var>plural( animal( 1 ) )</var> at your zoo? What is the variance?
You may round your answers to the nearest tenth.
</p>
<div class="solution" data-type="multiple">
<p>
Average age:<br><code>\quad</code>
<span class="sol short" data-type="decimal" data-inexact data-max-error="0.15"><var>mean( DATA )</var></span> years old
</p>
<p>
Variance:<br><code>\quad</code>
<span class="sol short" data-type="decimal" data-inexact data-max-error="0.15"><var>sum( SQR_DEV ) / DATA_POINTS</var></span> years<code>^2</code>
</p>
<div class="example">decimals, like <code>7.5</code></div>
<div class="example">answers within <code>\pm 0.15</code> are accepted to allow for rounding part-way through</div>
</div> <!-- solution -->
<div class="hints">
<p>
Because we have data for all <var>plural( DATA_POINTS, animal( 1 ) )</var> at the zoo, we are able
to calculate the <span class="hint_blue">population mean</span>
<code>(\color{<var>BLUE</var>}{\mu})</code> and
<span class="hint_pink">population variance</span> <code>(\color{<var>PINK</var>}{\sigma^2})</code>.
</p>
<div>
<p>
To find the <span class="hint_blue">population mean</span>, add up the values of all <code class="hint_green"><var>DATA_POINTS</var></code>
ages and divide by <code class="hint_green"><var>DATA_POINTS</var></code>.
</p>
<p>
<code>
\color{<var>BLUE</var>}{\mu} \quad = \quad
\dfrac{\sum\limits_{i=1}^{\color{<var>GREEN</var>}{N}} x_i}{\color{<var>GREEN</var>}{N}} \quad = \quad
\dfrac{\sum\limits_{i=1}^{\color{<var>GREEN</var>}{<var>DATA_POINTS</var>}} x_i}{\color{<var>GREEN</var>}{<var>DATA_POINTS</var>}}
</code>
</p>
</div>
<p>
<code>
\color{<var>BLUE</var>}{\mu} \quad = \quad
\dfrac{<var>plus.apply( KhanUtil, DATA )</var>}{\color{<var>GREEN</var>}{<var>DATA_POINTS</var>}} \quad = \quad
\color{<var>BLUE</var>}{<var>MEAN</var>\text{ <var>plural( "year", MEAN )</var> old}}
</code>
</p>
<div>
<p>
Find the <span class="hint_purple">squared deviations from the mean</span> for each <var>animal(1)</var>.
</p>
<div class="fake_header">
<span style="width: 100px;">
Age<br/>
<code>x_i</code>
</span><span style="width: 150px;">
<span class="hint_gray">Distance from the mean</span>
<code>(x_i - \color{<var>BLUE</var>}{\mu})</code>
</span><span style="width: 150px;">
<code>(x_i - \color{<var>BLUE</var>}{\mu})^2</code>
</span>
</div>
<div class="fake_row" data-each="DATA as i, POINT">
<span style="width: 100px;">
<code><var>POINT</var></code> <var>plural( "year", POINT )</var>
</span><span style="width: 150px;" class="hint_gray">
<code><var>roundTo( 2, POINT - MEAN )</var></code> <var>plural( "year", roundTo( 2, POINT - MEAN ) )</var>
</span><span style="width: 150px;" class="hint_purple">
<code><var>SQR_DEV[ i ]</var></code> <var>plural( "year", SQR_DEV[ i ] )</var><code>^2</code>
</span>
</div>
</div>
<div>
<p>
Because we used the <span class="hint_blue">population mean</span><code>(\color{<var>BLUE</var>}{\mu})</code> to compute the
<span class="hint_purple">squared deviations from the mean</span>, we can find the <span class="hint_red">variance</span>
<code>(\color{red}{\sigma^2})</code>, without introducing any bias, by simply averaging the
<span class="hint_purple">squared deviations from the mean</span>:
</p>
<p>
<code>
\color{red}{\sigma^2} \quad = \quad
\dfrac{\sum\limits_{i=1}^{\color{<var>GREEN</var>}{N}} (x_i - \color{<var>BLUE</var>}{\mu})^2}{\color{<var>GREEN</var>}{N}}
</code>
</p>
</div>
<p>
<code>
