Z-Score Calculator
Find how many standard deviations a value sits from the mean and its percentile under the normal curve.
Example
An IQ test has a mean of 100 and a standard deviation of 15. A score of 130 gives:
z = (130 - 100) / 15 = 2.00
percentile = Φ(2.00) = 97.72%
So a score of 130 is 2 standard deviations above the mean, higher than about 97.72% of the population.
How it works
Enter your value x, the population mean, and the standard deviation. The tool computes z = (x - mean) / SD and the percentile via the normal CDF.
Good to know
This Z-Score Calculator converts a raw value into a standard score by measuring how many standard deviations it lies from the mean, using the formula z = (x − mean) / SD. Alongside the z-score itself, it reports the percentile under the standard normal curve plus the share of values that fall below and above your value, and a plain-language readout of its position (for example, "2.00 SD above mean"). It's aimed at students, researchers, teachers, and anyone comparing a single data point against a known mean and standard deviation.
Reach for it whenever you need to put one number in context: judging a test score against a class average, flagging an unusually high or low measurement, comparing values that use different scales, or spotting outliers. Because a z-score is unitless, it lets you compare apples to oranges, such as how a height ranks versus how a weight ranks within their own distributions.
To read the output, focus on sign and size. A positive z means above the mean and a negative z means below it; roughly 68% of values fall between z = −1 and +1, about 95% between −2 and +2, and about 99.7% between −3 and +3. The percentile tells you the rank: a z of 2.00 maps to roughly the 97.7th percentile, meaning the value exceeds about 97.7% of the distribution.
A key caveat: the percentile, "below," and "above" figures assume your data are approximately normally distributed, so treat them as estimates for skewed or small samples. Also make sure you use the population mean and standard deviation that the value actually belongs to, and note that the standard deviation must be greater than zero or the calculator returns no result.
Frequently asked questions
What does a negative z-score mean?
A negative z-score means the value is below the mean. For example, z = -1 means the value sits one standard deviation below the mean, around the 15.87th percentile.
Why does the percentile assume a normal distribution?
The percentile is computed from the standard normal cumulative distribution (the normal CDF). It is only accurate when your data are approximately normally distributed; for skewed data treat the percentile as an estimate.
Is my data uploaded anywhere?
No — this calculator runs entirely in your browser; nothing is uploaded.
Is it free?
Yes, completely free with no sign-up and no limits.
People also ask
How do you calculate a z-score by hand?
Subtract the mean from your value, then divide by the standard deviation: z = (x − mean) / SD. For example, with x = 130, mean = 100, and SD = 15, z = (130 − 100) / 15 = 2.00.
What is a good z-score?
There is no universally "good" z-score; it depends on context. Values between about −2 and +2 are common (covering roughly 95% of a normal distribution), while scores beyond ±3 are rare and often treated as outliers.
What is the difference between a z-score and a percentile?
A z-score measures how many standard deviations a value is from the mean, while a percentile tells you the percentage of the distribution that falls below that value. The percentile is derived from the z-score using the normal cumulative distribution.
Can a z-score be greater than 3 or less than -3?
Yes. Z-scores have no fixed limit, but in a normal distribution only about 0.3% of values fall beyond ±3, so such scores are uncommon and usually indicate extreme or outlier values.
Should I use sample or population standard deviation for a z-score?
A standard z-score uses the population mean and standard deviation. If you only have a sample, you can use the sample statistics as estimates, but recognize that the resulting z-score and percentile are approximations.
What does a z-score of 0 mean?
A z-score of 0 means the value is exactly equal to the mean. It corresponds to the 50th percentile in a normal distribution, with half the values below it and half above.
How do you convert a z-score to a percentile?
Apply the standard normal cumulative distribution function (the normal CDF) to the z-score, which returns the proportion of values below it; multiply by 100 to get the percentile. For instance, z = 1.00 corresponds to about the 84.13th percentile.
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