Z-Score Calculator
Enter a value with its mean and standard deviation to get the z-score, the percentile and the probability of a lower or higher value. Or start from a z-score to get the tail probabilities and the two-tailed p-value, or from a percentile to get the z-score and, if you give a mean and standard deviation, the value itself. Everything runs in your browser.
A value at z = 1.5 is at about the 93.3th percentile, if the data follow a normal distribution.
A z-score is the number of standard deviations a value lies from the mean: z = (x - mean) / standard deviation. For the mean of a sample of size n, the divisor is the standard error, the standard deviation divided by the square root of n. The probabilities assume a normal (bell-shaped) distribution, and they are computed from the error function in full double precision, so a far tail such as z = 5 keeps its digits (about 2.87e-7). The percentile-to-z tab gives the inverse, for example 1.95996 for 97.5%. Everything is worked out in your browser and nothing is uploaded.
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Math & MoneyAbout the Z-Score Calculator
A z-score says how many standard deviations a value lies from the mean: z = (x - mean) / standard deviation. A z of 1.5 is one and a half standard deviations above the mean, and a z of -1 is one below. For the mean of a sample of n values, tick the sample option, and the divisor becomes the standard error, which is the standard deviation divided by the square root of n. The probabilities use the standard normal distribution, the bell curve with mean 0 and standard deviation 1. The tool gives the chance of a value below z, above z, outside plus or minus z (the two-tailed p-value) and between minus z and plus z. They are computed from the error function in full double precision, so the common reference points come out right: z = 1.96 leaves 2.5 percent in each tail, 1.645 gives the 95th percentile and 2.576 gives the 99.5th, and a far tail such as z = 5 is still accurate, about 0.000000287. The percentile tab works backward with the inverse normal function, so 97.5 percent gives 1.95996. These numbers describe a normal distribution. If your data are skewed or have heavy tails, a z-score still measures distance in standard deviations, but the probabilities will not match what you see in the data. Nothing is uploaded.
How do I calculate a z-score?▾
Subtract the mean from the value, then divide by the standard deviation. A score of 85 on a test with a mean of 70 and a standard deviation of 10 has z = (85 - 70) / 10 = 1.5.
How do I turn a z-score into a percentile?▾
Look up the share of the normal curve below it. The tool does this for you: z = 1.5 gives 0.9332, so the value is at about the 93rd percentile. A z of 0 is the 50th percentile, because half the data lie below the mean.
What z-score gives 95 percent confidence?▾
For a two-sided 95 percent interval the z-score is 1.96, which leaves 2.5 percent in each tail. For 90 percent it is 1.645, and for 99 percent it is 2.576. Use the percentile tab with 97.5%, 95% and 99.5% to see them.
What is a two-tailed p-value?▾
It is the probability of a result at least as far from the mean as yours, in either direction. For z = 1.96 it is about 0.05. Double the one-tailed probability, since the normal curve is symmetric.
Is a z-score the same for any data?▾
The z-score itself can be worked out for any data. But the probabilities and percentiles shown here are only right if the data follow a normal distribution. Nothing you enter is uploaded.