<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons.</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">To construct a frequency polygon, first examine the data and decide on the number of intervals, or class intervals, to use on the&nbsp;<em>x</em>-axis and&nbsp;<em>y</em>-axis. After choosing the appropriate ranges, begin plotting the data points. After all the points are plotted, draw line segments to connect them.</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;2.11</strong></span></span></p>

<table cellspacing="0" class="Table" style="border-collapse:collapse; border:none; width:800px">
	<caption>
	<p><span style="font-size:20px"><strong>A frequency polygon was constructed from the frequency table below.</strong></span></p>
	</caption>
	<thead>
		<tr>
			<td colspan="4" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; height:16px; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Frequency distribution for calculus final test scores</strong></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:25px; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Lower bound</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Upper bound</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Frequency</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Cumulative frequency</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:26px">
			<div style="text-align:center"><span style="font-size:28px">49.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:26px">
			<div style="text-align:center"><span style="font-size:28px">59.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:26px">
			<div style="text-align:center"><span style="font-size:28px">5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:26px">
			<div style="text-align:center"><span style="font-size:28px">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:25px">
			<div style="text-align:center"><span style="font-size:28px">59.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px">
			<div style="text-align:center"><span style="font-size:28px">69.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px">
			<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:25px">
			<div style="text-align:center"><span style="font-size:28px">15</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:26px">
			<div style="text-align:center"><span style="font-size:28px">69.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:26px">
			<div style="text-align:center"><span style="font-size:28px">79.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:26px">
			<div style="text-align:center"><span style="font-size:28px">30</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:26px">
			<div style="text-align:center"><span style="font-size:28px">45</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:25px">
			<div style="text-align:center"><span style="font-size:28px">79.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px">
			<div style="text-align:center"><span style="font-size:28px">89.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px">
			<div style="text-align:center"><span style="font-size:28px">40</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px">
			<div style="text-align:center"><span style="font-size:28px">85</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:26px">
			<div style="text-align:center"><span style="font-size:28px">89.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:26px">
			<div style="text-align:center"><span style="font-size:28px">99.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:26px">
			<div style="text-align:center"><span style="font-size:28px">15</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:26px">
			<div style="text-align:center"><span style="font-size:28px">100</span></div>
			</td>
		</tr>
	</tbody>
</table>

<p style="text-align:center"><span style="font-size:14px"><strong>Table2.14</strong></span></p>

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

<p><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238308331623830833.jpg" style="height:607px; width:800px" /></p>

<p style="text-align:center"><span style="font-size:11pt"><strong><span style="color:#424242">Figure&nbsp;</span></strong><strong><span style="color:#424242">2.8</span></strong></span></p>

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

<p style="text-align:justify"><span style="font-size:14px">The first label on the&nbsp;<em><span style="color:#424242">x</span></em>-axis is 44.5. This represents an interval extending from 39.5 to 49.5. Since the lowest test score is 54.5, this interval is used only to allow the graph to touch the&nbsp;<em><span style="color:#424242">x</span></em>-axis. The point labeled 54.5 represents the next interval, or the first &ldquo;real&rdquo; interval from the table, and contains five scores. This reasoning is followed for each of the remaining intervals with the point 104.5 representing the interval from 99.5 to 109.5. Again, this interval contains no data and is only used so that the graph will touch the&nbsp;<em><span style="color:#424242">x</span></em>-axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.</span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Frequency polygons are useful for comparing distributions. This is achieved by overlaying the frequency polygons drawn for different data sets.</span></span></p>

<p>&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>EXAMPLE&nbsp;2.12</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">We will construct an overlay frequency polygon comparing the scores from&nbsp;</span><a href="https://openstax.org/books/introductory-business-statistics/pages/2-1-display-data#eip-id1166383300854" style="color:blue; text-decoration:underline"><span style="color:black">Example 2.11</span></a><span style="color:black">&nbsp;with the students&rsquo; final numeric grade.</span></span></p>

<p>&nbsp;</p>

<p>&nbsp;</p>

<table cellspacing="0" class="Table" style="border-collapse:collapse; border:none; width:900px">
	<thead>
		<tr>
			<td colspan="4" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Frequency distribution for calculus final test scores</strong></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; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Lower bound</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Upper bound</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Frequency</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Cumulative frequency</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">
			<div style="text-align:center"><span style="font-size:28px">49.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">59.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">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">
			<div style="text-align:center"><span style="font-size:28px">59.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">69.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<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">
			<div style="text-align:center"><span style="font-size:28px">15</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">
			<div style="text-align:center"><span style="font-size:28px">69.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">79.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">30</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">45</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">
			<div style="text-align:center"><span style="font-size:28px">79.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">89.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">40</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">85</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">
			<div style="text-align:center"><span style="font-size:28px">89.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">99.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">15</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">100</span></div>
			</td>
		</tr>
	</tbody>
</table>

<p style="text-align:center"><span style="font-size:20px"><strong>Table2.16</strong></span></p>

<table cellspacing="0" class="Table" style="border-collapse:collapse; border:none; width:900px">
	<thead>
		<tr>
			<td colspan="4" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Frequency distribution for calculus final grades</strong></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; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Lower bound</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Upper bound</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Frequency</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Cumulative frequency</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">
			<div style="text-align:center"><span style="font-size:28px">49.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">59.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<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">
			<div style="text-align:center"><span style="font-size:28px">10</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">
			<div style="text-align:center"><span style="font-size:28px">59.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">69.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<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">
			<div style="text-align:center"><span style="font-size:28px">20</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">
			<div style="text-align:center"><span style="font-size:28px">69.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">79.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">30</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">50</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">
			<div style="text-align:center"><span style="font-size:28px">79.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">89.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">45</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">95</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">
			<div style="text-align:center"><span style="font-size:28px">89.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">99.5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">5</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">100</span></div>
			</td>
		</tr>
	</tbody>
</table>

<p style="text-align:center"><span style="font-size:14px"><strong>Table</strong><strong>2.17</strong></span></p>

<p><span style="font-size:14px"><strong><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238310731623831073.jpg" style="height:668px; width:800px" /></strong></span></p>

<p style="text-align:center"><span style="font-size:11pt"><strong><span style="color:#424242">Figure&nbsp;</span></strong><strong><span style="color:#424242">2.9</span></strong> </span></p>