<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>EXAMPLE&nbsp;2.8</strong></span></span></p>

<p style="text-align:justify"><span style="font-size:14px">The following data are the heights (in inches to the nearest half inch) of 100 male semiprofessional soccer players. The heights are&nbsp;<strong><span style="color:#424242">continuous</span></strong>&nbsp;data, since height is measured.</span></p>

<p style="text-align:justify"><br />
<span style="font-size:14px">60; 60.5; 61; 61; 61.5<br />
63.5; 63.5; 63.5<br />
64; 64; 64; 64; 64; 64; 64; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5<br />
66; 66; 66; 66; 66; 66; 66; 66; 66; 66; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5<br />
68; 68; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69.5; 69.5; 69.5; 69.5; 69.5<br />
70; 70; 70; 70; 70; 70; 70.5; 70.5; 70.5; 71; 71; 71<br />
72; 72; 72; 72.5; 72.5; 73; 73.5<br />
74</span></p>

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

<p style="text-align:justify"><span style="font-size:14px">The smallest data value is 60. Since the data with the most decimal places has one decimal (for instance, 61.5), we want our starting point to have two decimal places. Since the numbers 0.5, 0.05, 0.005, etc. are convenient numbers, use 0.05 and subtract it from 60, the smallest value, for the convenient starting point.</span></p>

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

<p style="text-align:justify"><span style="font-size:14px">60 &ndash; 0.05 = 59.95 which is more precise than, say, 61.5 by one decimal place. The starting point is, then, 59.95.</span></p>

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

<p style="text-align:justify"><span style="font-size:14px">The largest value is 74, so 74 + 0.05 = 74.05 is the ending value.</span></p>

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

<p style="text-align:justify"><span style="font-size:14px">Next, calculate the width of each bar or class interval. To calculate this width, subtract the starting point from the ending value and divide by the number of bars (you must choose the number of bars you desire). Suppose you choose eight bars.</span></p>

<p style="text-align:justify"><span style="font-size:14px"><span class="math-tex">\(\frac{74.05-59.95}{8}=1.76\)</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>NOTE </strong></span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px">We will round up to two and make each bar or class interval two units wide. Rounding up to two is one way to prevent a value from falling on a boundary. Rounding to the next number is often necessary even if it goes against the standard rules of rounding. For this example, using 1.76 as the width would also work. A guideline that is followed by some for the width of a bar or class interval is to take the square root of the number of data values and then round to the nearest whole number, if necessary. For example, if there are 150 values of data, take the square root of 150 and round to 12 bars or intervals.</span></p>

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

<p style="text-align:justify"><span style="font-size:14px">The boundaries are:</span></p>

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

<p style="text-align:justify"><span style="font-size:14px">59.95</span></p>

<p style="text-align:justify"><span style="font-size:14px">59.95 + 2 = 61.95</span></p>

<p style="text-align:justify"><span style="font-size:14px">61.95 + 2 = 63.95</span></p>

<p style="text-align:justify"><span style="font-size:14px">63.95 + 2 = 65.95</span></p>

<p style="text-align:justify"><span style="font-size:14px">65.95 + 2 = 67.95</span></p>

<p style="text-align:justify"><span style="font-size:14px">67.95 + 2 = 69.95</span></p>

<p style="text-align:justify"><span style="font-size:14px">69.95 + 2 = 71.95</span></p>

<p style="text-align:justify"><span style="font-size:14px">71.95 + 2 = 73.95</span></p>

<p style="text-align:justify"><span style="font-size:14px">73.95 + 2 = 75.95</span></p>

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

<p style="text-align:justify"><span style="font-size:14px">The heights 60 through 61.5 inches are in the interval 59.95&ndash;61.95. The heights that are 63.5 are in the interval 61.95&ndash;63.95. The heights that are 64 through 64.5 are in the interval 63.95&ndash;65.95. The heights 66 through 67.5 are in the interval 65.95&ndash;67.95. The heights 68 through 69.5 are in the interval 67.95&ndash;69.95. The heights 70 through 71 are in the interval 69.95&ndash;71.95. The heights 72 through 73.5 are in the interval 71.95&ndash;73.95. The height 74 is in the interval 73.95&ndash;75.95.</span></p>

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

<p style="text-align:justify"><span style="font-size:14px">The following histogram displays the heights on the&nbsp;<em>x</em>-axis and relative frequency on the&nbsp;<em>y</em>-axis.</span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238249671623824967.jpg" style="height:566px; width:800px" /></span></p>

<p style="text-align:center"><span style="font-size:14px"><strong><span style="color:#424242">Figure&nbsp;2.5</span></strong></span></p>

<div>
<p>&nbsp;</p>

<p><span style="font-size:14px"><strong>Interval Frequency Relative Frequency</strong></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px">59.95&ndash;61.95&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;5&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;5/100 = 0.05</span></p>

<p><span style="font-size:14px">61.95&ndash;63.95 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;3 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3/100 = 0.03</span></p>

<p><span style="font-size:14px">63.95&ndash;65.95 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;15&nbsp; &nbsp; &nbsp; 15/100 = 0.15</span></p>

