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Standard Deviation Calculator

This standard deviation calculator finds the sample or population standard deviation of a list of numbers, along with the variance, mean, sum of squares and range. Type or paste your data, separated by commas, spaces or new lines.

Separate with commas or spaces. You can paste a column from a spreadsheet.

Your data is

Result

Sample standard deviation s = 2.13809

Variance4.571429
Mean5
Count (n)8
Sum40
Sum of squared deviations32
Range (min – max)2 – 9

Results update as you type.

Standard Deviation Formulas

Sample standard deviation:

s = √( Σ(x − x̄)² ÷ (n − 1) )

Population standard deviation:

σ = √( Σ(x − μ)² ÷ n )

  • x = each value
  • x̄ or μ = the mean (sum of the values ÷ n)
  • n = number of values
  • Σ(x − mean)² = the sum of squared deviations, or “sum of squares”

The variance is the same calculation without the square root.

Worked Example

For the data 2, 4, 4, 4, 5, 5, 7, 9:

  1. Mean: (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) ÷ 8 = 40 ÷ 8 = 5
  2. Deviations from the mean: −3, −1, −1, −1, 0, 0, 2, 4
  3. Squared: 9, 1, 1, 1, 0, 0, 4, 16, which add up to 32
  4. Population variance: 32 ÷ 8 = 4, so σ = √4 = 2
  5. Sample variance: 32 ÷ 7 = 4.571429, so s = √4.571429 = 2.13809

Tips and Common Mistakes

Pick the right formula

If you’re unsure, ask whether you have every value you care about (population) or a subset used to estimate a bigger group (sample). Survey results and experiments are almost always samples.

Don’t round the mean too early

Rounding the mean before working out the deviations can change the answer noticeably. The calculator keeps full precision until the final display.

Pasting from a spreadsheet

Copy a column of cells and paste it straight into the box; new lines count as separators. Don’t include thousands separators such as 1,234, because the comma is read as a separator between two numbers.

The standard deviation is a square root of the variance; the square root calculator shows roots in simplified form. To compare two values as a percentage, use the percentage calculator.

Frequently Asked Questions

How do you calculate standard deviation?

Find the mean, subtract it from each value and square the result, then add those squares. Divide by n − 1 for a sample or by n for a population to get the variance, and take the square root. For 2, 4, 4, 4, 5, 5, 7 and 9, the mean is 5, the squares add up to 32, and the population standard deviation is √(32 ÷ 8) = 2.

What is the difference between sample and population standard deviation?

Use the population formula when your data includes every member of the group you care about, such as the scores of everyone in one class. Use the sample formula when the data is a sample used to estimate a larger group. The sample formula divides by n − 1 instead of n, which gives a slightly larger answer.

Why does sample standard deviation divide by n − 1?

A sample's values sit closer, on average, to their own mean than to the true population mean, so dividing by n would tend to underestimate the spread. Dividing by n − 1, known as Bessel's correction, corrects for this.

What does standard deviation tell you?

How spread out the values are around the mean. A small standard deviation means the values are close together; a large one means they're widely spread. It's in the same units as the data, so a standard deviation of 2 cm is easy to interpret alongside a mean in centimeters.

What is the difference between variance and standard deviation?

The variance is the average squared distance from the mean, and the standard deviation is its square root. Standard deviation is usually easier to interpret because it's in the same units as the data, while variance is in squared units.

Can standard deviation be negative?

No. It's the square root of an average of squared numbers, so it's always 0 or more. It's 0 only when every value is the same.

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