Standard Deviation and Variance Calculator (sample and population)

This standard deviation calculator gives you the sample and population standard deviation, the variance and the mean of any list of numbers. Paste your data, choose whether it is a sample or the whole population, and see how the values spread around the mean.

The chart shows each value, the mean and the bands of ±1 and ±2 deviations. Under the result you will find a table with each deviation and its square, which is the full procedure step by step.

How to use the standard deviation calculator

Step 1: Type or paste your numbers. Separate them with spaces, commas, line breaks or semicolons. Use a point for decimals.

Step 2: Choose Sample if your data is part of a bigger group, or Population if it is every value in the group.

Step 3: Click Calculate. Results also update as you type.

Step 4: Read the standard deviation, the variance, the chart and the step table with the deviation of every value.

Standard deviation and variance formulas

First find the mean, x̄ = (sum of the values) ÷ n. Then add up the squared deviations of each value from the mean. With every value of the population (σ, variance σ²):

σ² = Σ(x - μ)² ÷ N, σ = √σ²

With a sample (s, variance s²) you divide by n - 1 instead of n:

s² = Σ(x - x̄)² ÷ (n - 1), s = √s²

You divide by n - 1 because the mean of a sample sits closer to its own values than the true population mean does. Dividing by n would make the spread come out a bit short. Variance is in squared units, and the standard deviation brings it back to the units of the data.

Standard deviation examples

Example 1: Data 2, 4, 4, 4, 5, 5, 7, 9

The mean is (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) ÷ 8 = 5. The deviations and their squares:

Sum of squares: 32. As a population, σ² = 32 ÷ 8 = 4 and σ = √4 = 2. As a sample, s² = 32 ÷ 7 = 4.5714 and s = 2.1381.

Example 2: Data 10, 12, 23, 23, 16, 23, 21, 16

Mean = 144 ÷ 8 = 18. Sum of squares = (-8)² + (-6)² + 5² + 5² + (-2)² + 5² + 3² + (-2)² = 192.

Sample: s² = 192 ÷ 7 = 27.4286, s = 5.2372. Population: σ² = 192 ÷ 8 = 24, σ = 4.899.

Example 3: Two groups with the same mean

Group A: 70, 72, 75, 78, 80. Group B: 50, 60, 75, 90, 100. Both have a mean of 75, but they are not alike.

Group A: s = √(68 ÷ 4) = 4.1231. Group B: s = √(1700 ÷ 4) = 20.6155. Group B is far more spread out, and the mean alone does not show it.

The 68-95-99.7 rule for bell-shaped data

If the data is spread in a bell shape (a normal distribution), almost all of it falls close to the mean:

Frequently asked questions

What is the difference between sample and population standard deviation?

The population one (σ) is for when you have every value in the group, and it divides by n. The sample one (s) is for when your data is only part of the group, and it divides by n - 1. With samples it is almost always the one you need.

Why divide by n - 1?

Because the sample mean was computed from the same data and sits closer to it than the true mean does. Dividing by n - 1 corrects that bias.

What does a high or low standard deviation mean?

High means the values are widely spread, and low means they sit close to the mean. In the two-group example, 4.1231 against 20.6155 shows which group is more consistent.

What is the difference between variance and standard deviation?

Variance is the average of the squared deviations and is in squared units. The standard deviation is its square root and is in the same units as the data.

Can the standard deviation be negative or zero?

It is never negative. It is zero when all the values are equal. With a single value the sample version is undefined, because there is nothing to compare.

Learn more

• Series and sigma notation examples
• Examples of square roots

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