<p style="text-align:justify"><span style="font-size:14px">The <em>center </em>of a data set is also a way of describing location. The two most widely used measures of the <em>center </em>of the data are the <strong>mean </strong>(average) and the <strong>median</strong>. To calculate the <strong>mean weight </strong>of 50 people, add the 50 weights together and divide by 50. To find the <strong>median weight </strong>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></p>

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

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

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

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

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

<p style="text-align:justify"><span style="font-size:14px">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 <strong>sample mean </strong>is an <em>x </em>with a bar over it (pronounced &ldquo;x bar&rdquo;): &nbsp;</span></p>

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

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

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

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

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

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

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

<p style="text-align:justify"><span style="font-size:14px">1, 1, 1, 2, 2, 3, 4, 4, 4, 4, 4</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/16238352551623835255.png" /></span></p>

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

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

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

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

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

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

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

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

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

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

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>Example 2.27</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">Data indicating the number of months a patient with a specific disease lives after taking a new antibody drug are as follows (smallest to largest):</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">3, 4, 8, 8, 10, 11, 12, 13, 14, 15, 15, 16, 16, 17, 17, 18, 21, 22, 22, 24, 24, 25, 26, 26, 27, 27, 29, 29, 31, 32, 33,33, 34, 34, 35, 37, 40, 44, 44, 47</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Calculate the mean and the median.</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 2.27</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">The calculation for the mean is</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black"><span class="math-tex">\(\bar{x}\)</span></span>&nbsp;<span style="color:black">= [3 + 4 + (8)(2) + 10 + 11 + 12 + 13 + 14 + (15)(2) + (16)(2) + (17)(2) + 18 + 21 + (22)(2) + (24)(2) + 25 + (26)(2)+ (27)(2) + (29)(2) + 31 + 32 + (33)(2) + (34)(2) + 35 + 37 + 40 + (44)(2) + 47] / 40 = 23.6.</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 find the median, </span><em><span style="color:black">M</span></em><span style="color:black">, first use the formula for the location. The location is</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black"><span class="math-tex">\(\frac{n+1}{2}=\frac{40+1}{2}=20.5\)</span></span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Start from the smallest value and count up; the median is located between the 20th and 21st values (the two 24s):</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">3, 4, 8, 8, 10, 11, 12, 13, 14, 15, 15, 16, 16, 17, 17, 18, 21, 22, 22, 24, 24, 25, 26, 26, 27, 27, 29, 29, 31, 32, 33, 33, 34, 34, 35, 37, 40, 44, 44, 47</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span class="math-tex">\(M=\frac{24+24}{2}\)</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 2.28</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">Suppose that in a small town of 50 people, one person earns Rs.5,000,000 per year and the other 49 each earn Rs.30,000. Which is the better measure of the </span><em><span style="color:black">center</span></em><span style="color:black">: the mean or the median?</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 2.28</strong></span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span class="math-tex">\(\bar{x}=\frac{5,000,000+49(30,000)}{50}=129,400\)</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">M </span></em><span style="color:black">= 30,000</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">There are 49 people who earn Rs.30,000 and one person who earns Rs.5,000,000.</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 median is a better measure of the </span><em><span style="color:black">center </span></em><span style="color:black">than the mean because 49 of the values are 30,000 and one is 5,000,000. The 5,000,000 is an outlier. The 30,000 gives us a better sense of the middle of the data.</span></span></p>

