Z Score Calculator
Find how many standard deviations a value sits from the mean, and the share of a normal distribution that falls below it, above it and between.
A z score is the value minus the mean, divided by the standard deviation. A score of 85 on a test with a mean of 70 and a standard deviation of 10 has a z score of 1.5, which puts it above 93.32% of a normal distribution. Enter a value, mean and standard deviation, or type a z score directly.
The percentiles assume the data follow a normal distribution; the calculator does not test that, and for skewed data the z score is still valid but the percentile is not. Figures are rounded to two decimals, and z to four.
Z score
1.5
- Below (percentile)
- 93.32%
- Above
- 6.68%
- Between -1.5 and 1.5
- 86.64%
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About the Z Score Calculator
A z score turns any measurement into a count of standard deviations from the average, so results on different scales can be compared. The formula is z = (x - mean) / standard deviation: a positive z sits above the mean, a negative one below, and 0 is exactly average. When the data follow a normal curve, the z score also gives a percentile. The calculator works out the area under the standard normal curve to the left of z from the error function, so a z of 2 lands above 97.72% of values and a z of -2 above only 2.28%. The familiar bands come from the same curve: 68.27% of a normal distribution lies within one standard deviation of the mean, 95.45% within two and 99.73% within three. A value of 130 on a scale with mean 100 and standard deviation 15 is a z of 2, while 60 against a mean of 70 and a standard deviation of 5 is a z of -2.
Frequently asked questions
How do I calculate a z score?+
Subtract the mean from your value, then divide by the standard deviation. For 112 with a mean of 100 and a standard deviation of 15, that is 12 divided by 15, a z score of 0.8.
How do I turn a z score into a percentile?+
The percentile is the area of the standard normal curve to the left of z. A z of -1 gives 15.87%, so the value is higher than about 16 of every 100 in a normal distribution, and a z of 0.8 gives 78.81%.
What z score covers the middle 95%?+
1.96. The area between -1.96 and 1.96 is 95%, leaving 2.5% in each tail, which is why 1.96 appears in 95% confidence intervals.
Can a z score be negative?+
Yes. A negative z means the value is below the mean. A z of -0.5 sits above only 30.85% of a normal distribution, so 69.15% of values are higher.
Formulas: NIST Digital Library of Mathematical Functions, chapter 7 (error function); NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.1 Normal Distribution.
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