| Count, n | 8 |
| Sum of values | 40.000000 |
| Mean, x̄ | 5.000000 |
| Sum of squared deviations | 32.000000 |
| Population variance, σ² = SS / n | 4.000000 |
| Sample variance, s² = SS / (n − 1) | 4.571429 |
| Population standard deviation, σ | 2.000000 |
| Sample standard deviation, s | 2.138090 |
| Standard error of the mean, s / √n | 0.755929 |
| Range (max − min) | 7.000000 |
Use the population standard deviation when your list is every value you care about, such as all scores in one class. Use the sample standard deviation when the list is a subset of a larger group and you want to estimate its spread. The sample version divides by n − 1, which corrects for the fact that a sample sits closer to its own mean than the whole population does. Both are shown above.
Find the standard deviation and variance of any list of numbers. The standard deviation measures how far a list of numbers typically sits from its mean. Paste any list, separated by commas, spaces or line breaks, and this calculator returns the mean, the sum of squared deviations, the population and sample variance, the population and sample standard deviation, and the standard error of the mean. The formulas follow the NIST/SEMATECH e-Handbook of Statistical Methods.
Standard deviation is the usual single number for spread. It is in the same units as your data, so a standard deviation of 2 points means a typical value sits about 2 points from the mean. 100% free, no registration, and complete privacy — everything runs locally in your browser, so your data never touches a server.
Both standard deviations appear together. The population form divides by n, the sample form by n − 1.
The mean, the sum of squared deviations and the count are listed, so you can check the arithmetic by hand.
The standard error of the mean, s divided by the square root of n, for confidence intervals and error bars.
Results update in your browser as you calculate. Your numbers are never sent to a server.
Mean, median, mode and range for the same list of numbers.
Standard normal probabilities and critical values for a z score.
Confidence intervals for a mean, using the standard error.
P-values from a z or t statistic, one-tailed and two-tailed.
It is the typical distance of a value from the mean. Square each deviation from the mean, average them, and take the square root. In a roughly normal data set, most values fall within two standard deviations of the mean.
Use the population form when the list is every member of the group you care about. Use the sample form when the list is a sample from a larger group and you want to estimate its spread. It divides by n − 1, which corrects for measuring spread from the sample’s own mean.
Variance is the average squared deviation, so its units are squared. The standard deviation is its square root and uses the original units. Each one converts into the other.
The sample standard deviation divides by n − 1, which is zero for a single value, so it is undefined. The population standard deviation of one value is simply zero.
No. The calculation runs in your browser, and nothing you enter is sent to a server or saved.
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