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<div id="header">
<h1 class="title toc-ignore">Numerical Summaries</h1>
</div>
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<p>There are many ways to numerically summarize data. The fundamental
idea is to describe the center, or most <em>probable</em> values of the
data, as well as the spread, or the <em>possible</em> values of the
data.</p>
<hr />
<div id="mean" class="section level3 tabset tabset-fade tabset-pills">
<h3 class="tabset tabset-fade tabset-pills">Mean</h3>
<div style="float:left;width:150px;">
<p><span class="math display">\[
\bar{x} = \frac{\sum_{i=1}^n x_i}{n}
\]</span></p>
</div>
<p><strong>Measure of Center | 1 Quantitative Variable</strong></p>
<div id="overview" class="section level4">
<h4>Overview</h4>
<div style="padding-left:150px;">
<p>The “balance point” or “center of mass” of quantitative data. It is
calculated by taking the numerical sum of the values divided by the
number of values. Typically used in tandem with the standard deviation.
Most appropriate for describing the most typical values for relatively
normally distributed data. Influenced by outliers, so it is not
appropriate for describing strongly skewed data.</p>
</div>
<hr />
</div>
<div id="r-instructions" class="section level4">
<h4>R Instructions</h4>
<div style="padding-left:150px;">
<p>To calculate a mean in R use the code:</p>
<p><code>mean(object)</code></p>
<ul>
<li><code>object</code> must be a quantitative variable, what R calls a
“numeric vector.” Usually this is a column from a data set.</li>
<li>Use <code>na.rm=TRUE</code> if there are missing values in
<code>object</code> so that the code reads
<code>mean(object, na.rm=TRUE)</code>.</li>
</ul>
<p><br></p>
<p><strong>Example Code</strong></p>
<p>Hover your mouse over the example codes to learn more.</p>
<a href="javascript:showhide('mean1')">
<div class="hoverchunk">
<p><span class="tooltipr"> mean <span class="tooltiprtext">“mean” is an
R function used to calculate the mean of data.</span> </span><span
class="tooltipr"> ( <span class="tooltiprtext">Parenthesis to begin the
function. Must touch the last letter of the function.</span>
</span><span class="tooltipr"> airquality <span
class="tooltiprtext">“airquality” is a dataset. Type “View(airquality)”
in R to see it.</span> </span><span class="tooltipr"> $ <span
class="tooltiprtext">The $ allows us to access any variable from the
airquality dataset.</span> </span><span class="tooltipr"> Temp <span
class="tooltiprtext">“Temp” is a quantitative variable (numeric vector)
from the “airquality” dataset.</span> </span><span class="tooltipr">
)<br />
<span class="tooltiprtext">Closing parenthsis for the mean
function.</span> </span><span class="tooltipr"> <br />
<span class="tooltiprtext">Press Enter to run the code if you have typed
it in yourself. You can also click here to view the output.</span>
</span><span class="tooltipr" style="float:right;"> … <span
class="tooltiprtext">Click to View Output.</span> </span></p>
</div>
</a>
<div id="mean1" style="display:none;">
<pre><code>## [1] 77.88235</code></pre>
<p>Note that the single number showing above is the average
<code>Temp</code> from the <code>airquality</code> dataset.</p>
</div>
<a href="javascript:showhide('mean3')">
<div class="hoverchunk">
<p><span class="tooltipr"> library(tidyverse) <span
class="tooltiprtext">tidyverse is an R Package that is very useful for
working with data.</span> </span><br><span class="tooltipr"> airquality
<span class="tooltipRtext"><code>airquality</code> is a dataset in
R.</span> </span><span class="tooltipr"> %>% <span
class="tooltipRtext">The pipe operator that will send the
<code>airquality</code> dataset down inside of the code on the following
line.</span> </span><br/><span class="tooltipr"> group_by( <span
class="tooltipRtext">“group_by” is a function from library(tidyverse)
