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<h1 class="title toc-ignore">Numerical Summaries</h1>
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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>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>
</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:24%;">
<colgroup>
<col width="11%" />
<col width="12%" />
</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>
</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>
<h5 id="symbols-in-the-formula">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>
<h5 id="population-mean">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>
<h5 id="physicalinterpretation">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-4-1.png" width="672" /></p>
<h5 id="deviations">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>
<h5 id="outliers">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-5-1.png" width="672" /></p>
</div>
<hr />
</div>
</div>
<div id="median" class="section level3 tabset tabset-fade tabset-pills">
<h3>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:24%;">
<colgroup>
<col width="11%" />
<col width="12%" />
</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 %>%
group_by(Month) %>%
summarise(medTemp = median(Temp)) %>%
pander()</code></pre>
</div>
</div>
<hr />
</div>
<div id="explanation-1" class="section level4">
<h4>Explanation</h4>
<div style="padding-left:150px;">
<p>The mathematical formula used to compute the median of data depends on whether <span class="math inline">\(n\)</span>, the number of data points in the sample, is even or odd.</p>
<p>If <span class="math inline">\(n\)</span> is even, then there is no “middle” data point, so the middle two values are averaged. <span class="math display">\[
\text{Median} = \frac{x_{(n/2)}+x_{(n/2+1)}}{2}
\]</span></p>
<p>If <span class="math inline">\(n\)</span> is odd, then the middle data point is the median. <span class="math display">\[
\text{Median} = x_{((n+1)/2)}
\]</span></p>
<h5 id="symbols-in-the-formula-1">Symbols in the Formula</h5>
<p>There is no generally accepted symbol for the median. Sometimes a capital <span class="math inline">\(M\)</span> or even lower-case <span class="math inline">\(m\)</span> is used, but generally the word median is just written out.</p>
<ul>
<li><p><span class="math inline">\(x_{(n/2)}\)</span> represents the data value that is in the <span class="math inline">\((n/2)^{th}\)</span> position in the ordered list of values. It only exists when <span class="math inline">\(n\)</span> is even.</p></li>
<li><p><span class="math inline">\(x_{(n/2+1)}\)</span> represents the data value that immediately follows the <span class="math inline">\((n/2)^{th}\)</span> value in the ordered list of values. It only exists when <span class="math inline">\(n\)</span> is even.</p></li>
<li><p><span class="math inline">\(x_{((n+1)/2)}\)</span> represents the data value that is in the <span class="math inline">\(((n+1)/2)^{th}\)</span> position in the ordered list of values. It only exists when <span class="math inline">\(n\)</span> is odd.</p></li>
<li><p><span class="math inline">\(n\)</span> represents the sample size, or number of data values in the sample.</p></li>
</ul>
<h5 id="population-median">Population Median</h5>
<p>When <strong>all</strong> of the data from a population is available, the <strong>population median</strong> is calculated by the above formulas with the slight change that <span class="math inline">\(N\)</span>, the total number of data values in the population, instead of <span class="math inline">\(n\)</span>, the number of values in the sample, is used.</p>
<p>If <span class="math inline">\(N\)</span> is even, then there is no “middle” data point, so the middle two values are averaged. <span class="math display">\[
\text{Median} = \frac{x_{(N/2)}+x_{(N/2+1)}}{2}
\]</span></p>
<p>If <span class="math inline">\(N\)</span> is odd, then the middle data point is the median. <span class="math display">\[
\text{Median} = x_{((N+1)/2)}
\]</span></p>
<h5 id="physical-interpretation">Physical Interpretation</h5>
