Statistics / Normal distribution
Z-Score Calculator (Z to Percentile and Back)
Turn a value into a z-score from the mean and standard deviation, a z-score into a percentile, or a percentile into a z-score, with the area under the normal curve shown and shaded.
Z-Score Calculator (Z to Percentile and Back): A z-score is (value − mean) ÷ standard deviation: a score of 85 where the mean is 70 and the SD 10 is z = 1.5. Under a normal distribution, the share below z is the standard normal CDF, computed here to about 15 significant digits: 93.32% for z = 1.5, 84.13% for z = 1, and 97.5% for z = 1.96. The reverse turns a percentile into its z-score. Runs 100% locally in your browser with zero server file uploads.
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- School & study tools
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A z-score says how many standard deviations a value lies from the mean: z = (value − mean) ÷ standard deviation. For a normal distribution, the percentile is the share of values below it: z = 1 is the 84th percentile, z = 1.96 the 97.5th, and z = −1 the 16th. Percentiles are only accurate when the data really are close to normal.
Worked examples
An IQ of 130 on a scale with mean 100 and SD 15 is z = 2, above about 97.7% of people. A height 1.5 SD below average is z = −1.5, the 6.7th percentile.
To find the mean and standard deviation of your own data first, use the statistics calculator.
The 68–95–99.7 rule
In a normal distribution, about 68% of values lie within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.
To compare two groups' means rather than place one value, use the t-test calculator.
How to use it
- Choose value to z, z to percentile, or percentile to z.
- Enter the value with the mean and standard deviation, or the z-score, or the percentile.
- Read the z-score, the share below and above, and see it on the curve.
Privacy & limitations
Everything is calculated in your browser.
Related tools
Frequently asked questions
What does a negative z-score mean?
The value is below the mean: z = −1 is one standard deviation below, at about the 16th percentile.
Is the percentile always right?
Only if the data follow a normal distribution; for skewed data, such as incomes, the percentile from z can be far off.
What is z = 1.96?
The value that leaves 2.5% in each tail, so 95% of a normal distribution lies within ±1.96 standard deviations: the basis of 95% confidence intervals.
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