Computing the Central Tendency of Data

Definitions of Mode, Median and Mean

Mode
The most frequently occurring score. For example, the May 89 QPE scores for Internal Medicine indicated that the score 26.2 was obtained by eight students in the Year 2 class. All other possible scores occurred less frequently. The mode is easily computed and provides an indication of the "typical" score; however, its value is limited.

Notice in the data for Basic Science scores that both 26.7 and 30.0 occurred nine times. This situation describes the term, bi-modal.

Median
The midpoint of a distribution, above which half of the scores occurred and below which half of the scores occurred. 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.

Mean
More accurately called the arithmetic mean, it is defined as the sum of scores divided by the number of scores. Or, put in other terms, the mean is the sum of measures observed divided by the number of observations. Note the difference in sum and count: the sum refers to the arithmetic total obtained by adding the scores and count refers to the number of observations or scores.

Computation of Mode, Median and Mean
The number of observations or scores is referred to as "n".

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
    • median = the (n+1) / 2 th observation
  4. if n is an even number, then
    • median = the average of the n / 2 th and (n /2)+1 th observations

For example, consider the observations 8,25,7,5,8,3,10,12,9

  • Arranged in order, the observations are 3,5,7,8,8,9,10,12,25
  • In this case, n=9 ( an odd number); therefore, the median is the (9+1)/2=5 th observation.

For another example, consider the observations 8,45,7,5,8,3,10,12,9

  • Arranged in order, the observations are 3,5,7,8,8,9,10,12,45
  • In this case, n=9 ( an odd number); therefore, the median is the (9+1)/2=5 th observation.

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

For example, consider the observations 8,25,7,5,8,3,10,12,9

  • In this case, n=9 and the sum=87; therefore, the mean = 87 / 9 = 9.67

For another example, consider the observations 8,45,7,5,8,3,10,12,9

  • In this case, n=9 and the sum=107; therefore, the mean = 107 / 9 = 11.89

Did you notice that the median was the same, 5, for both data examples? On the other hand, the mean changed from 9.67 to 11.89 with the one extreme score changing 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