Standard Deviation Calculator – Variance & SD

What is standard deviation?

This standard deviation calculator (or sd calculator) computes a single number that summarizes how spread out the values in a dataset are around the mean. The standard deviation, or std dev, is reported in the same units as the data. A standard deviation of zero means every value is identical; larger values mean wider spread.

For a longer overview written for first-time readers, see What Is Standard Deviation?.

Sample vs population standard deviation

There are two versions of standard deviation. The difference is the denominator inside the formula.

• Sample standard deviation (s) divides the sum of squared deviations by n − 1. Use it when your data is a sample drawn from a larger population.
• Population standard deviation (σ) divides by N. Use it only when your data is the entire population you care about.

The n − 1 in the sample formula is called Bessel's correction. It makes the sample variance an unbiased estimator of the population variance. The full comparison is in the Sample vs Population Standard Deviation guide.

When to use sample standard deviation

Use sample standard deviation when your data is a random sample drawn from a larger group, and you want to use the sample to say something about that larger group. Most real-world data fits this case: a class of 30 test scores standing in for all students, a survey of 500 voters standing in for the electorate, 12 measurements of a part standing in for the manufacturing run. Statistical software defaults to the sample version for the same reason.

When to use population standard deviation

Use population standard deviation only when your numbers describe the entire population you are interested in, with no inference to a larger group. Examples: every test score in a single class when the class is the population, the closing prices of a stock for every day this year, or every student in a school for a one-time school-wide report. If there is any larger group your data is meant to represent, switch to the sample version.

The standard deviation formula

The two formulas share the same shape. The numerator is the sum of squared deviations from the mean. The denominator differs.

For a longer breakdown of each piece of these formulas, see the Standard Deviation Formula guide. For the symbol cheat sheet (σ, s, μ, x̄, n, n − 1), see Standard Deviation Symbol.

How to calculate standard deviation step by step

1. 1. Compute the mean of the data set.
2. 2. Subtract the mean from each value to get the deviation of that value.
3. 3. Square each deviation.
4. 4. Sum the squared deviations.
5. 5. Divide by N for the population variance, or by n − 1 for the sample variance.
6. 6. Take the square root of the variance to get the standard deviation.

The calculator above runs all six steps as you type. To run them in Excel or Google Sheets, see How to Calculate Standard Deviation in Excel.

Worked example: 2, 4, 4, 4, 5, 5, 7, 9

A classic textbook example. Eight values whose mean is a clean integer, so the arithmetic stays easy to follow. Paste this list into the calculator above to verify each line.

• n = 8
• Sum: 2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 = 40
• Mean (x̄ = μ): 40 / 8 = 5
• Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16
• Sum of squared deviations: 32

From the sum of squared deviations:

• Population variance: 32 / 8 = 4
• Population standard deviation: √4 = 2
• Sample variance: 32 / 7 ≈ 4.5714
• Sample standard deviation: √(32 / 7) ≈ 2.1381

Variance vs standard deviation

Variance and standard deviation describe the same idea on two different scales. Variance is the average squared deviation from the mean; standard deviation is its square root. Variance is convenient for the math (it adds nicely when you combine independent variables); standard deviation is convenient for reading (same units as the data).

The full comparison, with worked numbers, is in Variance vs Standard Deviation. If you only want variance, the variance calculator runs the same math and reports just the squared figure.

Empirical rule (68-95-99.7)

For a normal distribution, the empirical rule (also known as the 68-95-99.7 rule) describes the percentage of data that falls within a specific number of standard deviations from the mean.

For example, if test scores average 75 with a standard deviation (σ) of 5, about 95% of scores will fall between 65 and 85 (within 2σ of the mean).

Common mistakes

• Dividing by N when you should divide by n − 1. If your data is a sample (almost always), use n − 1. Dividing by N gives you a slightly biased underestimate of the population standard deviation.
• Skipping the square step. The deviations from the mean must be squared before you add them. Without squaring, the positive and negative deviations cancel out and the sum is always zero.
• Reporting variance and calling it standard deviation. Variance is in squared units. Standard deviation is the square root of variance and is in the original units of the data.
• Treating standard deviation as a sign of accuracy. Standard deviation describes spread, not correctness. Precise measurements can be inaccurate, and accurate measurements can be imprecise. For comparing a value to an accepted reference, see the percent error calculator.
• Ignoring outliers. A single very large or very small value can pull standard deviation up sharply. Investigate outliers before reporting; do not silently drop them, but make sure they are not data-entry errors.