that allows us to split the airquality dataset into “little” datasets,
one dataset for each value in the “Month” column.</span> </span><span
class="tooltipr"> Month <span class="tooltiprtext">“Month” is a column
from the airquality dataset that can be treated as qualitative.</span>
</span><span class="tooltipr"> ) <span class="tooltiprtext">Functions
must always end with a closing parenthesis.</span> </span><span
class="tooltipr"> %>% <span class="tooltipRtext">The pipe operator
that will send the grouped version of the <code>airquality</code>
dataset down inside of the code on the following line.</span>
</span><br/><span class="tooltipr"> summarise( <span
class="tooltipRtext">“summarise” is a function from library(tidyverse)
that allows us to compute numerical summaries on data.</span>
</span><span class="tooltipr"> aveTemp = <span
class="tooltiprtext">“AveTemp” is just a name we made up. It will
contain the results of the mean(…) function.</span> </span><span
class="tooltipr"> mean( <span class="tooltiprtext">“mean” is an R
function used to calculate the mean.</span> </span><span
class="tooltipr"> Temp <span class="tooltiprtext">Temp is a quantitative
variable (numeric vector) from the airquality dataset.</span>
</span><span class="tooltipr"> ) <span class="tooltiprtext">Functions
must always end with a closing parenthesis.</span> </span><span
class="tooltipr"> ) <span class="tooltiprtext">Functions must always end
with a closing parenthesis.</span> </span><span class="tooltipr">
<br />
<span class="tooltiprtext">Press Enter to run the code.</span>
</span><span class="tooltipr" style="float:right;"> … <span
class="tooltiprtext">Click to View Output.</span> </span></p>
</div>
</a>
<div id="mean3" style="display:none;">
<table style="width:25%;">
<colgroup>
<col width="11%" />
<col width="13%" />
</colgroup>
<thead>
<tr class="header">
<th align="center">Month</th>
<th align="center">aveTemp</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center">5</td>
<td align="center">65.55</td>
</tr>
<tr class="even">
<td align="center">6</td>
<td align="center">79.1</td>
</tr>
<tr class="odd">
<td align="center">7</td>
<td align="center">83.9</td>
</tr>
<tr class="even">
<td align="center">8</td>
<td align="center">83.97</td>
</tr>
<tr class="odd">
<td align="center">9</td>
<td align="center">76.9</td>
</tr>
</tbody>
</table>
<p>Note that R calculated the mean <code>Temp</code> for each month in
<code>Month</code> from the <code>airquality</code> dataset.</p>
<p>May (5), June (6), July (7), August (8), and September (9),
respectively.</p>
<p>Further, note that to get the “nicely formatted” table, you would
have to use</p>
<pre><code>library(pander)
airquality %>%
group_by(Month) %>%
summarise(aveTemp = mean(Temp)) %>%
pander()</code></pre>
</div>
<p><br/></p>
<p>A note about missing values in data…</p>
<a href="javascript:showhide('mean1b')">
<div class="hoverchunk">
<p><span class="tooltipr"> mean <span class="tooltiprtext">“mean” is an
R function used to calculate the mean of data.</span> </span><span
class="tooltipr"> ( <span class="tooltiprtext">Parenthesis to begin the
function. Must touch the last letter of the function.</span>
</span><span class="tooltipr"> airquality <span
class="tooltiprtext">“airquality” is a dataset. Type “View(airquality)”
in R to see it.</span> </span><span class="tooltipr"> $ <span
class="tooltiprtext">The $ allows us to access any variable from the
airquality dataset.</span> </span><span class="tooltipr"> Ozone <span
class="tooltiprtext">“Ozone” is a quantitative variable (numeric vector)
from the “airquality” dataset.</span> </span><span class="tooltipr"> ,
<span class="tooltiprtext">The comma allows us to specify optional
commands.</span> </span><span class="tooltipr"> na.rm=TRUE <span
class="tooltiprtext">Missing values are called “NA” in R. If data
