Standard Deviation Calculator
Standard deviation calculator with steps on how to calculate standard deviation by hand. Computes standard deviation, variance, and mean for a list of numbers (sample or population).
Our free standard deviation calculator computes the standard deviation, variance, and mean for any list of numbers. Choose between sample and population standard deviation depending on your data.
How to calculate standard deviation
Standard deviation measures how spread out the numbers in a dataset are from the mean. Here are the steps:
- Calculate the mean (average) of the dataset.
- Subtract the mean from each number and square the result.
- Find the average of those squared differences — this is the variance. For a sample, divide by n − 1; for a population, divide by n.
- Take the square root of the variance to get the standard deviation.
For example, for the dataset 2, 4, 4, 4, 5, 5, 7, 9: mean = 5, variance (population) = 4, standard deviation = 2.
Sample vs population standard deviation
Population standard deviation (σ) is used when your data represents the entire group. You divide by n.
Sample standard deviation (s) is used when your data is a sample from a larger population. You divide by n − 1 (Bessel's correction) to avoid underestimating the true variability. In most real-world analyses, sample standard deviation is the right choice.
What does standard deviation tell you?
A low standard deviation means the numbers are clustered close to the mean. A high standard deviation means they are spread out widely. It is commonly used in finance (to measure volatility), science (to express measurement uncertainty), and statistics (to describe distributions).