Educational Psychology

12. Standardized and other formal assessments

Central tendency and variability

There are three common ways of measuring central tendency or which score(s) are typical. The mean is
calculated by adding up all the scores and dividing by the number of scores. In the example in Table 44, the mean is

24. The median is the “middle” score of the distribution—that is half of the scores are above the median and half are
    below. The median on the distribution is 23 because 15 scores are above 23 and 15 are below. The mode is the score
    that occurs most often. In Table 44 there are actually two modes 22 and 27 and so this distribution is described as
    bimodal. Calculating the mean, median and mode are important as each provides different information for
    teachers. The median represents the score of the “middle” students, with half scoring above and below, but does not
    tell us about the scores on the test that occurred most often. The mean is important for some statistical calculations
    but is highly influenced by a few extreme scores (called outliers) but the median is not. To illustrate this, imagine a
    test out of 20 points taken by 10 students, and most do very well but one student does very poorly. The scores might
    be 4, 18, 18, 19, 19, 19, 19, 19, 20, 20. The mean is 17.5 (170/10) but if the lowest score (4) is eliminated the mean is
    now is 1.5 points higher at 19 (171/9). However, in this example the median remains at 19 whether the lowest score
    is included. When there are some extreme scores the median is often more useful for teachers in indicating the
    central tendency of the frequency distribution.

The measures of central tendency help us summarize scores that are representative, but they do not tell us
anything about how variable or how spread out are the scores. Exhibit 20 illustrates sets of scores from two
different schools on the same test for fourth graders. Note that the mean for each is 40 but in School A the scores
are much less spread out. A simple way to summarize variability is the range, which is the lowest score subtracted
from the lowest score. In School A with low variability the range is (45—35) = 10; in the school B the range is ( 55-
22 = 33).

Exhibit 20: Fourth grade math scores in two different schools with the

same mean but different variability

However, the range is only based on two scores in the distribution, the highest and lowest scores, and so does
not represent variability in all the scores. The standard deviation is based on how much, on average, all the

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