contains missing values, <code>mean(...)</code> will give “NA” as the
result unless we “remove” (rm) the “NA” (na) values.</span> </span><span
class="tooltipr"> )<br />
<span class="tooltiprtext">Closing parenthsis for the mean
function.</span> </span><span class="tooltipr"> <br />
<span class="tooltiprtext">Press Enter to run the code if you have typed
it in yourself. You can also click here to view the output.</span>
</span><span class="tooltipr" style="float:right;"> … <span
class="tooltiprtext">Click to View Output.</span> </span></p>
</div>
</a>
<div id="mean1b" style="display:none;">
<pre><code>## [1] 42.12931</code></pre>
<p>Note that the single number showing above is the average
<code>Ozone</code> from the <code>airquality</code> dataset. Because the
<code>Ozone</code> column had missing values, we had to use the option
<code>na.rm=TRUE</code> to get the mean to calculate. If we had left it
off, we would have gotten an “NA” result:</p>
<div class="sourceCode" id="cb4"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="fu">mean</span>(airquality<span class="sc">$</span>Ozone)</span></code></pre></div>
<pre><code>## [1] NA</code></pre>
</div>
</div>
<hr />
</div>
<div id="explanation" class="section level4">
<h4>Explanation</h4>
<div style="padding-left:150px;">
<p>The mathematical formula used to compute the mean of data is given by
the formula to the left. Although the formula looks complicated, all it
states is “add all the data values up and divide by the total number of
values.” Read on to learn what all the symbols in the formula
represent.</p>
<div id="symbols-in-the-formula" class="section level5">
<h5>Symbols in the Formula</h5>
<ul>
<li><p><span class="math inline">\(\bar{x}\)</span> is read “x-bar” and
is the symbol typically used for the <strong>sample mean</strong>, the
mean computed on a <em>sample</em> of data from a population.</p></li>
<li><p><span class="math inline">\(\Sigma\)</span>, the capital Greek
letter “sigma,” is the symbol used to imply “add all of the data values
up.”</p></li>
<li><p>The <span class="math inline">\(x_i\)</span>’s are the data
values. The <span class="math inline">\(i\)</span> in the <span
class="math inline">\(x_i\)</span> is stated to go from <span
class="math inline">\(i=1\)</span> all the way up to <span
class="math inline">\(n\)</span>. In other words, data value 1 is
represented by <span class="math inline">\(x_1\)</span>, data value 2:
<span class="math inline">\(x_2\)</span>, <span
class="math inline">\(\ldots\)</span>, up through the last data value
<span class="math inline">\(x_n\)</span>. In general, we just write
<span class="math inline">\(x_i\)</span>.</p></li>
<li><p><span class="math inline">\(n\)</span> represents the sample
size, or number of data values.</p></li>
</ul>
</div>
<div id="population-mean" class="section level5">
<h5>Population Mean</h5>
<p>When <strong>all</strong> of the data from a population is available,
the <strong>population mean</strong> is calculated instead of the sample
mean. The mathematical formula for the <strong>population mean</strong>
is the same as the formula for the sample mean, but is written with
slightly different notation. <span class="math display">\[
\mu = \frac{\sum_{i=1}^N x_i}{N}
\]</span> Notice that the symbol for the population mean is <span
class="math inline">\(\mu\)</span>, pronounced “mew,” another Greek
letter. (Review your <a
href="https://en.wikipedia.org/wiki/Greek_alphabet#Letters">Greek
alphabet</a>.) The only other difference between the two formulas is
that the sample mean uses a sample of data, denoted by <span
class="math inline">\(n\)</span>, while the population mean uses all the
population data, denoted by <span class="math inline">\(N\)</span>.</p>
</div>
<div id="physicalinterpretation" class="section level5">
<h5>Physical Interpretation</h5>
<p>The mean is sometimes described as the “balance point” of the data.
