<p style="text-align:justify"><span style="font-size:14px">Recall that for grouped data we do not know individual data values, so we cannot describe the typical value of the data with precision. In other words, we cannot find the exact mean, median, or mode. We can, however, determine the best estimate of the measures of center by finding the mean of the grouped data with the formula&nbsp; Mean &nbsp;of &nbsp;Frequency&nbsp;<span class="math-tex">\(Table=\frac{\sum{fm}}{\sum{f}}\)</span><span style="background-color:white"><span style="color:black">, where&nbsp;f=&nbsp;interval frequencies and&nbsp;<em>m</em>&nbsp;= interval midpoints.</span></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:black">Just as we could not find the exact mean, neither can we find the exact standard deviation. Remember that standard deviation describes numerically the expected deviation a data value has from the mean. In simple English, the standard deviation allows us to compare how&nbsp;<em>unusual</em>&nbsp;individual data are when compared to the mean.</span></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>EXAMPLE&nbsp;2.35</strong></span></span></p>

<p>&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Find the standard deviation for the data in&nbsp;</span><a href="https://openstax.org/books/statistics/pages/2-7-measures-of-the-spread-of-the-data#fs-idm67330768" style="color:blue; text-decoration:underline"><span style="color:#027eb5">Table 2.34</span></a><span style="color:#424242">.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<table cellspacing="0" class="Table" style="border-collapse:collapse; border:none; width:900px">
	<thead>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; height:49px; vertical-align:bottom; width:72px">
			<div style="text-align:center"><span style="font-size:28px"><strong>Class</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:49px; vertical-align:bottom; width:123px">
			<div style="text-align:center"><span style="font-size:28px"><strong>Frequency,&nbsp;<em>f</em></strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:49px; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Midpoint,&nbsp;<em>m</em></strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:49px; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><em><strong>m</strong></em><strong><sup>2</sup></strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:49px; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><span class="math-tex">\(\bar{x}^2\)</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:49px; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><em><strong>fm</strong></em><strong><sup>2</sup></strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:49px; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Standard Deviation</strong></span></div>
			</td>
		</tr>
	</thead>
	<tbody>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:30px; width:72px">
			<div style="text-align:center"><span style="font-size:28px">0&ndash;2</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px; width:123px">
			<div style="text-align:center"><span style="font-size:28px">1</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px">
			<div style="text-align:center"><span style="font-size:28px">1</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px">
			<div style="text-align:center"><span style="font-size:28px">1</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px">
			<div style="text-align:center"><span style="font-size:28px">7.58</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px">
			<div style="text-align:center"><span style="font-size:28px">1</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px">
			<div style="text-align:center"><span style="font-size:28px">3.5</span></div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:30px; width:72px">
			<div style="text-align:center"><span style="font-size:28px">3&ndash;5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px; width:123px">
			<div style="text-align:center"><span style="font-size:28px">6</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px">
			<div style="text-align:center"><span style="font-size:28px">4</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px">
			<div style="text-align:center"><span style="font-size:28px">16</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px">
			<div style="text-align:center"><span style="font-size:28px">7.58</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px">
			<div style="text-align:center"><span style="font-size:28px">96</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px">
			<div style="text-align:center"><span style="font-size:28px">3.5</span></div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:30px; width:72px">
			<div style="text-align:center"><span style="font-size:28px">6&ndash;8</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px; width:123px">
			<div style="text-align:center"><span style="font-size:28px">10</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px">
			<div style="text-align:center"><span style="font-size:28px">7</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px">
			<div style="text-align:center"><span style="font-size:28px">49</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px">
			<div style="text-align:center"><span style="font-size:28px">7.58</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px">
			<div style="text-align:center"><span style="font-size:28px">490</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px">
			<div style="text-align:center"><span style="font-size:28px">3.5</span></div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:30px; width:72px">
			<div style="text-align:center"><span style="font-size:28px">9&ndash;11</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px; width:123px">
			<div style="text-align:center"><span style="font-size:28px">7</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px">
			<div style="text-align:center"><span style="font-size:28px">10</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px">
			<div style="text-align:center"><span style="font-size:28px">100</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px">
			<div style="text-align:center"><span style="font-size:28px">7.58</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px">
			<div style="text-align:center"><span style="font-size:28px">700</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px">
			<div style="text-align:center"><span style="font-size:28px">3.5</span></div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:49px; width:72px">
			<div style="text-align:center"><span style="font-size:28px">12&ndash;14</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px; width:123px">
			<div style="text-align:center"><span style="font-size:28px">0</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px">
			<div style="text-align:center"><span style="font-size:28px">13</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px">
			<div style="text-align:center"><span style="font-size:28px">169</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px">
			<div style="text-align:center"><span style="font-size:28px">7.58</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px">
			<div style="text-align:center"><span style="font-size:28px">0</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px">
			<div style="text-align:center"><span style="font-size:28px">3.5</span></div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:49px; width:72px">
			<div style="text-align:center"><span style="font-size:28px">15&ndash;17</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px; width:123px">
			<div style="text-align:center"><span style="font-size:28px">2</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px">
			<div style="text-align:center"><span style="font-size:28px">16</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px">
			<div style="text-align:center"><span style="font-size:28px">256</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px">
			<div style="text-align:center"><span style="font-size:28px">7.58</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px">
			<div style="text-align:center"><span style="font-size:28px">512</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px">
			<div style="text-align:center"><span style="font-size:28px">3.5</span></div>
			</td>
		</tr>
	</tbody>
</table>

