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Free Standard Deviation Calculator Online

Paste a list of numbers to get the sample and population standard deviation, variance, mean, count and sum.

Separate with commas, spaces or new lines. Decimals and negatives are fine; anything else is ignored.

Sample standard deviation (s)

2.13809

Divides by n − 1. Use when your numbers are a sample of a larger group.
Sample standard deviation (s)
2.13809
Population standard deviation (σ)
2
Sample variance (s²)
4.571429
Population variance (σ²)
4
Mean
5
Count (n)
8
Sum
40

Formulas

Standard deviation measures how spread out numbers are around their mean (x̄). With n values:

Population:  σ = √( Σ(x − x̄)² / n )
Sample:      s = √( Σ(x − x̄)² / (n − 1) )

Variance is the same expression without the square root (σ² or s²). The sample version divides by n − 1 (Bessel's correction) because a sample's mean is itself estimated from the data, which would otherwise make the spread look slightly too small.

How to calculate standard deviation by hand

Use the five values 6, 8, 10, 12 and 14:

  1. Add the values and divide by the count to get the mean: 50 ÷ 5 = 10.
  2. Subtract the mean from each value: −4, −2, 0, 2, 4.
  3. Square each difference: 16, 4, 0, 4, 16.
  4. Add the squares to get the sum of squares: 40.
  5. Divide by n for the population variance (40 ÷ 5 = 8) or by n − 1 for the sample variance (40 ÷ 4 = 10).
  6. Take the square root: σ = √8 ≈ 2.828 and s = √10 ≈ 3.162.
The deviations always add up to zero, which is a handy check on your mean.
xx − x̄(x − x̄)²
6−416
8−24
1000
1224
14416
Sum: 50040

Sample or population?

Use population when your numbers are the entire group you care about, such as the test scores of every student in one class. Use sample when your numbers are a subset used to estimate a larger group, such as a survey of 200 customers. When in doubt, the sample standard deviation is the usual choice.

In spreadsheets, Excel and Google Sheets use STDEV.S for the sample standard deviation and STDEV.P for the population version. The older STDEV function is the sample version.

What the result tells you

A small standard deviation means most values sit close to the mean; a large one means they are spread out. A value of 0 means every number in the list is the same.

For data that roughly follows a bell-shaped (normal) distribution, the 68–95–99.7 rule applies: about 68% of values fall within one standard deviation of the mean, about 95% within two and about 99.7% within three. With a mean of 10 and a standard deviation of 3, roughly two-thirds of values would fall between 7 and 13. The rule does not hold for strongly skewed data.

Frequently asked questions

Why does the sample standard deviation need at least two numbers?

It divides by n − 1, which is zero for a single value. The population standard deviation of one value is simply 0.

What is a worked example?

For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5 and the squared deviations add up to 32. The population variance is 32 ÷ 8 = 4, so σ = 2. The sample variance is 32 ÷ 7 ≈ 4.571, so s ≈ 2.138.

What is the difference between variance and standard deviation?

Variance is the average squared distance from the mean, so it is in squared units. Standard deviation is its square root, which puts it back in the same units as your data and makes it easier to interpret.

Can standard deviation be negative?

No. It is the square root of an average of squared numbers, so it is always zero or positive. It is zero only when every value in the data set is identical.

Is a high standard deviation good or bad?

Neither on its own. It depends on what you measure. A standard deviation of 5 is small for house prices in thousands of dollars but large for the diameter of machined parts in millimeters. Compare it with the mean and with what is normal for your kind of data.

Results are estimates for informational purposes and are not financial, medical or professional advice. See our methodology and disclaimer.