<p>The median is the <span class="math inline">\(50^{th}\)</span> percentile of the data.</p>
<p>Say there are <span class="math inline">\(n=5\)</span> data points in the sample 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 median is calculated as follows. Note that <span class="math inline">\(n=5\)</span> is odd. <span class="math display">\[
\text{Median} = x_{((n+1)/2)} = x_{((5+1)/2)} = x_{(3)} = 6
\]</span> When these values are plotted it is clear that exactly 50% of the data (excluding the median) is to either side of the median.</p>
<p><img src="NumericalSummaries_files/figure-html/unnamed-chunk-8-1.png" width="672" /></p>
<h6 id="second-example">Second Example</h6>
<p>Say there was a sixth value in the data set equal to 10, so that <span class="math inline">\(n=6\)</span> is even.</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>
<li><span class="math inline">\(x_6 = 10\)</span></li>
</ul>
<p><span class="math display">\[
\text{Median} = \frac{x_{(n/2)}+x_{(n/2+1)}}{2} = \frac{x_{(6/2)}+x_{(6/2+1)}}{2} = \frac{x_{(3)}+x_{(4)}}{2} = \frac{6+7}{2} = 6.5
\]</span></p>
<p><img src="NumericalSummaries_files/figure-html/unnamed-chunk-9-1.png" width="672" /></p>
<h5 id="effect-of-outliers">Effect of Outliers</h5>
<p>The median is not greatly influenced by <em>outliers</em>. It is said to be <em>robust</em>. This is shown below by changing <span class="math inline">\(x_6\)</span> to be 20, which does not change the value of the median.</p>
<p><img src="NumericalSummaries_files/figure-html/unnamed-chunk-10-1.png" width="672" /></p>
</div>
<hr />
</div>
</div>
<div id="mode" class="section level3 tabset tabset-fade tabset-pills">
<h3>Mode</h3>
<div style="float:left;width:150px;">
<center>
<p><strong>Most</strong></p>
<p><strong>Frequent</strong></p>
<strong>Value</strong>
</center>
</div>
<p><strong>Measure of Center | 1 Quantitative or Qualitative Variable</strong></p>
<div id="overview-2" class="section level4">
<h4>Overview</h4>
<div style="padding-left:150px;">
<p>The most commonly occurring value. There may be more than one mode. Seldom used, but sometimes useful.</p>
</div>
<hr />
</div>
<div id="r-instructions-2" class="section level4">
<h4>R Instructions</h4>
<div style="padding-left:150px;">
<p>R will not calculate a mode directly. However, to tabulate the number of times each value occurs in a dataset, use the code:</p>
<p><code>table(object)</code></p>
<ul>
<li><code>object</code> can be quantitative or qualitative, but should contain at least one repeated value or <code>table()</code> is not useful.</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('table1')">
<div class="hoverchunk">
<p><span class="tooltipr"> table <span class="tooltiprtext">“table” is an R function used to count how many times each observation occurs in a list 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"> Month <span class="tooltiprtext">“Month” is a qualitative variable (technically a numeric vector) from the “airquality” dataset that contains repeated values.</span> </span><span class="tooltipr"> )<br />
<span class="tooltiprtext">Closing parenthsis for the 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="table1" style="display:none;">
<pre><code>##
## 5 6 7 8 9
## 31 30 31 31 30</code></pre>
<p>Note that the modes would be 5, 7, and 8 because these months all have the most (31) days in them.</p>
</div>
<a href="javascript:showhide('mode2')">
<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 = mean(Temp), <span class="tooltiprtext">Computes the mean of the Temp column.</span> </span><span class="tooltipr"> medTemp = median(Temp), <span class="tooltiprtext">Computes the median of the Temp column.</span> </span><span class="tooltipr"> sampleSize = n( ) <span class="tooltiprtext">Counts how many times each Month (the group_by statement) occurs in the dataset.</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="mode2" style="display:none;">
<table style="width:54%;">
<colgroup>
<col width="11%" />
<col width="13%" />
<col width="13%" />
<col width="15%" />
</colgroup>
<thead>
<tr class="header">
