Measures of Central Tendency

Mean

The mean is the average of a set of values, calculated by adding all of the values together and dividing by the total number of values. The formula can be written as

$$\overline{x}={\displaystyle \sum x/n,}$$

where ∑x is the sum of all of the values in the data set, n is the number of values in the data set, and $\overline{x}$ is the mean or average of those values. For example, a teacher may want to determine the average test score on the first exam of the semester. For 10 students in the class, the data set (test scores out of 100 points) is

$$60,65,68,74,85,85,85,85,92,98.$$

To find the mean, add all of the test scores together for a total of 743, and then divide by 10, which is the total number of scores. The average test score for Exam 1 is 74.3.

Median

The median is the middle value in a data set; it lies at the middle position when all the values in a set are arranged in an ascending or descending order. Because of its position in the data set, it literally divides the frequency distribution into two equal parts, resulting in half of the values in a data set lying at or above the median and half of the values lying at or below the median. Therefore, the median represents the 50th percentile in a distribution. To calculate the median, simply identify the value at the middle of the ordered distribution. If the number of values in a data set are odd, then (n + 1) / 2 value is the median. If the number of values in a data set are even, it is found by calculating the average of n / 2 and (n / 2 + 1) value. Note that the values in the data set are not part of the calculation, only the place the value holds in the ordered distribution. To find the median of the test scores provided earlier in this entry, you would use the calculation for the even number of values because there are 10 test scores. Therefore, the average of the value found at 10 / 2 (the fifth value is 85) and the value found at 10 / 2 + 1 (the sixth value is 85) in the ordered set is the median.

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