Skip to content

HTTPS clone URL

Subversion checkout URL

You can clone with HTTPS or Subversion.

Download ZIP
Fetching contributors…

Cannot retrieve contributors at this time

328 lines (295 sloc) 18.87 kb
<!DOCTYPE html>
<html data-require="math math-format word-problems stat">
<head>
<meta charset="UTF-8" />
<title>Standard deviation</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">$.map( randGaussian( TGT_MEAN, TGT_STDDEV, DATA_POINTS ), function( lifespan ) {
lifespan = lifespan &lt; 1 ? 1 : round( lifespan );
return randRange( 1, lifespan );
} )</var>
<var id="MEAN">roundTo( 1, mean( DATA ) )</var>
<var id="SQR_DEV">$.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>
<var id="STDDEV">roundTo( 1, stdDev( DATA ) )</var>
<var id="STDDEV_POP">roundTo( 1, stdDevPop( DATA ) )</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 standard deviation?
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>
Standard deviation:<br><code>\quad</code>
<span class="sol short" data-type="decimal" data-inexact data-max-error="0.15"><var>stdDevPop( DATA )</var></span> years
</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 standard deviation</span> <code>(\color{<var>PINK</var>}{\sigma})</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>
<div>
<p>
As you might guess from the notation, the <span class="hint_pink">population standard deviation</span>
<code>(\color{<var>PINK</var>}{\sigma})</code> is found by taking the square root of the <span class="hint_red">population variance</span>
<code>(\color{red}{\sigma^2})</code>.
</p>
<p>
<code>\color{<var>PINK</var>}{\sigma} = \sqrt{\color{red}{\sigma^2}}</code>
</p>
</div>
<p>
<code>
\color{<var>PINK</var>}{\sigma} = \sqrt{\color{red}{<var>VARIANCE_POP</var>\text{ <var>plural( "year", VARIANCE_POP )</var>}^2}} =
\color{<var>PINK</var>}{<var>STDDEV_POP</var>\text{ <var>plural( "year", STDDEV_POP )</var>}}
</code>
</p>
<p><strong>
The average <var>animal( 1 )</var> at the zoo is <var>plural( MEAN, "year" )</var> old with a standard deviation
of <var>plural( STDDEV_POP, "year" )</var>.
</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 standard deviation?
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>
Standard deviation:<br><code>\quad</code>
<span class="sol short" data-type="decimal" data-inexact data-max-error="0.15"><var>stdDev( DATA )</var></span> years
</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 standard deviation by finding the <span class="hint_blue">sample mean</span>
<code>(\color{<var>BLUE</var>}{\overline{x}})</code> and
<span class="hint_pink">sample standard deviation</span> <code>(\color{<var>PINK</var>}{s})</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&mdash;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>
<div>
<p>
As you might guess from the notation, the sample standard deviation <code>(\color{<var>PINK</var>}{s})</code> is
found by taking the square root of the sample variance <code>(\color{red}{s^2})</code>.
</p>
<p>
<code>\color{<var>PINK</var>}{s} = \sqrt{\color{red}{s^2}}</code>
</p>
</div>
<p>
<code>
\color{<var>PINK</var>}{s} = \sqrt{\color{red}{<var>VARIANCE</var>\text{ <var>plural( "year", VARIANCE )</var>}^2}} =
\color{<var>PINK</var>}{<var>STDDEV</var>\text{ <var>plural( "year", STDDEV )</var>}}
</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 standard deviation
of <var>plural( STDDEV, "year" )</var>.
</strong></p>
</div> <!-- hints -->
</div> <!-- sample -->
</div> <!-- problems -->
</div> <!-- exercise -->
</body>
</html>
Jump to Line
Something went wrong with that request. Please try again.