Standard Deviation Calculator
Calculate standard deviation, variance, and mean for sample or population data, with full step-by-step solution
Accepts: comma separated, space separated, or one value per line
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Standard Deviation Formulas
Population Standard Deviation
Sample Standard Deviation
Where xᵢ are individual data values, μ (or x̄) is the mean, n is the count, and Σ denotes summation. The key difference is dividing by n for population and by n−1 for sample (Bessel’s correction).
What Is Standard Deviation?
Standard deviation is a statistical measure that quantifies the amount of variation or spread in a set of data values. Standard deviation tells you how much individual values typically differ from the mean (average) of the data set.
A low standard deviation means the values tend to be close to the mean, forming a tight cluster. A high standard deviation means the values are spread out over a much wider range. Standard deviation is the most widely used measure of dispersion in statistics.
When to Use Sample vs Population SD
Use population standard deviation (σ) when your data represents the entire group you are studying, such as the exact heights of all 30 students in one class. Use sample standard deviation (s) when your data is a sample drawn from a larger population, which is the far more common situation in surveys, experiments, and research. The sample formula uses n−1 in the denominator to correct for the bias that comes from estimating the population mean.
How to Interpret Standard Deviation
In a normal (bell curve) distribution, approximately 68% of all values fall within one standard deviation of the mean, about 95% fall within two standard deviations, and about 99.7% fall within three standard deviations. This rule is called the empirical rule or the 68-95-99.7 rule and is widely used in quality control, finance, and science.
Frequently Asked Questions
Standard deviation is a number that measures how spread out the values in a data set are from the mean (average). A small standard deviation means values are clustered tightly around the mean. A large standard deviation means values are scattered widely. It is the square root of the variance and is expressed in the same units as the original data.
Population standard deviation (σ) is used when you have data for every member of the entire group and divides by n. Sample standard deviation (s) is used when you have data for a subset of a larger group and divides by n−1. The n−1 denominator (Bessel’s correction) removes bias in the estimate of the true population spread. In most real-world situations, you are working with a sample, so use s.
Calculate standard deviation in six steps: (1) Find the mean by summing all values and dividing by n. (2) Subtract the mean from each value to find each deviation. (3) Square each deviation. (4) Sum all squared deviations to get SS (sum of squares). (5) Divide SS by n (population) or n−1 (sample) to get the variance. (6) Take the square root of the variance to get the standard deviation.
A high standard deviation means the values in your data set are widely spread out from the mean. There is more variability or inconsistency in the data. For example, a class where students scored 20, 50, 80, and 90 would have a much higher standard deviation than a class where everyone scored between 60 and 70, even if both had the same mean.
Variance is the average of the squared deviations from the mean. Standard deviation is simply the square root of the variance. Variance is in squared units (e.g., meters squared), which can be hard to interpret. Standard deviation converts back to the original units (e.g., meters), making it much easier to understand in context. Both measure spread, but standard deviation is more practically useful.