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.