About the Standard Deviation Calculator
This standard deviation calculator measures how spread out a set of numbers is. Paste or type your data — separated by commas, spaces or new lines — and choose whether it is a sample or an entire population. You get the standard deviation, the variance, the mean, the median, the range, the coefficient of variation and the standard error of the mean, plus both the sample (s) and population (σ) versions side by side.
It is designed for statistics students checking homework, researchers summarising measurements, teachers analysing test scores, quality engineers monitoring consistency and investors gauging volatility. The step-by-step table lists each value’s deviation from the mean and its squared deviation, exactly as you would lay it out by hand, so you can follow or verify every step.
Use “Sample” when your numbers are a subset drawn from a larger group (the usual case) — it divides by n − 1 (Bessel’s correction) to avoid underestimating the spread. Use “Population” only when your list contains every member of the group you care about. A low standard deviation means values cluster tightly around the mean; a high one means they are widely scattered.
With the default inputs, the standard deviation is 5.237229. Change any value above to recalculate instantly.
How to use the standard deviation calculator
- 1Paste or type your numbers, separated by commas, spaces or line breaks.
- 2Choose Sample if the data is a subset of a larger group, or Population if it is the whole group.
- 3Read the standard deviation, variance and mean in the results.
- 4Use the step-by-step table to check each deviation and squared deviation.
- 5Compare the coefficient of variation across data sets with different units or scales.
Formula and method
First find the mean of the data. Subtract the mean from every value to get its deviation, square each deviation so negatives do not cancel positives, and add them up to get the sum of squares. Divide by n − 1 for a sample (Bessel’s correction, which gives an unbiased estimate of the population variance) or by N for a full population. That quotient is the variance; its square root is the standard deviation, which is back in the original units of the data.
The coefficient of variation is the standard deviation divided by the absolute mean, expressed as a percentage, so you can compare spread between data sets on different scales. The standard error of the mean is the standard deviation divided by √n and describes how precisely the sample mean estimates the true mean.
- s
- Sample standard deviation
- σ
- Population standard deviation
- xᵢ
- Each data value
- x̄, μ
- Sample mean, population mean
- n, N
- Number of values in the sample or population
Worked examples
Sample of 8 values
The mean is 144 ÷ 8 = 18. The squared deviations add up to 192. Dividing by n − 1 = 7 gives a variance of 27.43, and √27.43 ≈ 5.237 is the sample standard deviation.
Textbook population: 2, 4, 4, 4, 5, 5, 7, 9
The mean is 40 ÷ 8 = 5 and the squared deviations sum to 32. As a population, 32 ÷ 8 = 4 and √4 = 2. Treated as a sample, 32 ÷ 7 ≈ 4.571 gives s ≈ 2.138.
Body temperature readings
Five readings average 98.44 °F. Their squared deviations total 0.812; dividing by 4 gives a variance of 0.203, so the sample standard deviation is about 0.45 °F — the readings are tightly clustered.
Frequently asked questions
Should I use sample or population standard deviation?+
Use the sample standard deviation (divide by n − 1) whenever your data is a subset used to learn about a larger group — this is the usual case in research, surveys and experiments. Use the population formula (divide by N) only when you have every member of the group.
What does standard deviation tell you?+
It tells you the typical distance of values from the mean, in the same units as the data. For roughly normal data, about 68% of values lie within one standard deviation of the mean and about 95% within two.
What is the difference between variance and standard deviation?+
Variance is the average squared deviation from the mean, so it is in squared units (e.g. cm²). Standard deviation is the square root of the variance, which puts it back into the original units and makes it easier to interpret.
Why divide by n − 1 for a sample?+
A sample’s values are, on average, closer to the sample mean than to the true population mean, so dividing by n would underestimate the spread. Dividing by n − 1 (Bessel’s correction) makes the sample variance an unbiased estimate of the population variance.
What is a good coefficient of variation?+
It depends on the field, but as a rough guide a CV under 10% indicates low variability and over 30% indicates high variability. It is most useful for comparing the consistency of data sets measured on different scales.