Z-Scores and the 68-95-99.7 Rule: Formula and Examples
Calculate a z-score, interpret it, and use the 68-95-99.7 rule for normal distributions, with two worked examples and a common mistake.
The z-score
- z = (x − μ) / σ
- μ is the mean
- σ is the standard deviation
A z-score counts standard deviations from the mean.
68-95-99.7 rule
- About 68% within 1 standard deviation
- About 95% within 2
- About 99.7% within 3
This applies to approximately normal distributions.
Example 1
Scores have mean 75 and standard deviation 5. Find z for 85.
- z = (85 − 75) / 5
- z = 10 / 5
z = 2
Example 2
What percent of scores fall between 70 and 80?
- 70 and 80 are 1 standard deviation from 75
- About 68% lie within 1 standard deviation
About 68%
Common mistake
A z-score of 2 means 2 points above the mean
It means 2 standard deviations above the mean.
Divide by the standard deviation to get z.
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