Standard deviation explained
Standard deviation measures how spread out a set of numbers is around its mean. A small standard deviation means the values cluster tightly near the average; a large one means they are scattered widely. This guide explains what it measures, how to calculate it step by step, and how to read it.
What standard deviation measures
Two data sets can share the same mean yet look completely different: one tightly bunched, the other widely scattered. Standard deviation captures that difference. It is, roughly, the typical distance of a value from the mean. The larger it is, the more variable the data. It is measured in the same units as the data, which makes it easy to interpret.
How to calculate it
Standard deviation is the square root of the variance, and the variance is the average of the squared distances from the mean.
- Find the mean of the data set.
- Subtract the mean from each value to get its deviation.
- Square each deviation so negatives do not cancel positives.
- Average the squared deviations. This is the variance.
- Take the square root of the variance to get the standard deviation.
For 2, 4, 6, 8, 10 the mean is 6. The deviations are −4, −2, 0, 2, 4; squared they are 16, 4, 0, 4, 16, which sum to 40. Dividing by 5 gives a variance of 8, and the square root is about 2.83.
Population versus sample
There are two versions. When your data is the entire population, divide the sum of squared deviations by N, the number of values. When your data is only a sample from a larger population, divide by N minus 1 instead. This correction, called Bessel correction, makes the sample standard deviation a better estimate of the true population value. Most real-world statistics use the sample version.
Reading a standard deviation
For data that follows the familiar bell curve, the standard deviation has a clean interpretation through the 68-95-99.7 rule: about 68 percent of values lie within one standard deviation of the mean, about 95 percent within two, and about 99.7 percent within three. So if test scores average 70 with a standard deviation of 10, roughly two thirds of scores fall between 60 and 80.
- Standard deviation measures the typical spread of data around the mean.
- It is the square root of the variance, the average squared distance from the mean.
- Divide by N for a full population, or by N minus 1 for a sample.
- It is in the same units as the data, so it is easy to interpret.
- Under the bell curve, about 68 percent of values lie within one standard deviation.
Go deeper
Use the Standard Deviation Calculator, or read the Normal Distribution Guide for the bell curve in detail.
FAQ
What does standard deviation tell you?
How spread out the data is. A small value means the numbers cluster near the mean; a large value means they are widely scattered.
What is the difference between variance and standard deviation?
Variance is the average of the squared deviations from the mean; standard deviation is its square root, which returns the measure to the original units.
Why divide by N minus 1 for a sample?
It corrects a bias that would otherwise make the sample underestimate the true spread. This is known as Bessel correction.