Standard Deviation Calculator
Standard deviation, variance, mean, median and more, with step-by-step working.
How to use Standard Deviation Calculator
- Paste or type your numbers, separated by commas, spaces or new lines.
- Say whether the data is a sample of a larger group or the whole population.
- Read the standard deviation and the other statistics. For up to 50 values, open the step-by-step working.
How it works
The mean is the sum divided by the count. Each value’s distance from the mean is squared and the squares are added. Population variance divides that sum by n; sample variance divides by n − 1 (Bessel’s correction), which compensates for a sample underestimating the spread of the group it came from. The standard deviation is the square root of the variance: σ = √(Σ(x − μ)² ÷ n) for a population, s = √(Σ(x − x̄)² ÷ (n − 1)) for a sample.
Also shown: median (middle value), mode (most frequent value or values), range, quartiles by linear interpolation (the method used by Excel’s QUARTILE.INC and NumPy’s default), the standard error of the mean (s ÷ √n) and the coefficient of variation (s ÷ |mean|).
Example
For 2, 4, 4, 4, 5, 5, 7, 9: the mean is 5 and the squared distances add up to 32. Population: √(32 ÷ 8) = 2. Sample: √(32 ÷ 7) = 2.138.
Frequently asked questions
Should I use sample or population standard deviation?
Population if your numbers are the entire group you care about (every student in one class). Sample if they are a subset used to estimate a larger group (a survey). When unsure, sample is the usual choice.
Why divide by n − 1?
A sample’s values sit closer to their own mean than to the true population mean, so dividing by n gives an answer that is too small on average. Dividing by n − 1 corrects that.
What does standard deviation tell me?
How spread out the values are around the mean, in the same units as the data. For roughly bell-shaped data, about 68% of values lie within one standard deviation of the mean and about 95% within two.
Which quartile method is used?
Linear interpolation between the closest ranks, the same as Excel’s QUARTILE.INC and NumPy’s default. Other methods can differ slightly on small data sets.
Limitations
- Up to 100,000 values; step-by-step working for up to 50.
- Results are shown to about 10 significant digits.
- Text or missing values must be removed first: the tool tells you which entry is not a number.
Last reviewed November 9, 2026. How we build and test tools.