Statistical Inference: A Short Course

3.3 The Median

For a variable X (representing a sample or a population), let us define the median of X as the value that divides the observations on X into two equal parts; it is a positional value—half of the observations on X lie below the median, and half lie above the median. Since there is no standard notation in statistics for the median, we shall invent our own: the population median will be denoted as “Med;” the sample median will appear as “med.”

Since the median is not a calculated value (as was the mean) but only a positional value that locates the middle of our data set, we need some rules for determining the median. To this end, we have rules for finding the median:

1. Arrange the observations on a variable X in an increasing sequence.

2.

a. For an odd number of observations, there is always a middle term whose value is the median.

b. For an even number of observations, there is no specific middle term. Hence take as the median the average of the two middle terms.

Note that for either case 2a or 2b, the median is the term that occupies the position (or ) in the increasing sequence of data values.

Example 3.3

Given the variable X: 8, 7, 12, 8, 6, 2, 4, 3, 5, 11, 10, locate the median. Arranging the observations in an increasing sequence yields 2, 3, 4, 5, 6, 7, ...