<th align="center">Month</th>
<th align="center">aveTemp</th>
<th align="center">medTemp</th>
<th align="center">sampeSize</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center">5</td>
<td align="center">65.55</td>
<td align="center">66</td>
<td align="center">31</td>
</tr>
<tr class="even">
<td align="center">6</td>
<td align="center">79.1</td>
<td align="center">78</td>
<td align="center">30</td>
</tr>
<tr class="odd">
<td align="center">7</td>
<td align="center">83.9</td>
<td align="center">84</td>
<td align="center">31</td>
</tr>
<tr class="even">
<td align="center">8</td>
<td align="center">83.97</td>
<td align="center">82</td>
<td align="center">31</td>
</tr>
<tr class="odd">
<td align="center">9</td>
<td align="center">76.9</td>
<td align="center">76</td>
<td align="center">30</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 %>%
group_by(Month) %>%
summarise(aveTemp = mean(Temp), medTemp = median(Temp), sampeSize = n()) %>%
pander()</code></pre>
</div>
</div>
<hr />
</div>
</div>
<div id="minimum" class="section level3 tabset tabset-fade tabset-pills">
<h3>Minimum</h3>
<div style="float:left;width:150px;">
<p><span class="math display">\[
x_{(1)}
\]</span></p>
</div>
<p><strong>Measure of Spread | 1 Quantitative Variable</strong></p>
<div id="overview-3" class="section level4">
<h4>Overview</h4>
<div style="padding-left:150px;">
<p>The smallest occurring data value. One of the numerical summaries in the five-number summary. Typically not useful on its own. Gives a good feel for the spread in the left tail of the distribution when used with the five-number summary.</p>
</div>
<hr />
</div>
<div id="r-instructions-3" class="section level4">
<h4>R Instructions</h4>
<div style="padding-left:150px;">
<p>To calculate a minimum in R use the code:</p>
<p><code>min(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>
<p>Hover your mouse over the example codes to learn more.</p>
<a href="javascript:showhide('min1')">
<div class="hoverchunk">
<p><span class="tooltipr"> min <span class="tooltiprtext">“min” is an R function used to calculate the minimum 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 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="min1" style="display:none;">
<pre><code>## [1] 56</code></pre>
<p>Note that the single number showing above is the minimum <code>Temp</code> from the <code>airquality</code> dataset.</p>
</div>
<a href="javascript:showhide('min2')">
<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"> minTemp = <span class="tooltiprtext">“minTemp” is just a name we made up. It will contain the results of the median(…) function.</span> </span><span class="tooltipr"> min( <span class="tooltiprtext">“min” is an R function used to calculate the minimum.</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="min2" style="display:none;">
<table style="width:24%;">
<colgroup>
<col width="11%" />
<col width="12%" />
</colgroup>
<thead>
<tr class="header">
<th align="center">Month</th>
<th align="center">minTemp</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center">5</td>
<td align="center">56</td>
</tr>
<tr class="even">
<td align="center">6</td>
<td align="center">65</td>
</tr>
<tr class="odd">
<td align="center">7</td>
<td align="center">73</td>
</tr>
<tr class="even">
<td align="center">8</td>
<td align="center">72</td>
</tr>
<tr class="odd">
<td align="center">9</td>
<td align="center">63</td>
</tr>
</tbody>
</table>
<p>Note that R calculated the minimum <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 %>%
group_by(Month) %>%
summarise(minTemp = min(Temp)) %>%
pander()</code></pre>
</div>
</div>
<hr />
</div>
</div>
<div id="maximum" class="section level3 tabset tabset-fade tabset-pills">
<h3>Maximum</h3>
<div style="float:left;width:150px;">
<p><span class="math display">\[
x_{(n)}
\]</span></p>
</div>
<p><strong>Measure of Spread | 1 Quantitative Variable</strong></p>
<div id="overview-4" class="section level4">
<h4>Overview</h4>
<div style="padding-left:150px;">
