Standard Deviation Calculator

Standard deviation at a glance#

Standard deviation measures how spread out a set of numbers is around the mean. It is the square root of the variance: find the mean, square each value's distance from the mean, average those squared distances, then take the square root. A low standard deviation means values cluster near the mean; a high one means they are more spread out.

There are two versions. Population standard deviation divides the sum of squared deviations by N, the count of all values, and is used when your data is the whole group. Sample standard deviation divides by N minus 1 instead, which corrects for the fact that a sample tends to underestimate spread, and is used when your data is a sample drawn from a larger population.

How to calculate standard deviation#

Take the data set 2, 4, 4, 4, 5, 5, 7, 9. There are 8 values that sum to 40, so the mean is 40 divided by 8, which is 5. Subtract the mean from each value to get the deviations: -3, -1, -1, -1, 0, 0, 2, and 4. Square each deviation: 9, 1, 1, 1, 0, 0, 4, and 16. These add up to 32.

So this data set has a population standard deviation of 2 and a sample standard deviation of about 2.138. The two differ because the sample version divides by a smaller number, which makes it slightly larger.

Enter your numbers in the calculator above for the exact mean, variance and standard deviation, with each step shown. Pick the population formula when your data covers the whole group, or the sample formula when it is a subset, since that choice changes the result.

Standard deviation vs variance#

Variance and standard deviation measure the same spread, but in different units. Variance is the average of the squared deviations, so its units are squared, like dollars squared or centimeters squared. Standard deviation is the square root of the variance, which returns the result to the original units. That is why standard deviation is easier to read: for the data set 2, 4, 4, 4, 5, 5, 7, 9, the population variance is 4 but the standard deviation is 2, in the same units as the data.

Reading the result with the empirical rule#

In a roughly normal, bell-shaped distribution, the standard deviation tells you where most values fall. About 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. This is the empirical rule, also called the 68-95-99.7 rule.

So with a mean of 100 and a standard deviation of 15, about 95% of values fall between 70 and 130. A value beyond three standard deviations is rare and often flagged as an outlier.

Where standard deviation is used#

In finance, standard deviation measures how much an investment's returns swing around their average, a common gauge of volatility and risk. In quality control, Six Sigma uses standard deviation to set control limits and flag a process that drifts out of range. In any survey or test, it shows how tightly responses or scores cluster around the average.

Standard deviation FAQ#

What is the difference between population and sample standard deviation?#

The denominator differs. Population standard deviation divides by N, the full count, and sample standard deviation divides by N minus 1. For the data set 2, 4, 4, 4, 5, 5, 7, 9, that gives 2 for the population and about 2.138 for the sample.

Why is the sample standard deviation larger?#

Dividing by N minus 1 instead of N uses a smaller denominator, so the result is slightly larger. This correction offsets the tendency of a sample to understate the true spread of the whole population.

Should I use the population or sample formula?#

Use the population formula when your numbers are the entire group you care about. Use the sample formula when your numbers are a subset drawn from a larger population you want to estimate.

Can standard deviation be zero?#

Yes. If every value is identical, there is no spread, so both the variance and the standard deviation are zero.