Standard deviation and z-score calculator
Find the spread of a data set, or convert a value to a z-score and back.
Leave one box empty and it solves for that value.
Standard deviation
Standard deviation measures how far values typically sit from the mean. Use the sample version (divide by n − 1) when your numbers are a sample of a larger group, and the population version (divide by n) when you have every value.
Z-scores
A z-score says how many standard deviations a value is above or below the mean: z = (x − μ) ÷ σ. The percentile shown assumes a normal distribution and uses a numerical approximation that is accurate to about six decimal places.
Worked example
For the data 2, 4, 4, 4, 5, 5, 7, 9, the mean is 40 ÷ 8 = 5. The squared deviations from the mean add up to 32. The population standard deviation is √(32 ÷ 8) = 2, and the sample standard deviation is √(32 ÷ 7) = 2.138.
Using the population values, the z-score of 9 is (9 − 5) ÷ 2 = 2: it is two standard deviations above the mean.
Frequently asked questions
Should I use the sample or the population standard deviation?
Use the sample formula (divide by n − 1) when your data is a sample used to estimate a larger population, which is the usual case in labs and surveys. Use the population formula (divide by n) only when you have every member of the group.
What does a z-score tell me?
How many standard deviations a value is from the mean. Positive is above the mean and negative is below. For roughly normal data, about 95% of values have z between −2 and 2.
How do I turn a z-score into a percentile?
For a normal distribution, use the standard normal table or this calculator: z = 1 is about the 84th percentile, and z = 1.96 is the 97.5th.
What is the difference between variance and standard deviation?
Variance is the average squared deviation. Standard deviation is its square root, which puts it back in the same units as the data.