<p><span style="font-size:14px">65.95&ndash;67.95 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;40&nbsp; &nbsp; &nbsp; &nbsp;40/100 = 0.40</span></p>

<p><span style="font-size:14px">67.95&ndash;69.95&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;17&nbsp; &nbsp; &nbsp; &nbsp;17/100 = 0.17</span></p>

<p><span style="font-size:14px">69.95&ndash;71.95 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;12 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;12/100 = 0.12</span></p>

<p><span style="font-size:14px">71.95&ndash;73.95&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;7&nbsp; &nbsp; &nbsp; 7/100 = 0.077</span></p>

<p><span style="font-size:14px">3.95&ndash;75.95&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp;1&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;1/100 = 0.01</span></p>
</div>

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

<div>
<p>&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>EXAMPLE&nbsp;2.9</strong></span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px">Create a histogram for the following data: the number of books bought by 50 part-time college students at ABC College. The number of books is&nbsp;<strong><span style="color:#424242">discrete data</span></strong>, since books are counted.</span></p>

<p style="text-align:justify"><br />
<span style="font-size:14px">1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1<br />
2; 2; 2; 2; 2; 2; 2; 2; 2; 2<br />
3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3<br />
4; 4; 4; 4; 4; 4<br />
5; 5; 5; 5; 5<br />
6; 6</span></p>

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

<p style="text-align:justify"><span style="font-size:14px">Eleven students buy one book. Ten students buy two books. Sixteen students buy three books. Six students buy four books. Five students buy five books. Two students buy six books.</span></p>

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

<p style="text-align:justify"><span style="font-size:14px">Calculate the width of each bar/bin size/interval size.</span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><strong>Solution 2.10</strong></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">The smallest data value is 1, and the largest data value is 6. To make sure each is included in an interval, we can use 0.5 as the smallest value and 6.5 as the largest value by subtracting and adding 0.5 to these values. We have a small range here of 6 (6.5 &ndash;&ndash; 0.5), so we will want a fewer number of bins; let&rsquo;&#39;s say six this time. So, six divided by six bins gives a bin size (or interval size) of one.</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Notice that we may choose different rational numbers to add to, or subtract from, our maximum and minimum values when calculating bin size. In the previous example, we added and subtracted .05, while this time, we added and subtracted .5. </span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Given a data set, you will be able to determine what is appropriate and reasonable.</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px">The following histogram displays the number of books on the&nbsp;<em><span style="color:#424242">x</span></em>-axis and the frequency on the&nbsp;<em><span style="color:#424242">y</span></em>-axis.</span></p>

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

<p style="text-align:justify"><span style="font-size:14px">Calculate the number of bars as follows:</span></p>

<p style="text-align:justify"><span style="font-size:14px"><span class="math-tex">\(\frac{6.5-0.5}{Number of bars}=1\)</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px">where 1 is the width of a bar. Therefore, bars = 6.</span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:#f9f9f9"><span style="color:black">The following histogram displays the number of books on the&nbsp;<em>x</em>-axis and the frequency on the&nbsp;<em>y</em>-axis.</span></span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="background-color:#f9f9f9"><span style="color:black"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238301781623830178.jpg" style="height:503px; width:800px" /></span></span></span></p>

<p style="text-align:center"><span style="font-size:14px"><strong><span style="color:#424242">Figure&nbsp;2.6</span></strong></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;2.10</strong></span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px">Using this data set, construct a histogram.</span></p>

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

<table cellspacing="0" class="Table" style="border-collapse:collapse; width:900px">
	<caption>
	<p><span style="font-size:20px"><strong>Number of hours my classmates spent playing video games on weekends</strong></span></p>
	</caption>
	<tbody>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black">
			<div style="text-align:center"><span style="font-size:28px">9.95</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black">
			<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:1px solid black">
			<div style="text-align:center"><span style="font-size:28px">2.25</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black">
			<div style="text-align:center"><span style="font-size:28px">16.75</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black">
			<div style="text-align:center"><span style="font-size:28px">0</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">19.5</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">22.5</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">7.5</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">12.75</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">5.5</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">11</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">20.75</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">17.5</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">23</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">21.9</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">24</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">23.75</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">18</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">20</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">22.9</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">18.8</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">20.5</span></div>
			</td>
		</tr>
	</tbody>
</table>

<p style="text-align:center"><span style="font-size:14px"><strong><span style="color:#424242">Table2.13</span></strong></span></p>

<p>&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><strong><span style="color:black">Solution&nbsp;1</span></strong></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><strong><span style="color:black"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238304301623830430.jpg" style="height:686px; width:800px" /></span></strong></span></p>

<p style="text-align:center"><span style="font-size:14px"><strong><span style="color:#424242">Figure&nbsp;2.7</span></strong></span></p>

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

<p style="text-align:justify"><span style="font-size:14px">Some values in this data set fall on boundaries for the class intervals. A value is counted in a class interval if it falls on the left boundary, but not if it falls on the right boundary. Different researchers may set up histograms for the same data in different ways. There is more than one correct way to set up a histogram.</span></p>
</div>