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

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

<p style="text-align:justify"><span style="font-size:14px"><strong>Example 2.29</strong></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Statistics exam scores for 20 students are as follows:</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">50, 53, 59, 59, 63, 63, 72, 72, 72, 72, 72, 76, 78, 81, 83, 84, 84, 84, 90, 93</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Find the mode.</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 2.29</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">The most frequent score is 72, which occurs five times. Mode = 72.</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:#000000"><strong>Example 2.29</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">Five real estate exam scores are 430, 430, 480, 480, 495. The data set is bimodal because the scores 430 and 480 each occur twice.</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">When is the mode the best measure of the </span><em><span style="color:black">center</span></em><span style="color:black">? Consider a weight loss program that advertises a mean weight loss of six pounds the first week of the program. The mode might indicate that most people lose two pounds the first week, making the program less appealing.</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>The Law of Large Numbers and the Mean</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">The Law of Large Numbers says that if you take samples of larger and larger size from any population, then the mean of the sample is very likely to get closer and closer to </span><em><span style="color:black">&mu;</span></em><span style="color:black">. This law is discussed in more detail later in the text.</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>Sampling Distributions and Statistic of a Sampling Distribution</strong></span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px">You can think of a sampling distribution as a relative frequency distribution with a great many samples. </span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><strong><span style="color:black">1: Sampling and Data </span></strong><span style="color:black">for a review of relative frequency. Suppose 30 randomly selected students were asked the number of movies they watched the previous week. The results are in the </span><strong><span style="color:black">relative frequency table </span></strong><span style="color:black">shown below.</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/16238359891623835989.png" style="height:575px; width:800px" /></span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><strong>A relative frequency distribution includes the relative frequencies of a number of samples.</strong></span></p>

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

<p style="text-align:justify"><span style="font-size:14px">Recall that a <strong>statistic </strong>is a number calculated from a sample. Statistic examples include the mean, the median, and the mode as well as others. The sample mean <em><span class="math-tex">\(\bar{x}\)</span></em>&nbsp;is an example of a statistic that estimates the population mean <em>&mu;</em>.</span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>Calculating the Mean of Grouped Frequency Tables</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">When only grouped data is available, you do not know the individual data values (we know only intervals and interval frequencies); therefore, you cannot compute an exact mean for the data set. What we must do is estimate the actual mean by calculating the mean of a frequency table. A frequency table is a data representation in which grouped data is displayed along with the corresponding frequencies. To calculate the mean from a grouped frequency table, we can apply the basic definition of mean <img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238370791623837079.png" style="height:25px; width:159px" /></span>.&nbsp;We simply need to modify the definition to fit within the restrictions of a frequency table.</span></p>

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

<p style="text-align:justify"><span style="font-size:14px">Since we do not know the individual data values, we can instead find the midpoint of each interval. The midpoint is&nbsp;<img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238371081623837108.png" style="height:26px; width:176px" />We can now modify the mean definition to be <em>Mean of &nbsp;Frequency Table </em>=&nbsp;</span><span style="font-size:10px"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238371661623837166.png" style="height:38px; width:37px" /></span><span style="font-size:14px">&nbsp;where <em>f </em>= the frequency of the interval, <em>m </em>= the midpoint of the interval, and sigma (&Sigma;) is read as &quot;sigma&quot; and means to sum up. So this formula says that we will sum the products of each midpoint and the corresponding frequency and divide by the sum of all of the frequencies.</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">A frequency table displaying Professor Blount&rsquo;s last statistic test is shown. Find the best estimate of the class mean.</span></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/16238372181623837218.png" style="height:706px; width:800px" /></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 2.31</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">&bull; Find the midpoints for all intervals.</span></span></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/16238372841623837284.png" style="height:871px; width:800px" /></span></span></p>

<p style="text-align:justify"><span style="font-size:14px">Calculate the sum of the product of each interval frequency and midpoint. &Sigma; <em>f m</em></span></p>

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

<p style="text-align:justify"><span style="font-size:14px">53.25(1) + 59.5(0) + 65.5(4) + 71.5(4) + 77.5(2) + 83.5(3) + 89.5(4) + 95.5(1) = 1460.25</span></p>

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

<p style="text-align:justify"><span style="font-size:14px">&micro;<span class="math-tex">\(=\frac{\sum{fn}}{\sum{f}}=\frac{1460.25}{19}=76.86\)</span></span></p>

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

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