Standard deviation tells you how much a data set varies around its mean — the typical distance of a value from the average. It is the square root of the variance (the mean squared deviation). There are two versions: the population SD divides the sum of squared deviations by N, while the sample SD divides by n − 1 to correct the bias that arises when you only have a sample. A small SD means the data is consistent; a large SD means it is spread out.
Reviewed: June 20, 2026 · Author: Naveen P N, Founder — AI Calculator · Verified against: the standard variance & SD definitions (Bessel's correction).
The standard deviation equations
Both start by finding the mean, then summing the squared deviations Σ(x − mean)². The only difference is the divisor: N for a population, n − 1 for a sample. Squaring the deviations stops positive and negative differences cancelling and weights large departures more heavily; the final square root returns the answer to the original units. Variance is the same calculation without that final square root, so variance = SD², expressed in squared units.
Worked example — eight values
Scenario: the data set 2, 4, 4, 4, 5, 5, 7, 9.
The mean is 5 and the sum of squared deviations is 32. Dividing by N = 8 gives a population variance of 4 and a population standard deviation of 2; dividing by n − 1 = 7 gives a sample variance of about 4.571 and a sample standard deviation of about 2.1381. The sample value is slightly larger — that is Bessel's correction at work, inflating the estimate to remove the bias of using a sample mean. For comparison, the tighter set 1, 2, 3, 4, 5 has a population SD of just 1.4142, and the wider 10, 12, 14, 16, 18 reaches 2.8284, exactly mirroring how scattered each set is.
Frequently Asked Questions
The typical distance of values from the mean — √variance. Small = tight, large = scattered.
Population ÷ N; sample ÷ (n − 1). Sample SD is slightly larger (Bessel's correction).
Mean → squared deviations → sum → ÷ N or n−1 → square root. 2,4,4,4,5,5,7,9 → 2 or 2.1381.
The mean squared deviation — SD². Here population variance 4, sample variance ≈ 4.571.
Sample unless your data is the whole population. With large n the two nearly match.