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Outline and Review of Chapter 5
Introduction to z scores
In the previous chapters we discussed different measures of central tendency and different ways
to describe the variability in a distribution. For the next few chapters, we will be discussing
techniques that only apply to normal distributions and therefore will be using the mean as the
primary measure of central tendency and the standard deviation as the primary measure of
variability.
Normal distributions have a standard shape; the most common scores are clustered around a
central mean and as the scores deviate from the mean in either direction, they become less and
less common. If you know the standard deviation, you can safely predict that about two thirds of
the scores (actually p = 0.68) can be found within one standard deviation of the mean, about 95%
(p = 0.95) can be found within two standard deviations of the mean and almost all of the scores
(p = 0.997) are within three standard deviations of the mean. With the help of the unit normal
table (Appendix B) you can precisely state what proportion of the distribution lies above or
below any score (X value) , provided you know the distance between the mean and the X value in
standard deviations. In this chapter, we will only discuss population data and therefore use the
population symbols (μ, σ, N) and formulae for population parameters.
z-scores and Location in a Distribution
I would guess that you already appreciate the significance of standard scores although you might
not realize it. Imagine that, after taking an exam, you discover that you have received a grade of
77 (X = 77). If you know that 100 was the highest possible score, you might be disappointed to
find that you have not received the best possible score but you might also wonder, How did
everyone else do? If you discover that the class mean was 75 (μ = 75) you know that you did
better than at least half the class. However, you still do not know if the class distribution consists
of a very tight group of scores clustered between 72 and 77 (in which case, your grade is among
the best) or if it consists of a broad range of scores between 52 and 98 (in which case, your grade
is, quite literally, mediocre). The standard deviation (σ) is the last piece of the puzzle that you
need in order to precisely determine how many students scored better than you and how many
scored worse than you. If σ= 2 then you are one full standard deviation above the mean
In order to understand where your score
Using z-scores to standardize a distribution
IQ as an example of a Standardized distribution