\color{red}{\sigma^2} \quad = \quad
\dfrac{<var>plus.apply( KhanUtil, $.map( SQR_DEV, function( x ) { return "\\color{purple}{" + x + "}"; }) )</var>}
{\color{<var>GREEN</var>}{<var>DATA_POINTS</var>}}
</code>
</p>
<p>
<code>
\color{red}{\sigma^2} \quad = \quad
\dfrac{\color{purple}{<var>roundTo( 2, sum( SQR_DEV ) )</var>}}{\color{<var>GREEN</var>}{<var>DATA_POINTS</var>}} \quad = \quad
\color{red}{<var>VARIANCE_POP</var>\text{ <var>plural( "year", VARIANCE_POP )</var>}^2}
</code>
</p>
<p><strong>
The average <var>animal( 1 )</var> at the zoo is <var>plural( MEAN, "year" )</var> old, and the population variance
is <var>plural( VARIANCE_POP, "year" )</var><code>^2</code>.
</strong></p>
</div> <!-- hints -->
</div> <!-- population -->
<div id="sample">
<div class="problem" data-else>
<p>
You have found the following ages (in years) of <var>plural( DATA_POINTS, animal( 1 ) )</var>
randomly selected from the <var>plural( POPULATION, animal( 1 ) )</var> at your local zoo:
</p>
<p><code>\qquad<var>DATA.join( ",\\enspace " )</var></code></p>
</div>
<p class="question">
Based on your sample, what is the average age of the <var>plural( animal( 1 ) )</var>? What is the variance?
You may round your answers to the nearest tenth.
</p>
<div class="solution" data-type="multiple">
<p>
Average age:<br><code>\quad</code>
<span class="sol short" data-type="decimal" data-inexact data-max-error="0.15"><var>mean( DATA )</var></span> years old
</p>
<p>
Variance:<br><code>\quad</code>
<span class="sol short" data-type="decimal" data-inexact data-max-error="0.15"><var>sum( SQR_DEV ) / ( DATA_POINTS - 1 )</var></span> years<code>^2</code>
</p>
<div class="example">decimals, like <code>0.75</code></div>
<div class="example">answers within <code>\pm 0.15</code> are accepted to allow for rounding part-way through</div>
</div> <!-- solution -->
<div class="hints">
<p>
Because we only have data for a small sample of the <var>plural( POPULATION, animal( 1 ) )</var>, we are only able
to estimate the population mean and variance by finding the <span class="hint_blue">sample mean</span>
<code>(\color{<var>BLUE</var>}{\overline{x}})</code> and
<span class="hint_pink">sample variance</span> <code>(\color{<var>PINK</var>}{s^2})</code>.
</p>
<div>
<p>
To find the <span class="hint_blue">sample mean</span>, add up the values of all <code class="hint_green"><var>DATA_POINTS</var></code>
samples and divide by <code class="hint_green"><var>DATA_POINTS</var></code>.
</p>
<p>
<code>
\color{<var>BLUE</var>}{\overline{x}} \quad = \quad
\dfrac{\sum\limits_{i=1}^{\color{<var>GREEN</var>}{n}} x_i}{\color{<var>GREEN</var>}{n}} \quad = \quad
\dfrac{\sum\limits_{i=1}^{\color{<var>GREEN</var>}{<var>DATA_POINTS</var>}} x_i}{\color{<var>GREEN</var>}{<var>DATA_POINTS</var>}}
</code>
</p>
</div>
<p>
<code>
\color{<var>BLUE</var>}{\overline{x}} \quad = \quad
\dfrac{<var>plus.apply( KhanUtil, DATA )</var>}{\color{<var>GREEN</var>}{<var>DATA_POINTS</var>}} \quad = \quad
\color{<var>BLUE</var>}{<var>MEAN</var>\text{ <var>plural( "year", MEAN )</var> old}}
</code>
</p>
<p>
Find the <span class="hint_purple">squared deviations from the mean</span> for each sample. Since we don't know the
population mean, estimate the mean by using the <span class="hint_blue">sample mean</span> we just calculated
<code>(\color{<var>BLUE</var>}{\overline{x}} = \color{<var>BLUE</var>}{<var>MEAN</var>\text{ <var>plural( "year", MEAN )</var>}})</code>.
