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Z-Score Calculator
A z-score says how many standard deviations a value is from the mean. Enter a raw score with the mean and standard deviation to get its z-score and percentile, or switch modes and turn a percentile back into a z-score and a raw score.
- Percentile (share below)
- 93.3193%
- Share above
- 6.6807%
- Raw score
- 85
Saved setups
Save a set of inputs you reuse — your usual rate, your loan, your room sizes — and load it back in one tap.
Your recent calculations
Results you calculate here are kept on this device so you can come back to them.
Formula
How to use it
- Choose whether you are starting from a raw score or from a percentile.
- Enter the mean and standard deviation of the distribution.
- Enter the score or the percentile.
- Read the z-score, the share of values below it and the share above.
Worked examples
A score of 85 when the mean is 70 and the standard deviation is 10
- Z-score
- 1.5
- Percentile (share below)
- 93.3193%
- Share above
- 6.6807%
An IQ of 130 (mean 100, SD 15)
- Z-score
- 2
- Percentile (share below)
- 97.725%
The 95th percentile of IQ scores
- Z-score
- 1.6449
- Raw score
- 124.6728
Reference values
z = −2 is the 2.28th percentile, −1 the 15.87th, 0 the 50th, 1 the 84.13th, 1.645 the 95th, 1.96 the 97.5th, 2 the 97.72nd, 2.326 the 99th and 3 the 99.87th.
A z-score between −1 and 1 is within the middle 68% of a normal distribution; beyond ±2 is unusual, and beyond ±3 is rare.
Assumptions and accuracy
The z-score itself needs no assumptions, but converting it to a percentile assumes the data follows a normal (bell-shaped) distribution. For skewed data the percentile will be off.
The normal curve is evaluated with a series for the error function and a continued fraction in the tails, which agrees with reference values to about 12 significant digits. Percentiles are converted back to z-scores by Newton’s method on the same function.
Questions people ask
What is the z-score of 85 if the mean is 70 and the standard deviation is 10?
1.5. That is the 93.32nd percentile, so about 6.68% of values are higher.
What z-score is the 95th percentile?
1.645. For IQ scores (mean 100, standard deviation 15) that is a score of about 124.7.
What percentile is an IQ of 130?
About the 97.7th. An IQ of 130 is two standard deviations above the mean, a z-score of 2.