<p>The largest occurring data value. One of the numerical summaries in the five-number summary. Typically not useful on its own. Gives a good feel for the spread in the right tail of the distribution when used in the five-number summary.</p>
</div>
<hr />
</div>
<div id="r-instructions-4" class="section level4">
<h4>R Instructions</h4>
<div style="padding-left:150px;">
<p>To calculate a maximum in R use the code:</p>
<p><code>max(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>
<p>Hover your mouse over the example codes to learn more.</p>
<a href="javascript:showhide('max1')">
<div class="hoverchunk">
<p><span class="tooltipr"> max <span class="tooltiprtext">“max” is an R function used to calculate the maximum 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 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="max1" style="display:none;">
<pre><code>## [1] 97</code></pre>
<p>Note that the single number showing above is the maximum <code>Temp</code> from the <code>airquality</code> dataset.</p>
</div>
<a href="javascript:showhide('max2')">
<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"> maxTemp = <span class="tooltiprtext">“maxTemp” is just a name we made up. It will contain the results of the median(…) function.</span> </span><span class="tooltipr"> max( <span class="tooltiprtext">“max” is an R function used to calculate the maximum.</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="max2" style="display:none;">
<table style="width:24%;">
<colgroup>
<col width="11%" />
<col width="12%" />
</colgroup>
<thead>
<tr class="header">
<th align="center">Month</th>
<th align="center">maxTemp</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center">5</td>
<td align="center">81</td>
</tr>
<tr class="even">
<td align="center">6</td>
<td align="center">93</td>
</tr>
<tr class="odd">
<td align="center">7</td>
<td align="center">92</td>
</tr>
<tr class="even">
<td align="center">8</td>
<td align="center">97</td>
</tr>
<tr class="odd">
<td align="center">9</td>
<td align="center">93</td>
</tr>
</tbody>
</table>
<p>Note that R calculated the maximum <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 %>%
group_by(Month) %>%
summarise(maxTemp = max(Temp)) %>%
pander()</code></pre>
</div>
</div>
<hr />
</div>
</div>
<div id="quartiles-five-number-summary" class="section level3 tabset tabset-fade tabset-pills">
<h3>Quartiles (five-number summary)</h3>
<div style="float:left;width:150px;">
<center>
<p><strong>25</strong><span class="math inline">\(^{th}\)</span>, <strong>50</strong><span class="math inline">\(^{th}\)</span>, <strong>75</strong><span class="math inline">\(^{th}\)</span></p>
<p><em>and</em> <strong>100</strong><span class="math inline">\(^{th}\)</span></p>
<strong><em>Percentiles</em></strong>
</center>
</div>
<p><strong>Measure of Center & Spread | 1 Quantitative Variable</strong></p>
<div id="overview-5" class="section level4">
<h4>Overview</h4>
<div style="padding-left:150px;">
<p>Good for describing the spread of data, typically for skewed distributions. There are four quartiles. They make up the <strong><em>five-number summary</em></strong> when combined with the minimum. The second quartile is the median (50<span class="math inline">\(^{th}\)</span> percentile) and the fourth quartile is the maximum (100<span class="math inline">\(^{th}\)</span> percentile). The first quartile (<span class="math inline">\(Q_1\)</span> or lower quartile) and third quartile (<span class="math inline">\(Q_3\)</span> or upper quartile) show the spread of the “middle 50%” of the data, which is often called the <strong><em>interquartile range</em></strong>. Comparing the interquartile range to the minimum and maximum shows how the possible values spread out around the more probable values.</p>
</div>
<hr />
</div>
<div id="r-instructions-5" class="section level4">
<h4>R Instructions</h4>
<div style="padding-left:150px;">
<p>To calculate a five-number summary (and mean) in R use the code:</p>
<p><code>quantile(object, percentile)</code></p>
<ul>
<li><code>object</code> must be a quantitative variable, what R calls a “numeric vector.”</li>
