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Z-Score

Statistics & Risk Management

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Standardization

• A “Standard” Normal Distribution has a Mean of ZERO and a Standard Deviation of 1.0.

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Visualize the Standard Normal Distribution

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The Ζ-Score• Is based upon Standard Deviation divisions.• Gives you a standardized way to measure the

distance left(-) or right(+) of the Mean value.• Works regardless of the value of the mean,

because we make the mean ZERO and the Standard Deviation 1.0

• Related PERCENTILES• Related T-SCORES

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Formula for a Ζ-Score

Mean of the PopulationSample Score

Population Standard Deviation

z-Score

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Application

We have a 100 test scores. The Mean is 79 and the standard deviation is 4.

If you earned a score of 85, where do you sit within the distribution?

z = (85 - 79) / 4 = 6 / 4 = 1.5 You are sitting around the 94th

percentile…very good.

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Formula for a Z-Score

Mean of the Population

Sample Score Population Standard Deviation

z-Score

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Application

We have a 100 test scores. The Mean is 79 and the standard deviation is 4.

What score do you need to earn a 90% to get that “A”?

X = (1.2 x 4) + 79 = 83.8 You need to earn a score of 83.8 or

above for the “A”.