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# Section 2.5

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# Measures of the Center of the Data

The “center” of a data set is also a way of describing location. The two most widely used measures of the “center” of the data are the mean (average) and the median. To calculate the mean weight of 50 people, add the 50 weights together and divide by 50. To find the median weight 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.

The letter used to represent the sample mean is an x with a bar over it (pronounced “x bar”): x¯ . The Greek letter μ (pronounced “mew”) represents the population mean. One of the requirements for the sample mean to be a good estimate of the population mean is for the sample taken to be truly random.

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. A To see that both ways of calculating the mean are the same, consider the sample:

1; 1; 1; 2; 2; 3; 4; 4; 4; 4; 4

x¯=1+1+1+2+2+3+4+4+4+4+411=2.7

x¯=3(1)+2(2)+1(3)+5(4)11=2.7

In the second calculation, the frequencies are 3, 2, 1, and 5.

You can quickly find the location of the median by using the expression n+12 .

The letter n is the total number of data values in the sample. If n is an odd number, the median is the middle value of the ordered data (ordered smallest to largest). If n is an even number, the median is equal to the two middle values added together and divided by two after the data has been ordered. For example, if the total number of data values is 97, then n+12 = 97+12 = 49. The median is the 49th value in the ordered data. If the total number of data values is 100, then n+12 = 100+12 = 50.5. The median occurs midway between the 50th and 51st values. The location of the median and the value of the median are not the same. The upper case letter M is often used to represent the median. The next example illustrates the location of the median and the value of the median.

### Example 2.26

AIDS data indicating the number of months a patient with AIDS lives after taking a new antibody drug are as follows (smallest to largest):
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;
Calculate the mean and the median.

Solution:

The calculation for the mean is:

x¯=[3+4+(8)(2)+10+11+12+13+14+(15)(2)+(16)(2)++35+37+40+(44)(2)+47]40=23.6

To find the median, M, first use the formula for the location. The location is:

n+12=40+12=20.5

Starting at the smallest value, the median is located between the 20th and 21st values (the two 24s):

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;

M=24+242=24

Another measure of the center is the mode. The mode 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.

Example. Statistics exam scores for 20 students are as follows:

50; 53; 59; 59; 63; 63; 72; 72; 72; 72; 72; 76; 78; 81; 83; 84; 84; 84; 90; 93

Solution: To find the mean:

x – = 50 + 53 + 2 ( 59 ) + 2 ( 63 ) + 5 ( 72 ) + 76 + 78 + 81 + 83 + 3 ( 84 ) + 90 + 93 20 = 73

So, Mean=73

To find the median:

First we find its location. There are 20 data values. n+12=20+12=10.5

Starting at the smallest value, the median is located between the 10th and 11st values (the two 72s):

m e d i a n = 72 + 72 2 = 72

So, Median =72.

The most frequent score is 72, which occurs five times. Mode = 72.