Calculating the Median

The arithmetic mean of a set of numbers is commonly called the average of the numbers. Its method of calculation is well known. A synonym for 'average' is 'typical' which may be a helpful word to use given the computational baggage that is associated with average. Average also becomes a troublesome word when it brings to mind 'mediocre' as a synonym.

The median is another 'typical' value which may be used to represent a larger set of numbers. A simple definition of median is that it is the middle value in a set of numbers which have been ordered by magnitude. This definition causes a problem when there is no middle value because there are an even number of values, or when there are several numbers having the same value around the point where the middle should occur. A more general definition of median is that it is the 50th percentile of the frequency distribution formed by counting the number of times each value occurs.

When the distribution of numbers is symmetrical, such as that of the so-called bell curve, the mean and median are equal. When the distribution is not symmetrical, debate arises concerning which is the more typical number. An example often used concerns average income where the arithmetic mean may be quite different than the income of the typical wage earner because of the skewed distribution of values. In this case, and often in the case of course/instructor rating scales, the median more closely approximates the income of the typical worker or the rating given by the typical student.

Calculation of the median using the idea of a grouped frequency distribution allows one to recognize that, for example, a 5-point rating scale constrains responses to a small set of discrete values when the underlying attribute being measured is really a continuous scale. Evidence of this is observed in the collection of students' ratings of instruction when we observe that some students mark 2 consecutive values in an attempt to communicate that they're not sure if they want to award, for example, a 4 or a 5. We have also observed on a number of occasions that a respondent will make a mark, for example, between the 4 and the 5. Neither of these types of responses provide valid data but they do illustrate the presence of a continuous scale underlying the small set of discrete values. Calculation of the median in such situations proceeds as follows.

If the distribution of responses given by a class of 25 students is

1 Strongly Disagree,
1 Disagree,
4 Neutral,
8 Agree, and
11 Strongly Agree

and the values 1 through 5 are assigned as ratings corresponding to Strongly Disagree through Strongly Agree, the mean is 4.08. The median is computed as the value attributed to the 50th percentile point in the distribution of ratings given by the 25 respondents. Six responses are Neutral or lower while 14 indicate Agree or lower. The point 12.5, 50% of 25, is thus in the interval corresponding to Agree which ranges from 3.5 to 4.5 when the distribution is considered to be continuous rather than consisting of the discrete values 1 through 5. We need to travel (12.5 - 6) = 6.5 out of 8 units along the interval between 3.5 and 4.5. Therefore the median is computed as 3.5 + (6.5 / 8) = 4.31 which is a value that more closely reflects the consensus of the raters; almost .25 of a 'rating' higher than the mean.

The above can be summarized by the formula:

 Median = L + I * N/2 - F f

where:

L = lower limit of the interval containing the median (3.5 in the example above)
I = width of the interval containing the median (1.0 in the example above)
N = total number of respondents (25 in the example above)
F = cumulative frequency corresponding to the lower limit (6 in the example above)
f = number of cases in the interval containing the median (8 in the example above).

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