Standard Deviation Calculator

Standard deviation measures how spread out values are from the mean (average) of a data set. A low standard deviation means values cluster tightly around the mean, while a high standard deviation indicates wide dispersion. It is one of the most commonly used statistics in science, business and social research, partly because it is expressed in the same units as the data itself, whether that is pounds, kilograms or exam scores.

Two data sets can share a mean and behave completely differently. Take two classes both averaging 60% in a maths exam. If the first has a standard deviation of 5%, most scores sit between 55-65% and the class is predictable. If the second has a standard deviation of 20%, some pupils are on 20% and others on 95%, which points to an entirely different teaching problem despite the identical average. That contrast is the reason the mean on its own is rarely enough.

Working it out is a fixed sequence. Calculate the mean by dividing the sum by the count, subtract the mean from each value to get the deviations, square each deviation to remove the negatives, add the squares, divide by n for the population figure or n−1 for the sample figure, then take the square root. With the scores 4, 8, 6, 10 and 7, the mean is 7, the deviations are -3, 1, -1, 3 and 0, and the squares are 9, 1, 1, 9 and 0, which sum to 20. Dividing by 5 gives a variance of 4 and a standard deviation of √4 = 2.

This calculator computes both population standard deviation (σ) and sample standard deviation (s), along with variance, mean, range and sum, and shows each stage of the working. Enter your numbers separated by commas or spaces and it handles data sets of any size. For the set {4, 8, 6, 5, 3} the mean is 5.2, the deviations are −1.2, 2.8, 0.8, −0.2 and −2.2, and the squared deviations are 1.44, 7.84, 0.64, 0.04 and 4.84. That gives a population figure of √(14.8 ÷ 5) = 1.72 and a sample figure of √(14.8 ÷ 4) = 1.92.

Which of the two you want depends on what your numbers represent. Use the population version, dividing by n, when you hold every value, such as the heights of every Premier League player this season. Use the sample version, dividing by n−1, when you are estimating a wider population from a subset, such as measuring 50 randomly selected UK adults to say something about the country. Bessel's correction exists because a sample mean sits closer to its own values than the true population mean does, so without the adjustment the sample figure would understate the real spread. Scientific research almost always uses the sample version.

For normally distributed data, the spread follows the 68/95/99.7 rule, where 68% of values fall within ±1 SD of the mean, 95% within ±2 SD and 99.7% within ±3 SD. IQ scores make this concrete, with a mean of 100 and a standard deviation of 15, so 68% of people score 85-115, 95% score 70-130 and 99.7% score 55-145. Mensa's threshold of the top 2% corresponds to an IQ of 132+, roughly 2 SD above the mean. A-level grade boundaries work similarly, with A at 90%+ and A* at 95%+ marking out the top 5-10%, around 1.3-1.5 SD above average.

Variance is the same calculation stopped one step early, before the square root, so it carries squared units. Data measured in pounds gives a variance in pounds squared and a standard deviation in pounds, which is why the standard deviation is what gets reported and the variance is what gets used in further mathematics, since variances of independent variables add and standard deviations do not. Dividing the standard deviation by the mean and multiplying by 100 gives the coefficient of variation, which compares variability across different scales. UK hospital wait times averaging 30 days with a standard deviation of 15 give a CV of 50%. The same machinery drives quality control, where Six Sigma means 6 SD or 3.4 defects per million, as well as Value at Risk in finance, psychology and medicine.