What does standard deviation tell you?

Standard deviation is the square root of variance and is expressed in the same units as your original data, which makes it easier to interpret. A low standard deviation means the values are clustered tightly around the mean; a high standard deviation means they are spread widely. About 68% of values in a normal distribution fall within one standard deviation of the mean.

Can variance be negative?

No. Variance is calculated from squared deviations, so it is always zero or positive. A variance of zero means all values in the data set are identical. Standard deviation, as the square root of a non-negative number, is also always zero or positive.

Variance Calculator

This calculator finds the variance and standard deviation of a data set, and lets you choose whether to treat the data as a complete population or as a sample drawn from a larger group. Variance measures how far the values in a data set are spread from their mean: a small variance means the numbers are tightly clustered, and a large variance means they are widely dispersed. Standard deviation is the square root of variance and is more commonly reported because it is expressed in the same units as the original data, making interpretation more intuitive. The difference between population and sample variance comes down to the denominator: population variance divides by N (the total count) and is correct when your data contains every member of the group; sample variance divides by N minus 1, applying Bessel's correction so that the result is an unbiased estimate of the true population variance. To use the calculator, enter your numbers separated by commas and select whether they represent a population or a sample. You receive the variance, the standard deviation, the mean and the count. Decimals and negative numbers are accepted. This tool suits students studying statistics, researchers checking the spread of their data, quality control engineers measuring process variability, and finance professionals computing the volatility of returns. Results are calculated from the values you enter and do not account for how the data was collected.

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Data set (comma-separated numbers)

Type

4.00

variance

Standard deviation2.00

Mean5.00

Count8

How it works

The calculator parses your comma-separated list and computes the mean by summing all values and dividing by the count. It then finds the squared deviation of each value from the mean, sums those squared deviations, and divides by N for population variance or by N minus 1 for sample variance. Standard deviation is the positive square root of that result. The mean and count are also displayed so you can cross-check your data entry.

Worked example

Using the default data set 2, 4, 4, 4, 5, 5, 7, 9: the count is 8 and the sum is 40, so the mean is 5.00. The squared deviations from the mean are: (2-5)²=9, (4-5)²=1, (4-5)²=1, (4-5)²=1, (5-5)²=0, (5-5)²=0, (7-5)²=4, (9-5)²=16. The sum of squared deviations is 32. Population variance is 32 divided by 8 = 4.00. Population standard deviation is the square root of 4 = 2.00. Sample variance would be 32 divided by 7 = 4.57.

Related calculators

Mean Median Mode: find all core descriptive statistics for a data set at once.
Z-Score Calculator: use your mean and standard deviation to find how many SDs a value is from the mean.
Confidence Interval Calculator: build a confidence interval around a sample mean using standard deviation.
Percent Error Calculator: measure how far a result deviates from an accepted value.

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