Calculating medians is not as simple as calculating arithmetic averages. We calculate medians for TQ ratings, however, because distribution of TQ ratings tend to be negatively skewed. That is, most students rate teachers at the high end of the scale - "3," "4," and "5" - but a few students mark "2" and, "1" giving the distribution a tail pointing in the negative direction. Statisticians usually recommend the median rather than the arithmetic average as a measure of typical response when distributions are skewed.

The procedure that E&E uses to calculate TQ medians is one that statisticians commonly give for finding the median from a frequency distribution ¹ :

where the lower real limit in the computational formula belongs to the interval containing the middle score. In the case of TQ responses, the lower real limits are 0.5, 1.5, 2.5, 3.5 and 4.5, f is the frequency of the interval containing the middle score, cf is the frequency up the lower limit of interval, and i is the width of the interval (1.0 for TQ distributions).

For a TQ item with the following distribution,

the calculation would be as follows:

Another way of calculating the median from TQ data treats the scores as an ungrouped array. When N is odd and there are no tied scores around the middle, the median is taken to be the middle score in the array. When N is even and there are no tied scores around the middle of an array, the median is customarily taken to be the average of the middle two scores.

When there are tied scores around the middle of the distribution, the median is customarily found by interpolation. Consider the following array of twenty scores: 2, 3, 3, 4, 5, 7, 7, 8, 8, 8, 8, 9, 10, 12, 14, 15, 17, 19, 19, 20. The median is the tenth score in the array. Counting over from left to right, we find the tenth score to be 8. But so also are the 8th, 9th, and 11th scores. Clearly, the median cannot be a score of 8. There are 7 scores below and 9 scores above a score of 8. The accepted practice under these circumstances is to ask: The median is what proportion of 8? In other words, what proportion of the distance is it between 7.5 and 8.5? Since the tenth score is the third 8 among the four tied scores, the median is three-quarters of the distance between 7.5 and 8.5. It is 7.5 + 0.75 = 8.25.

For TQ responses, there will almost always be tied scores around the middle or an ordered array. The TQ distribution that was used in the example earlier would look like this as an ordered array: 2, 2, 3, 4, 4, 5, 5. The middle score in the array appears to be 4, but a media of 4 would not divide the array into two equal halves. There are three scores below 4 and only two above. The accepted practice then is to ask: The midpoint of the distribution is what proportion of the distance between the lower and upper real limits of a score of 4? In other words, what proportion of the distance is it between 3.5 and 4.5? Since counting 3.5 scores up the array take us one-half score into an interval containing two tied scores, the median is one-quarter (0.5 times 2) of the way between 3.5 and 4.5. It is 3.5 + 0.25 = 3.75. ²

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1

See Hays, Statistics for the Social Sciences (Second Edition), pp. 217-219.

2

Runyan and Haber, Fundamentals of Behavioral Statistics (Third Edition), p.84.