Standard Deviation Calculator

What Standard Deviation Tells You

The mean tells you the center of a data set; the standard deviation tells you how spread out it is - roughly, the typical distance of a value from the mean.

A small standard deviation means the numbers huddle close to the mean (everyone scored about the same). A large one means they are spread far apart. If every value is identical, there is no spread at all and the standard deviation is 0.

To find it: measure how far each value is from the mean, square those gaps (so being below the mean does not cancel being above it), average them - that average is the variance - and take the square root to return to the original units. The plot under the calculator shades the band one standard deviation either side of the mean, where most of the data usually sits.

The Formulas

The mean and the population standard deviation:

$$\mu = \frac{\sum x}{N} \qquad \sigma = \sqrt{\frac{\sum (x - \mu)^2}{N}}$$

For a sample, divide by $n - 1$ instead of $N$ and use the sample mean $\bar{x}$:

$$s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}}$$

Use population when the data is the entire group; use sample when it is a sample drawn from a larger group (it divides by $n - 1$).

How to Use It

1. Choose population or sample.
2. Enter the numbers, separated by commas.
3. Read the standard deviation, with the mean and variance in the steps.

Worked Examples

Data (population)	Mean	Standard deviation
2, 4, 4, 4, 5, 5, 7, 9	5	2
10, 12, 14, 16, 18	14	2.8284
5, 5, 5, 5	5	0

For the first set as a sample, dividing by 7 instead of 8 gives a standard deviation of about 2.1381.

FAQ

What is the difference between population and sample standard deviation?

Population divides by the count (N); sample divides by one less (n − 1). Use sample when your data is only a portion of a larger group.

How do you calculate standard deviation?

Find the mean, subtract it from each value and square the result, average those squares (the variance), then take the square root.

What does a standard deviation of 0 mean?

Every value is identical, so there is no spread — as with 5, 5, 5, 5.

What is variance?

Variance is the average of the squared differences from the mean; the standard deviation is its square root, in the same units as the data.