<p>&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">For this data set, we have the mean,&nbsp;</span><span class="math-tex">\(\bar{x}=7.58\)</span>&nbsp;<span style="color:black">and the standard deviation,&nbsp;<em>s<sub>x</sub></em>&nbsp;= 3.5. This means that a randomly selected data value would be expected to be 3.5 units from the mean. If we look at the first class, we see that the class midpoint is equal to one. This is almost two full standard deviations from the mean since 7.58 &ndash; 3.5 &ndash; 3.5 = .58. While the formula for calculating the standard deviation is not complicated,s<sub>x</sub>=<span class="math-tex">\(\sqrt{\frac{f(m-\bar{x})^2}{n-1}}\)</span>&nbsp;where&nbsp;<em>s<sub>x</sub></em>&nbsp;= sample standard deviation&nbsp;<span class="math-tex">\(\bar{x}=\)</span>&nbsp;</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">sample mean; the calculations are tedious. It is usually best to use technology when performing the calculations.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">As explained before, a <em>z</em>-score allows us to compare statistics from different data sets. If the data sets have different means and standard deviations, then comparing the data values directly can be misleading.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">&bull; For each data value, calculate how many standard deviations away from its mean the value is.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">&bull; In symbols, the formulas for calculating <em>z</em>-scores become the following.</span></p>

<p style="text-align:justify">&nbsp;</p>

<table border="1" cellpadding="1" cellspacing="1" style="width:900px">
	<tbody>
		<tr>
			<td>
			<div style="text-align:center"><strong><span style="font-size:28px"><span style="color:#333333">Sample</span></span></strong></div>
			</td>
			<td>
			<div style="text-align:center"><span style="font-size:28px"><span class="math-tex">\(Z=\frac{x-\bar{x}}{s}\)</span></span></div>
			</td>
		</tr>
		<tr>
			<td>
			<div style="text-align:center"><strong><span style="font-size:28px"><span style="color:#333333">Population</span></span></strong></div>
			</td>
			<td>
			<div style="text-align:center"><span style="font-size:28px"><span class="math-tex">\(Z=\frac{x-\mu}{\sigma}\)</span></span></div>
			</td>
		</tr>
	</tbody>
</table>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#333333">Table 2.36</span></span></p>

