<p style="text-align:justify"><span style="font-size:14px">Consider the following data set:</span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">4, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 10</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">This data set can be represented by the following histogram. Each interval has width 1, and each value is located in the&nbsp;</span></span><span style="font-size:14px"><span style="color:black">middle of an interval.</span></span></p>

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

<p style="text-align:justify"><br />
<span style="font-size:14px"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238376001623837600.png" style="height:437px; width:800px" /></span></p>

<p style="text-align:justify"><span style="font-size:14px"><strong><span style="color:#26397f">Figure 2.18</span></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 histogram displays a </span><strong><span style="color:black">symmetrical </span></strong><span style="color:black">distribution of data. A distribution is symmetrical if a vertical line can be drawn at some point in the histogram such that the shape to the left and the right of the vertical line are mirror images of each other.</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">The mean, the median, and the mode are each seven for these data. </span><strong><span style="color:black">In a perfectly symmetrical distribution, the mean and the median are the same. </span></strong><span style="color:black">This example has one mode (unimodal), and the mode is the same as the mean and median.</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">In a symmetrical distribution that has two modes (bimodal), the two modes would be different from the mean and median.</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">The histogram for the data: 4, 5, 6, 6, 6, 7, 7, 7, 7, 8 is not symmetrical. The right-hand side seems </span><em><span style="color:black">chopped off </span></em><span style="color:black">compared to the left-hand side. A distribution of this type is called </span><strong><span style="color:black">skewed to the left </span></strong><span style="color:black">because it is pulled out to the left. A skewed left distribution has more high values.</span></span></p>

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

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238376511623837651.png" style="height:579px; width:800px" /></span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><strong><span style="color:#26397f">Figure 2.19</span></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 mean is 6.3, the median is 6.5, and the mode is seven. </span><strong><span style="color:black">Notice that the mean is less than the median, and they are both less than the mode</span></strong><span style="color:black">. The mean and the median both reflect the skewing, but the mean reflects it more so. The mean is pulled toward the tail in a skewed distribution.</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">The histogram for the data: 6, 7, 7, 7, 7, 8, 8, 8, 9, 10 is also not symmetrical. It is </span><strong><span style="color:black">skewed to the right</span></strong><span style="color:black">. A skewed right distribution has more low values.</span></span></p>

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

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

<p style="text-align:justify"><span style="font-size:14px">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238377051623837705.png" style="height:575px; width:800px" /></span></p>

<p style="text-align:justify"><span style="font-size:14px"><strong><span style="color:#26397f">Figure 2.20</span></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 mean is 7.7, the median is 7.5, and the mode is seven. Of the three statistics, </span><strong><span style="color:black">the mean is the largest, while the mode is the smallest</span></strong><span style="color:black">. Again, the mean reflects the skewing the most.</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 summarize, generally if the distribution of data is skewed to the left, the mean is less than the median, which is often less than the mode. If the distribution of data is skewed to the right, the mode is often less than the median, which is less than the 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">Skewness and symmetry become important when we discuss probability distributions in later chapters.</span></span></p>

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