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Position of a Term in a Distribution for AP Statistics (page 2)

By — McGraw-Hill Professional
Updated on Feb 5, 2011

z-Scores

One way to identify the position of a term in a distribution is to note how many standard deviations the term is above or below the mean. The statistic that does this is the z-score:

The z-score is positive when x is above the mean and negative when it is below the mean.

example: z3 = 1.5 tells us that the value 3 is 1.5 standard deviations above the mean. z3 = –2 tells us that the value 3 is two standard deviations below the mean.

example: For the first test of the year, Harvey got a 68. The class average (mean) was 73, and the standard deviation was 3. What was Harvey's z-score on this test?

Thus, Harvey was 1.67 standard deviations below the mean.

Suppose we have a set of data with mean and standard deviation s. If we subtract from every term in the distribution, it can be shown that the new distribution will have a mean of If we divide every term by s, then the new distribution will have a standard deviation of s/s = 1. Conclusion: If you compute the z-score for every term in a distribution, the distribution of z scores will have a mean of 0 and a standard deviation of 1.

Practice problems for these concepts can be found at:

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