Standard Deviation Calculator
Standard deviation measures how spread out data is from the mean. This calculator computes both population standard deviation (σ) and sample standard deviation (s) for any dataset. Enter a comma-separated list of numbers, and the calculator displays population and sample standard deviations, variance, mean, and sum of squared deviations. The population formula divides by N; the sample formula divides by (N-1) for better accuracy when working with samples. Understanding standard deviation is essential in statistics, quality control, finance, and science.
Standard deviation formulas
mean = sum of values / N
Population variance (σ²) = sum of (x - mean)² / N
Sample variance (s²) = sum of (x - mean)² / (N - 1)
Population std dev (σ) = sqrt(σ²)
Sample std dev (s) = sqrt(s²)
Worked example
Dataset: 10, 12, 14, 16, 18
- Mean = (10 + 12 + 14 + 16 + 18) / 5 = 70 / 5 = 14
- Deviations: -4, -2, 0, 2, 4
- Squared deviations: 16, 4, 0, 4, 16. Sum = 40
- Population variance = 40 / 5 = 8. Population std dev = sqrt(8) = 2.83
- Sample variance = 40 / 4 = 10. Sample std dev = sqrt(10) = 3.16
Population or sample: the n versus n-1 question
The commonest standard-deviation mistake is dividing by the wrong number. If your data is the entire group you care about, every student in one class, you divide by n and compute the population standard deviation. If your data is a sample drawn from a larger group you want to describe, you divide by n minus 1. That single adjustment, Bessel's correction, is not a technicality; using the wrong one biases the result, and the error is largest on small samples.
The reason is subtle but intuitive. A sample's own mean sits closer to its own points than the true population mean does, so measuring spread around the sample mean systematically underestimates the real spread. Dividing by n minus 1 rather than n inflates the estimate just enough to correct that bias on average. As samples grow large the two formulas converge, which is why the choice matters most for small datasets.
Beyond the formula, remember what standard deviation does and does not tell you. It summarises spread around the mean and is most meaningful for roughly bell-shaped data, where about two-thirds of values fall within one standard deviation. On skewed data or with strong outliers it can mislead, and a median with an interquartile range often describes the distribution more honestly.
Standard deviation calculator: frequently asked questions
What is standard deviation?
Standard deviation measures how spread out data is from the mean. A low standard deviation means data points cluster near the mean. A high standard deviation means data is scattered widely. Two formulas exist: population standard deviation (sigma) uses division by N; sample standard deviation (s) uses division by (N-1).
What is the difference between population and sample standard deviation?
Population standard deviation is used when you have data for an entire population. Sample standard deviation is used when you have a sample from a larger population. Sample standard deviation divides by (N-1) instead of N to correct for bias. This Bessel correction makes sample estimates more accurate.
What is variance?
Variance is the square of standard deviation. It represents the average squared deviation from the mean. Population variance = sigma²; sample variance = s². Variance is useful in statistics but harder to interpret because it is in squared units.
When should I use population vs. sample standard deviation?
Use population standard deviation when analyzing an entire population. Use sample standard deviation when analyzing a sample from a larger population (the usual case in experiments and surveys). Sample standard deviation is slightly larger, accounting for uncertainty from sampling.
What does a standard deviation value tell me?
For normally distributed data, approximately 68% of values fall within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations (the 68-95-99.7 rule).
Official sources
- Statistical definitions: National Institute of Standards and Technology.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.