Educational Psychology

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In this example, 34 per cent of the scores are between 100 and 115 and as well, 34 per cent of the scores lie
between 85 and 100. This means that 68 per cent of the scores are between -1 and +1 standard deviations of the
mean (i.e. 85 and 115). Note than only 14 per cent of the scores are between +1 and +2 standard deviations of the
mean and only 2 per cent fall above +2 standard deviations of the mean.

In a normal distribution a student who scores the mean value is always in the fiftieth percentile because the
mean and median are the same. A score of +1 standard deviation above the mean (e.g. 115 in the example above) is
the 84 per cent tile (50 per cent and 34 per cent of the scores were below 115). In Exhibit 10 we represent the
percentile equivalents to the normal curve and we also show standard scores.^1

Kinds of test scores

A standard score expresses performance on a test in terms of standard deviation units above of below the
mean (Linn & Miller, 2005). There are a variety of standard scores:

Z-score: One type of standard score is a z-score, in which the mean is 0 and the standard deviation is 1. This
means that a z-score tells us directly how many standard deviations the score is above or below the mean. For
example, if a student receives a z score of 2 her score is two standard deviations above the mean or the eighty-
fourth percentile. A student receiving a z score of -1.5 scored one and one half deviations below the mean. Any score
from a normal distribution can be converted to a z score if the mean and standard deviation is known. The formula
is:

Z−score=ScoreStandard deviation−mean score

So, if the score is 130 and the mean is 100 and the standard deviation is 15 then the calculation is:

Z=^13015 −^100 = 2

If you look at Exhibit 15 you can see that this is correct—a score of 130 is 2 standard deviations above the mean
and so the z score is 2.

T-score: A T-score has a mean of 50 and a standard deviation of 10. This means that a T-score of 70 is two
standard deviations above the mean and so is equivalent to a z-score of 2.

Stanines: Stanines (pronounced staynines) are often used for reporting students’ scores and are based on a
standard nine point scale and with a mean of 5 and a standard deviation of 2. They are only reported as whole
numbers and Figure 11-10 shows their relation to the normal curve.

Grade equivalent sores

A grade equivalent score provides an estimate of test performance based on grade level and months of the school
year (Popham, 2005, p. 288). A grade equivalent score of 3.7 means the performance is at that expected of a third
grade student in the seventh month of the school year. Grade equivalents provide a continuing range of grade levels
and so can be considered developmental scores. Grade equivalent scores are popular and seem easy to understand
however they are typically misunderstood. If, James, a fourth grade student, takes a reading test and the grade
equivalent score is 6.0; this does not mean that James can do sixth grade work. It means that James performed on

1 Exhibit 11.10 must be re-drawn. Please contact the Associate Editor for the original.

Educational Psychology 297 A Global Text