<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">The </span><em><span style="color:black">center </span></em><span style="color:black">of a data set is also a way of describing location. The two most widely used measures of the </span><em><span style="color:black">center </span></em><span style="color:black">of the data are the </span><strong><span style="color:black">mean </span></strong><span style="color:black">(average) and the </span><strong><span style="color:black">median</span></strong><span style="color:black">. To calculate the </span><strong><span style="color:black">mean weight </span></strong><span style="color:black">of 50 people, add the 50 weights together and divide by 50. To find the </span><strong><span style="color:black">median weight </span></strong><span style="color:black">of the 50 people, order the data and find the number that splits the data into two equal parts. The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers. The mean is the most common measure of the center.</span></span></p>

<p><span style="font-size:14px"><strong>NOTE</strong></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">The words </span><em><span style="color:black">mean </span></em><span style="color:black">and </span><em><span style="color:black">average </span></em><span style="color:black">are often used interchangeably. The substitution of one word for the other is common practice. The technical term is </span><em><span style="color:black">arithmetic mean </span></em><span style="color:black">and </span><em><span style="color:black">average </span></em><span style="color:black">is technically a center location. However, in practice among non-statisticians, </span><em><span style="color:black">average </span></em><span style="color:black">is commonly accepted for </span><em><span style="color:black">arithmetic mean</span></em><span style="color:black">.</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">When each value in the data set is not unique, the mean can be calculated by multiplying each distinct value by its frequency and then dividing the sum by the total number of data values. The letter used to represent the </span><strong><span style="color:black">sample mean </span></strong><span style="color:black">is an </span><em><span style="color:black">x </span></em><span style="color:black">with a bar over it (pronounced &ldquo;x bar&rdquo;): &nbsp;</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">The sample mean is a statistic.</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">The Greek letter </span><em><span style="color:black">&mu; </span></em><span style="color:black">(pronounced &quot;mew&quot;) represents the </span><strong><span style="color:black">population mean</span></strong><span style="color:black">. The population mean is a parameter. One of the requirements for the </span><strong><span style="color:black">sample mean </span></strong><span style="color:black">to be a good estimate of the </span><strong><span style="color:black">population mean </span></strong><span style="color:black">is for the sample taken to be truly random.</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">To see that both ways of calculating the mean are the same, consider the following sample:</span></span></p>

<p><span style="font-size:14px"><span style="color:black">1, 1, 1, 2, 2, 3, 4, 4, 4, 4, 4</span></span></p>

<p><span style="font-size:14px"><span style="color:black"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238352551623835255.png" style="height:212px; width:800px" /></span></span></p>

<p><span style="font-size:14px">In the second example, the frequencies are 3(1) + 2(2) + 1(3) + 5(4).</span></p>

<p><span style="font-size:14px">You can quickly find the location of the median by using the expression (<em>n </em>+ 1)/2</span></p>

<p style="text-align:justify"><span style="font-size:14px">The letter <em>n </em>is the total number of data values in the sample. As discussed earlier, if <em>n </em>is an odd number, the median is the middle value of the ordered data (ordered smallest to largest). If <em>n </em>is an even number, the median is equal to the two middle values added together and divided by two after the data have been ordered. For example, if the total number of data values is 97, then <em>n </em>+ 1</span></p>

<p><span style="font-size:14px">2 = 97 + 1</span></p>

<p style="text-align:justify"><span style="font-size:14px">2 = 49. The median is the 49th value in the ordered data. If the total number of data values is 100,</span></p>

<p><span style="font-size:14px">then <em>n </em>+ 1</span></p>

<p><span style="font-size:14px">2 = 100 + 1</span></p>

<p style="text-align:justify"><span style="font-size:14px">2 = 50.5. The median occurs midway between the 50th and 51st values. The location of the median and the value of the median are <strong>not </strong>the same. The uppercase letter <em>M </em>is often used to represent the median. </span></p>

<p style="text-align:justify"><span style="font-size:14px">The next example illustrates the location of the median and the value of the median.</span></p>

<p style="text-align:justify"><span style="font-size:14px">Another measure of the center is the mode. The <strong>mode </strong>is the most frequent value. There can be more than one mode in a data set as long as those values have the same frequency and that frequency is the highest. A data set with two modes is called bimodal.</span></p>