</p>
<div>
<div class="fake_header">
<span style="width: 100px;">
Age<br/>
<code>x_i</code>
</span><span style="width: 150px;">
<span class="hint_gray">Distance from the mean</span>
<code>(x_i - \color{<var>BLUE</var>}{\overline{x}})</code>
</span><span style="width: 150px;">
<code>(x_i - \color{<var>BLUE</var>}{\overline{x}})^2</code>
</span>
</div>
<div class="fake_row" data-each="DATA as i, POINT">
<span style="width: 100px;">
<code><var>POINT</var></code> <var>plural( "year", POINT )</var>
</span><span style="width: 150px;" class="hint_gray">
<code><var>roundTo( 2, POINT - MEAN )</var></code> <var>plural( "year", roundTo( 2, POINT - MEAN ) )</var>
</span><span style="width: 150px;" class="hint_purple">
<code><var>SQR_DEV[ i ]</var></code> <var>plural( "year", SQR_DEV[ i ] )</var><code>^2</code>
</span>
</div>
</div>
<div>
<p>
Normally we can find the variance <code>(\color{red}{s^2})</code> by averaging the
<span class="hint_purple">squared deviations from the mean</span>. But remember we don't know the real
population mean—we had to estimate it by using the <span class="hint_blue">sample mean</span>.
</p>
<p>
The age of any particular <var>animal( 1 )</var> in our sample is likely to be closer to the average age
of the <var>plural( DATA_POINTS, animal( 1 ) )</var> we sampled than it is to the average age
of all <var>plural( POPULATION, animal( 1 ) )</var> in the zoo.
Because of that, the <span class="hint_purple">squared deviations from the mean</span> we calculated will
probably underestimate the actual deviations from the population mean.
</p>
<p>
To compensate for this underestimation, rather than simply averaging the <span class="hint_purple">squared deviations from the mean</span>,
we total them and divide by <code class="hint_green">n - 1</code>.
</p>
<p>
<code>
\color{red}{s^2} \quad = \quad
\dfrac{\sum\limits_{i=1}^{\color{<var>GREEN</var>}{n}} (x_i - \color{<var>BLUE</var>}{\overline{x}})^2}{\color{<var>GREEN</var>}{n - 1}}
</code>
</p>
</div>
<p>
<code>
\color{red}{s^2} \quad = \quad
\dfrac{<var>plus.apply( KhanUtil, $.map( SQR_DEV, function( x ) { return "\\color{purple}{" + x + "}"; }) )</var>}
{\color{<var>GREEN</var>}{<var>DATA_POINTS</var> - 1}}
</code>
</p>
<p>
<code>
\color{red}{s^2} \quad = \quad
\dfrac{\color{purple}{<var>roundTo( 2, sum( SQR_DEV ) )</var>}}{\color{<var>GREEN</var>}{<var>DATA_POINTS - 1</var>}} \quad = \quad
\color{red}{<var>VARIANCE</var>\text{ <var>plural( "year", VARIANCE )</var>}^2}
</code>
</p>
<p><strong>
We can estimate that the average <var>animal( 1 )</var> at the zoo is <var>plural( MEAN, "year" )</var> old with a variance
of <var>plural( VARIANCE, "year" )</var><code>^2</code>.
</strong></p>
</div> <!-- hints -->
</div> <!-- sample -->
</div> <!-- problems -->
</div> <!-- exercise -->
</body>
</html>