<li><code>percentile</code> must be a value between 0 and 1. For the first quartile, it would be 0.25. For the third, it would be 0.75.</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('summary1')">
<div class="hoverchunk">
<p><span class="tooltipr"> summary <span class="tooltiprtext">“summary” is an R function used to calculate the five-number summary (and 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 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="summary1" style="display:none;">
<table style="width:69%;">
<colgroup>
<col width="9%" />
<col width="13%" />
<col width="12%" />
<col width="11%" />
<col width="13%" />
<col width="8%" />
</colgroup>
<thead>
<tr class="header">
<th align="center">Min.</th>
<th align="center">1st Qu.</th>
<th align="center">Median</th>
<th align="center">Mean</th>
<th align="center">3rd Qu.</th>
<th align="center">Max.</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center">56</td>
<td align="center">72</td>
<td align="center">79</td>
<td align="center">77.88</td>
<td align="center">85</td>
<td align="center">97</td>
</tr>
</tbody>
</table>
<p>Showing above are the five-number summary and mean of <code>Temp</code> from the <code>airquality</code> dataset.</p>
<p>Note, you must use <code>pander(summary(airquality$Temp))</code> to get the nicely formatted output.</p>
</div>
<a href="javascript:showhide('summary2')">
<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"> min = min(Temp), <span class="tooltiprtext">Computes the min of the Temp column.</span> </span><span class="tooltipr"> Q1 = quantile(Temp, 0.25), <span class="tooltiprtext">Computes the first quartile of the Temp column.</span> </span><span class="tooltipr"> med = median(Temp), <span class="tooltiprtext">Computes the second quartile of the Temp column, known as the median.</span> </span><span class="tooltipr"> Q3 = quantile(Temp, 0.75), <span class="tooltiprtext">Computes the third quartile of the Temp column.</span> </span><span class="tooltipr"> max = max(Temp) <span class="tooltiprtext">Computes the max of the Temp column.</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="summary2" style="display:none;">
<table style="width:56%;">
<colgroup>
<col width="11%" />
<col width="8%" />
<col width="9%" />
<col width="8%" />
<col width="11%" />
<col width="6%" />
</colgroup>
<thead>
<tr class="header">
<th align="center">Month</th>
<th align="center">min</th>
<th align="center">Q1</th>
<th align="center">med</th>
<th align="center">Q3</th>
<th align="center">max</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center">5</td>
<td align="center">56</td>
<td align="center">60</td>
<td align="center">66</td>
<td align="center">69</td>
<td align="center">81</td>
</tr>
<tr class="even">
<td align="center">6</td>
<td align="center">65</td>
<td align="center">76</td>
<td align="center">78</td>
<td align="center">82.75</td>
<td align="center">93</td>
</tr>
<tr class="odd">
<td align="center">7</td>
<td align="center">73</td>
<td align="center">81.5</td>
<td align="center">84</td>
<td align="center">86</td>
<td align="center">92</td>
</tr>
<tr class="even">
<td align="center">8</td>
<td align="center">72</td>
<td align="center">79</td>
<td align="center">82</td>
<td align="center">88.5</td>
<td align="center">97</td>
</tr>
<tr class="odd">
<td align="center">9</td>
<td align="center">63</td>
<td align="center">71</td>
<td align="center">76</td>
<td align="center">81</td>
<td align="center">93</td>
</tr>
</tbody>
</table>
<p>Note that R calculated the five-number summary for <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 %>%
group_by(Month) %>%
summarise(min = min(Temp),
Q1 = quantile(Temp, c(.25)),
med = median(Temp),
Q3 = quantile(Temp, c(.75)),
max = max(Temp)) %>%
pander()</code></pre>
</div>
</div>
<hr />
</div>
</div>
<div id="standard-deviation" class="section level3 tabset tabset-fade tabset-pills">
<h3>Standard Deviation</h3>
<div style="float:left;width:150px;">
<p><span class="math inline">\(s = \sqrt{\frac{\sum_{i=1}^n(x_i-\bar{x})^2}{n-1}}\)</span></p>
</div>
<p><strong>Measure of Spread | 1 Quantitative Variable</strong></p>
<div id="overview-6" class="section level4">
<h4>Overview</h4>