The following example will demonstrate.</p>
<p>Say there are <span class="math inline">\(n=5\)</span> data points
with the following values.</p>
<ul>
<li><span class="math inline">\(x_1 = 2\)</span></li>
<li><span class="math inline">\(x_2 = 5\)</span></li>
<li><span class="math inline">\(x_3 = 6\)</span></li>
<li><span class="math inline">\(x_4 = 7\)</span></li>
<li><span class="math inline">\(x_5 = 10\)</span></li>
</ul>
<p>The sample mean is calculated as follows. <span
class="math display">\[
\bar{x} = \frac{\sum_{i=1}^n x_i}{n} = \frac{2 + 5 + 6 + 7 + 10}{5} =
6
\]</span> If these values were plotted, and an “infinitely thin bar”
connected the points, then the bar would “balance” at the mean (the
triangle) as shown below.</p>
<p><img src="NumericalSummaries_files/figure-html/unnamed-chunk-6-1.png" width="672" /></p>
</div>
<div id="deviations" class="section level5">
<h5>Middle of the Deviations</h5>
<p>The above plot demonstrates that there are equal, but opposite, “sums
of deviations” to either side of the mean. Note that a deviation is
defined as the distance from the mean to a given point. Thus, <span
class="math inline">\(x_1\)</span> has a deviation of -4 from the mean,
<span class="math inline">\(x_2\)</span> a deviation of -1, <span
class="math inline">\(x_3\)</span> a deviation of 0, <span
class="math inline">\(x_4\)</span> a deviation of 1, and <span
class="math inline">\(x_5\)</span> a deviation of 4. To the left there
is a sum of deviations equal to -5 and on the right, a sum of deviations
equal to 5. This can be verified to hold for any scenario.</p>
</div>
<div id="outliers" class="section level5">
<h5>Effect of Outliers</h5>
<p>The mean can be strongly influenced by <em>outliers</em>, points that
deviate abnormally from the mean. This is shown below by changing <span
class="math inline">\(x_5\)</span> to be 20. Note that the deviation of
<span class="math inline">\(x_5\)</span> is 12, and the sum of
deviations to the left of the mean (<span
class="math inline">\(\bar{x}=8\)</span>) is <span
class="math inline">\(-1 + -2 + -3 + -6 = -12\)</span>.</p>
<p>The mean of the altered data</p>
<ul>
<li><span class="math inline">\(x_1 = 2\)</span></li>
<li><span class="math inline">\(x_2 = 5\)</span></li>
<li><span class="math inline">\(x_3 = 6\)</span></li>
<li><span class="math inline">\(x_4 = 7\)</span></li>
<li><span class="math inline">\(x_5 = 20\)</span></li>
</ul>
<p>is now <span class="math inline">\(\bar{x} = \frac{\sum_{i=1}^n
x_i}{n} = \frac{2 + 5 + 6 + 7 + 20}{5} = 8\)</span>.</p>
<p><img src="NumericalSummaries_files/figure-html/unnamed-chunk-7-1.png" width="672" /></p>
</div>
</div>
<hr />
</div>
</div>
<div id="median" class="section level3 tabset tabset-fade tabset-pills">
<h3 class="tabset tabset-fade tabset-pills">Median</h3>
<div style="float:left;width:150px;">
<span class="math display">\[
\frac{x_{(n/2)}+x_{(n/2+1)}}{2}
\]</span>
<center>
<span class="math inline">\(\uparrow\)</span> <strong>even</strong>
<span class="math inline">\(n\)</span> <strong>odd</strong> <span
class="math inline">\(\downarrow\)</span>
</center>
<p><span class="math display">\[
x_{((n+1)/2)}
\]</span></p>
</div>
<p><strong>Measure of Center | 1 Quantitative Variable</strong></p>
<div id="overview-1" class="section level4">
<h4>Overview</h4>
<div style="padding-left:150px;">
<p>The “middle data point,” i.e., the 50<span
class="math inline">\(^{th}\)</span> percentile. Half of the data is
below the median and half is above the median. Typically used in tandem
with the five-number summary to describe skewed data because it is not
heavily influenced by outliers, i.e., it is <em>robust</em>. Can also be
used with normally distributed data, but the mean and standard deviation
are more useful measures in such cases.</p>
</div>
<hr />
</div>
<div id="r-instructions-1" class="section level4">
<h4>R Instructions</h4>
<div style="padding-left:150px;">
<p>To calculate a median in R use the code:</p>
<p><code>median(object)</code></p>
<ul>
<li><code>object</code> must be a quantitative variable, what R calls a
“numeric vector.”</li>
</ul>
<p><br></p>
<p><strong>Example Code</strong></p>
<a href="javascript:showhide('median1')">
<div class="hoverchunk">
<p><span class="tooltipr"> median <span class="tooltiprtext">“median” is
an R function used to calculate the median of data.</span> </span><span
class="tooltipr"> ( <span class="tooltiprtext">Parenthesis to begin the
