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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