What this standard deviation calculator does
The standard deviation calculator measures how spread out your numbers are around the mean. It returns both flavours — sample standard deviation (s, divided by n − 1) and population standard deviation (σ, divided by N) — along with the corresponding variances, the mean and the count, plus a step-by-step trace of the calculation. Statisticians use one or the other depending on whether the data represents a sample or the entire population, so showing both removes the most common point of confusion.
The formulas
Sample: s = √[ Σ(xᵢ − x̄)² ÷ (n − 1) ]
Population: σ = √[ Σ(xᵢ − μ)² ÷ N ]
The only difference is the divisor — n − 1 (Bessel's correction) for samples and N for populations. Variance is the value before taking the square root, in squared units.
How to calculate by hand
- Calculate the mean: sum the values and divide by the count.
- Subtract the mean from each value to get the deviations.
- Square each deviation.
- Sum all the squared deviations.
- Divide by N for population, or n − 1 for sample, to get variance.
- Take the square root to get standard deviation.
The calculator runs every step and the table above shows you the working.
Sample vs population: which to pick
If your data set is everything you care about — every customer, every student, every test result — use the population formula. If your data is a sample drawn from a larger group and you want to estimate the spread in that larger group, use the sample formula. The n − 1 divisor exists because using the sample mean (rather than the true unknown population mean) makes the squared-deviation sum slightly smaller than it should be; dividing by n − 1 instead of n compensates and gives an unbiased estimator.
Reading the result with the empirical rule
For roughly normal (bell-curve) data, about 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three. So if a test has a mean of 70 with standard deviation 10, you would expect most students between 60 and 80, with very few below 50 or above 90. This is a quick gut check on whether a result is unusual or routine.
Common mistakes
- Mixing variance and standard deviation — variance is in squared units (cm²), standard deviation in the original units (cm).
- Using the wrong divisor. Engineers and economists often use n − 1 by default; physics labs computing the spread of all their measurements use N.
- Forgetting that standard deviation is only meaningful for numerical data — it does not apply to categories.
Use with the other tools
For a simpler summary, the average calculator returns mean, median, mode, range and sum. For percentages on individual values, see the percentage calculator.