Why do we need z-scores? To answer that question, let's use a clever example
about a fictitious measure of attractiveness. This is an imaginary
scale that can be used to rate how attractive someone is. We use this scale
to rate three men on their level of attractiveness. Shifty scores 35, Mickey
scores 65, and Antonio scores 90. What do these scores tell us? Well, not very much.
We know that Antonio is more attractive than Mickey or Shifty, but we know really
nothing about how these three guys compared to everyone else who took the
test. They could all be really ugly, or really great looking, we don't know. So
what else would be helpful for us to interpret the scores? It would be nice to
know the average score on the test. We need the mean. So we go back to the
attractiveness test validation study. We read the directions, and we find that the
average score on this test is 60. Now we know a little more. We know who is above
and who is below the mean. This will allow us to compute their deviation
scores. For each guy take his score and then subtract the mean of 60. Shifty with
his score of 35 is 25 points below the mean. Mickey, who scored 65, is five points
above the mean, and Antonio who scored 90, is 30 points above the mean. Now what we
want to know is whether those deviations are
big or not? How can we interpret these scores? How large are these deviations? In
order to know whether the deviations are large or not, we need a standard to
compare each deviation score. We need a standard deviation. Just like the mean is
the average of all of the scores in the sample, the standard deviation is the
average amount of deviation of all of the scores in the sample. The standard
deviation can tell us whether the individual deviation scores are large or
not. So we go back to the attractiveness test validation paper, we read the
directions again, and we find that the standard deviation for this test is 10.
With this information, we can divide each deviation score by 10 to compute a
z-score. To apply the z-score formula we first subtract the mean from each raw
score, then we divide each deviation score by 10. The result is a z-score.
Let's try that now for Shifty, his z-score is a negative 2.50. Mickey's
z-score is a positive 0.50 and the z-score for Antonia is positive 3.00.
We can now look at a normal curve to get a better picture of where each score
falls relative to other scores. Using our normal distribution, we can see that
Shifty is way down here. Mickey is near the middle, and Antonio is on the upper
end of the bell curve. This gives us a pretty clear picture of where these guys
fall relative to other men who took the test.