Chapter 5 – Z-Scores for Individual Scores

We have discussed the idea that if you find out that you earned a 65 out of 100 on a test, you should next ask for the mean and standard deviation of the class in order to have a strong sense of how you performed on the test.

If the mean is μ = 75, and the standard deviation is σ = 5, then you know that the most typical score was 75, and that people in the class deviated from that mean of 75 by about 5 points on average. This would give you a sense that most people scored between 70 and 80. Your 65 is now not looking particularly good. In fact, your score of 65 is two standard deviations below the mean.

However, if the mean is μ = 75, and the standard deviation is σ = 20, then you know that the most typical score was 75, and that people deviated from that mean of 75 by about 20 points on average. This would give you a sense that most people scored between 55 and 95. Your 65 might not feel quite as bad then. Your score of 65 is only half a standard deviation below the mean.

In this chapter, we are going to explore the z-score, which is a statistic that can provide the type of information described above by measuring the number of standard deviations by which a score deviates from the mean.

You will find that the next few chapters all involve z-scores. That’s because z-scores can be used in a number of ways. In fact, z-scores can be used as descriptive statistics, which is the focus of this chapter, but they can also be used as inferential statistics, which we will work toward over the next chapters.

But for now, let’s learn how to calculate a z-score so that we can describe where an individual’s score lies within a group of scores (much like the exam score of 65 above).