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Standard Deviation Calculator

Paste a list of numbers to get the sample and population standard deviation, variance, mean, count and sum, with a step-by-step table of each value's deviation from the mean and the formula filled in with your numbers.

Free, no signupNo daily limitRuns in your browser - nothing you enter is uploadedUpdated

Separate with commas, spaces or new lines. You can paste a column straight from a spreadsheet.

-Sample SD (s)
-Population SD (σ)
-Mean
-Count (n)
-Sum
-Sample variance (s²)
-Population variance (σ²)
-Sum of squares Σ(x − x̄)²

Step-by-step table

#xx − x̄(x − x̄)²

The two formulas

population σ = √( Σ(x − μ)² ÷ N ) sample s = √( Σ(x − x̄)² ÷ (n − 1) )

Both start the same way: find the mean, measure how far each value is from it, square those distances (so negatives do not cancel positives) and add them up. The only difference is what you divide by.

Divide by N when your numbers are the entire group you care about - every student in one class, every sale last month. Divide by n − 1 when your numbers are a sample used to estimate a bigger group. A sample's values sit a little closer to their own mean than to the true mean, so dividing by one less corrects that underestimate (Bessel's correction).

Standard deviation is in the same units as your data, which is why it is easier to read than variance: if a test's scores are roughly bell-shaped with mean 70 and standard deviation 10, about two-thirds of them fall between 60 and 80.

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

mean = 40 ÷ 8 = 5 deviations: −3, −1, −1, −1, 0, 0, 2, 4 squares: 9, 1, 1, 1, 0, 0, 4, 16 → sum 32 population: σ = √(32 ÷ 8) = √4 = 2 sample: s = √(32 ÷ 7) = √4.571… = 2.138

What this calculator does not do

  • Raw data only: it does not accept a frequency table or grouped data.
  • Displayed values are rounded to six decimal places.
  • The step table shows up to 200 rows; larger data sets are still calculated in full.

Frequently asked questions

What is the difference between sample and population standard deviation?
Population standard deviation divides the sum of squared deviations by n, and is used when your data is the whole group. Sample standard deviation divides by n − 1, and is used when your data is a sample from a larger group. The n − 1 corrects the tendency of a sample to underestimate the spread.
Which one should I use?
If the numbers are every member of the group you care about (all students in this class), use population. If they are a sample meant to describe a bigger group (30 students out of a school), use sample. Most statistics courses default to sample.
How is standard deviation calculated?
Find the mean, subtract it from each value, square each difference, add the squares, divide by n (population) or n − 1 (sample), and take the square root. For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the squared deviations add to 32, and the population SD is √(32 / 8) = 2.
What is variance?
Variance is the standard deviation squared - the average squared deviation from the mean, before taking the square root.
Why is there no sample standard deviation for one number?
Sample standard deviation divides by n − 1, which is 0 when there is only one value, so it is undefined. Population standard deviation of a single value is 0.
How should I enter the data?
Separate numbers with commas, spaces, semicolons or new lines. Negative numbers and decimals are fine, and anything that is not a number is named in the error message.

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Step-by-step guide: How to calculate standard deviation →

Built and checked by Taimur Hassan Siddiqui, who has more than ten years in computer-science education. The formulas on this page are covered by an automated test suite, and the worked example on this page is one of its cases. Last updated 7 October 2026.