Standard Deviation Calculator
Paste or type a data set and get its standard deviation, variance and mean, with the full calculation shown step by step. Choose population or sample depending on what your data represents.
Formula and calculation
Worked example
For 2, 4, 4, 4, 5, 5, 7, 9: the mean is 5, and the squared deviations from it sum to 32. Dividing by n = 8 gives a variance of 4, so the population standard deviation is 2 — the classic textbook example.
Population or sample — does it really matter?
The two formulas differ only in the divisor (n vs n − 1), and the gap is largest for small data sets: for five values, the sample standard deviation is about 12% larger. The choice is about the question, not about preference — if your numbers are everything there is (all employees, all test scores, a census), use population. If they are measurements drawn from something larger and you want to extrapolate, use sample. When in doubt in the physical sciences, sample is the safer default, which is why it's often called the standard deviation of “n − 1”.
What the results mean
- Population standard deviation
- The standard deviation — how far values typically sit from the mean. Larger means more spread. Population and sample versions differ by the n vs n−1 divisor.
- Variance
- The variance — the standard deviation squared, in squared units. Statisticians use it in calculations; the standard deviation is what people interpret.
- Mean
- The arithmetic average of the data.
- Count (n)
- How many numbers were in the data set.
- Sum
- The sum of all values — the first step of the calculation.
- Σ(x − mean)²
- The sum of squared deviations from the mean, Σ(x − x̄)² — the heart of the formula. Variance is this divided by n (population) or n − 1 (sample).
Frequently asked questions
How do I calculate standard deviation by hand?
Four steps: find the mean; subtract it from each value and square the results; average those squares (÷ n or ÷ n−1); take the square root. The step-by-step table here shows every intermediate number for your data.
What is the difference between variance and standard deviation?
Variance is the average squared deviation; standard deviation is its square root. Because squaring changes units (dollars become squared dollars), the standard deviation is the number you can compare directly with the original data.
What does a standard deviation actually tell me?
For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean and 95% within two. For skewed data the spread number still describes variability but those percentages do not apply — a rule worth remembering before quoting “two sigma”.
How many numbers can I enter?
Up to 1,000 values. The step-by-step table is shown for data sets up to 200 values — beyond that the table is omitted to keep the page fast, but all the computed statistics are still exact.