function. Must touch the last letter of the function.</span>
</span><span class="tooltipr"> airquality <span
class="tooltiprtext">“airquality” is a dataset. Type “View(airquality)”
in R to see it.</span> </span><span class="tooltipr"> $ <span
class="tooltiprtext">The $ allows us to access any variable from the
airquality dataset.</span> </span><span class="tooltipr"> Temp <span
class="tooltiprtext">“Temp” is a quantitative variable (numeric vector)
from the “airquality” dataset.</span> </span><span class="tooltipr">
)<br />
<span class="tooltiprtext">Closing parenthsis for the median
function.</span> </span><span class="tooltipr"> <br />
<span class="tooltiprtext">Press Enter to run the code if you have typed
it in yourself. You can also click here to view the output.</span>
</span><span class="tooltipr" style="float:right;"> … <span
class="tooltiprtext">Click to View Output.</span> </span></p>
</div>
</a>
<div id="median1" style="display:none;">
<pre><code>## [1] 79</code></pre>
<p>Note that the single number showing above is the median
<code>Temp</code> from the <code>airquality</code> dataset.</p>
</div>
<a href="javascript:showhide('median2')">
<div class="hoverchunk">
<p><span class="tooltipr"> library(tidyverse) <span
class="tooltiprtext">tidyverse is an R Package that is very useful for
working with data.</span> </span><br><span class="tooltipr"> airquality
<span class="tooltipRtext"><code>airquality</code> is a dataset in
R.</span> </span><span class="tooltipr"> %>% <span
class="tooltipRtext">The pipe operator that will send the
<code>airquality</code> dataset down inside of the code on the following
line.</span> </span><br/><span class="tooltipr"> group_by( <span
class="tooltipRtext">“group_by” is a function from library(tidyverse)
that allows us to split the airquality dataset into “little” datasets,
one dataset for each value in the “Month” column.</span> </span><span
class="tooltipr"> Month <span class="tooltiprtext">“Month” is a column
from the airquality dataset that can be treated as qualitative.</span>
</span><span class="tooltipr"> ) <span class="tooltiprtext">Functions
must always end with a closing parenthesis.</span> </span><span
class="tooltipr"> %>% <span class="tooltipRtext">The pipe operator
that will send the grouped version of the <code>airquality</code>
dataset down inside of the code on the following line.</span>
</span><br/><span class="tooltipr"> summarise( <span
class="tooltipRtext">“summarise” is a function from library(tidyverse)
that allows us to compute numerical summaries on data.</span>
</span><span class="tooltipr"> medTemp = <span
class="tooltiprtext">“medTemp” is just a name we made up. It will
contain the results of the median(…) function.</span> </span><span
class="tooltipr"> median( <span class="tooltiprtext">“median” is an R
function used to calculate the median.</span> </span><span
class="tooltipr"> Temp <span class="tooltiprtext">Temp is a quantitative
variable (numeric vector) from the airquality dataset.</span>
</span><span class="tooltipr"> ) <span class="tooltiprtext">Functions
must always end with a closing parenthesis.</span> </span><span
class="tooltipr"> ) <span class="tooltiprtext">Functions must always end
with a closing parenthesis.</span> </span><span class="tooltipr">
<br />
<span class="tooltiprtext">Press Enter to run the code.</span>
</span><span class="tooltipr" style="float:right;"> … <span
class="tooltiprtext">Click to View Output.</span> </span></p>
</div>
</a>
<div id="median2" style="display:none;">
<table style="width:25%;">
<colgroup>
<col width="11%" />
<col width="13%" />
</colgroup>
<thead>
<tr class="header">
<th align="center">Month</th>
<th align="center">medTemp</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center">5</td>
<td align="center">66</td>
</tr>
<tr class="even">
<td align="center">6</td>
<td align="center">78</td>
</tr>
<tr class="odd">
<td align="center">7</td>
<td align="center">84</td>
</tr>
<tr class="even">
<td align="center">8</td>
<td align="center">82</td>
</tr>
<tr class="odd">
<td align="center">9</td>
<td align="center">76</td>
</tr>
</tbody>
</table>
<p>Note that R calculated the median <code>Temp</code> for each month in
<code>Month</code> from the <code>airquality</code> dataset.</p>
<p>May (5), June (6), July (7), August (8), and September (9),
respectively.</p>
<p>Further, to get the nicely formatted table you must use:</p>
<pre><code>library(pander)
airquality %>%