Computing the Central Tendency of Data

Definitions of Mode, Median and Mean, the Measures of Central Tendency

Mode
The mode is the score that occurs the most often. The mode is easily computed and provides an indication of the "typical" score; however, its value as a measure to describe a data set is very limited. It is also important to note that in some data sets there are multiple values which have the same frequency which is higher than the frequency of all the other values. Thus, there are multiple modes. For example, a data set with two modes is called
bi-modal.

Median
The median is the midpoint of a distribution; half the scores are above the median, and half the scores are below it. The median is also known as the 50th percentile. The median provides a measure which is less affected by extreme scores than is the mean; however, most statistical procedures depend on the mean, not the median.

To find the median, arrange all the scores from least to greatest, and find the score in the middle. We use "n" to indicate the number of elements in a data set. If "n" is odd, then the median is the term in the (n+1)/2 position. If "n" is even, then the median is the average of the terms in positions (n/2) and (n/2)+1.

Mean
More accurately called the arithmetic mean, it is defined as the sum of the scores divided by the number of scores (n).

Examples of the Computation of Mode, Median and Mean

Consider the observations {8, 25, 7, 5, 8, 3, 10, 12, 9}

First, since we are working by hand, we might wish to arrange the data in order from least to greatest.

{3, 5, 7, 8, 8, 9, 10, 12, 25}

It then becomes easy to "compute" the mode, which is 8.

To compute the median:

  1. arrange the scores in order from smallest to largest (ascending order)
  2. count the number of scores (determine n)
  3. if n is an odd number, then the median = the (n+1)/2 term; In this case, n=9; therefore, the median is the (9+1)/2=5 or the fifth term, which is 8, just like the mode. NOTE: Don't confuse the position of the term with the value of the term. The median in this example isn't 5, it is the fifth term when arranged in ascending order, which is 8.
  4. if n is an even number, then the median is the average of the n/2 term and the term that follows it, the (n /2)+1 term.

To compute the mean:

  1. count the number of scores (determine n),
  2. determine the sum of the scores by adding them,
  3. divide the sum by n.

Again, in this case n = 9 and the sum = 87; therefore, the mean = 87 / 9 = 9.67

Now practice by finding the mode, median, and mean for this similar example: {8, 45, 7, 5, 8, 3, 10, 12, 9}

In this example, the n is still 9, the mode is still 8, the median is still 8, but the mean = 107/9 = 11.89

As noted, the mode and median were the same, 8, for both data examples, but the mean changed from 9.67 to 11.89. Now, how were these data sets different? One score changed from 25 to 45.

Extreme scores in a set of data have a more pronounced effect on the mean than on the median.


Copyright 1997 © T. Lee Willoughby