Standard Deviation Calculator

Use this standard deviation calculator to find sample and population standard deviation, variance, mean and median for any data set.

What Is a Standard Deviation Calculator?

A standard deviation calculator measures how spread out a set of numbers is. A small standard deviation means the values cluster near the mean, while a large one means they are widely scattered. Paste your data into the box, separated by commas, spaces or new lines, and get the result instantly along with variance, mean, median and range.

Sample vs. Population Standard Deviation

Use population standard deviation (sigma) when your data includes every member of the group. Use sample standard deviation (s) when your data is only a sample from a larger population. The sample version divides by n - 1 instead of n, which corrects for the bias of working with a sample. This calculator shows both.

Standard Deviation Formula

Find the mean, subtract it from each value, square those differences and add them up. Divide the total by n for a population or by n - 1 for a sample to get the variance. The standard deviation is the square root of the variance.

Worked Example

Take the data set 10, 12, 23, 23, 16, 23, 21, 16. The mean is 18 and the sum of squared differences is 192. The sample variance is 192 / 7, about 27.43, so the sample standard deviation is about 5.24. The population standard deviation is the square root of 192 / 8, which is about 4.90.

How to Interpret the Result

For data that follows a normal bell curve, about 68 percent of values fall within one standard deviation of the mean, 95 percent within two and 99.7 percent within three. This is called the empirical rule, and it helps you spot unusual values.

Where Standard Deviation Is Used

Teachers use it to understand test score spread, investors use it to measure risk and volatility, and manufacturers use it for quality control. Scientists rely on it to describe variation in experiments.

Frequently Asked Questions

Can standard deviation be negative?

No. It is a square root, so it is always zero or positive. A value of zero means all numbers are identical.

What is the difference between variance and standard deviation?

Variance is in squared units, while standard deviation is in the original units, which makes it easier to interpret.

How many numbers do I need?

At least two. More data gives more reliable results.

Enter numbers separated by commas, spaces or new lines.