Sometimes we might need to compare one score to another. The comparison might involve scores from different instruments that have different means and standard deviations, or it might involve finding out the standing of a certain score in a group, or perhaps comparing a single score to the average. Usually this presents a problem, especially in the case of comparing two scores from two different instruments. But if we standardize the scores, then the comparison is easy to do. There are different types of standard scores, and two of the common ones are T scores and Z scores. For example, if you have a bone density test you will get a T score of something like -1.5. In schools, certain diagnostic scores are often reported in Z scores. For example, your child’s teacher may tell you that your child’s math comprehension standardized score is 2.25. In fact, Z scores are so common that I’ll use these to explain.
To turn a raw score into a Z score, we simply subtract the group (i.e. data set) mean from the raw score and divide by the standard deviation of the group. Like this:
Z = X – μ
This new score will usually be somewhere between -3 and 3, but in a few rare instances could reach beyond this. The reason I say rare is because over 99% of scores in any normal group of scores will be between these two values (more on this in the next post).
So how is the Z score useful? Z scores have a mean of zero and a standard deviation of 1. This never changes no matter what set of scores you used in your transformation, and no matter what the mean and standard deviation of your original scores were. This is what makes the Z scores standardized. Suppose your daughter scores a 40 on her math test (M = 30, s = 10), and a 60 on her history test (M = 50, s = 5). Since the two tests have different means and standard deviations, it’s hard to tell which one she did better on. But if we standardize her two scores like this:
Z = 40 – 30 = 1 Z = 60 – 50 = 2
Now it’s easy to see that the history test score (Z = 2) is higher than the math test score (Z = 1), and we can conclude that even though she surpassed the average on both tests (remember, the mean of Z is 0), she did better on the history test. In fact, she scored 2 standard deviations higher than the average on the history test, and one standard deviation higher than average on the math test. Both of which are very good.
How good is very good? We will see in the next post where I continue this with a discussion of the normal distribution.