<p>&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">As shown in the table, when only a sample mean and sample standard deviation are given, the top formula is used. When the population mean and population standard deviation are given, the bottom formula is used.</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:white">Example 2.36</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Two students, John and Ali, from different high schools, wanted to find out who had the highest GPA when compared to his school. Which student had the highest GPA when compared to his school?</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><strong><span style="color:black">Student GPA School Mean GPA School Standard Deviation</span></strong></span></p>

<table cellspacing="0" class="Table" style="background:#ededed; border-collapse:collapse; border:none; width:900px">
	<thead>
		<tr>
			<td style="background-color:#ededed; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; height:15px; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong><span style="color:#424242">Student</span></strong></span></div>
			</td>
			<td style="background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:15px; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong><span style="color:#424242">GPA</span></strong></span></div>
			</td>
			<td style="background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:15px; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong><span style="color:#424242">School Mean GPA</span></strong></span></div>
			</td>
			<td style="background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:15px; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong><span style="color:#424242">School Standard Deviation</span></strong></span></div>
			</td>
		</tr>
	</thead>
	<tbody>
		<tr>
			<td style="background-color:#ededed; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:#424242">John</span></span></div>
			</td>
			<td style="background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:#424242">2.85</span></span></div>
			</td>
			<td style="background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:#424242">3.0</span></span></div>
			</td>
			<td style="background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:#424242">.7</span></span></div>
			</td>
		</tr>
		<tr>
			<td style="background-color:#ededed; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:#424242">Ali</span></span></div>
			</td>
			<td style="background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:#424242">77</span></span></div>
			</td>
			<td style="background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:#424242">80</span></span></div>
			</td>
			<td style="background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:#424242">10</span></span></div>
			</td>
		</tr>
	</tbody>
</table>

<p>&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>Table 2.37</strong></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>Solution 2.36</strong></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">For each student, determine how many standard deviations (#ofSTDEVs) his GPA is away from the average, for his school. Pay careful attention to signs when comparing and interpreting the answer.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><em><span style="color:black">z </span></em><span style="color:black">= # of STDEVs</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span class="math-tex">\(=\frac{value-mean}{standard\hspace{0.1cm}deviation}=\frac{x+\mu}{\sigma}\)</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">For John, </span><em><span style="color:black">z </span></em><span style="color:black">= # </span><em><span style="color:black">o f STDEVs </span></em></span></p>

<p style="text-align:justify"><span style="font-size:14px"><em><span style="color:black">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</span></em><span style="color:black">= (2.85 &ndash; 3.0)/.07</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</span></span><span style="font-size:14px"><span style="color:black">= &minus; 0.21</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">For Ali, &nbsp;&nbsp;</span><em><span style="color:black">z </span></em><span style="color:black">= # </span><em><span style="color:black">o f STDEVs</span></em></span></p>

<p style="text-align:justify"><span style="font-size:14px"><em><span style="color:black">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp; &nbsp;</span></em><span style="color:black">= (77 &ndash; 80)/10</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;=&minus; 0.3</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">John has the better GPA when compared to his school because his GPA is 0.21 standard deviations </span><strong><span style="color:black">below </span></strong><span style="color:black">his school&#39;s mean, while Ali&#39;s GPA is .3 standard deviations </span><strong><span style="color:black">below </span></strong><span style="color:black">his school&#39;s mean.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">John&#39;s </span><em><span style="color:black">z</span></em><span style="color:black">-score of &ndash;.21 is higher than Ali&#39;s </span><em><span style="color:black">z</span></em><span style="color:black">-score of &ndash;.3. For GPA, higher values are better, so we conclude that John has the better GPA when compared to his school. The </span><em><span style="color:black">z</span></em><span style="color:black">-score representing John&#39;s score does not fall as far below the mean as the </span><em><span style="color:black">z</span></em><span style="color:black">-score representing Ali&#39;s score.</span></span></p>