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Standard Deviation Calculator
Enter your numbers separated by commas, spaces, tabs, or newlines. The calculator will instantly find the sample and population standard deviation, variance, mean, sum, and show the step-by-step math.
Calculate standard deviation
Results
| Count (N) | — |
|---|---|
| Sum (∑x) | — |
| Mean (Mean, μ or x̄) | — |
| Sum of Squares (SS) | — |
| Sample Std. Deviation (s) | — |
| Sample Variance (s²) | — |
| Population Std. Deviation (σ) | — |
| Population Variance (σ²) | — |
Understanding Standard Deviation and Variance
Standard Deviation measures how spread out the numbers in a dataset are. A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range of values.
Variance is the average of the squared differences from the mean. Standard deviation is simply the square root of the variance.
Sample vs. Population Formulas
When calculating standard deviation and variance, the formulas differ based on whether you are working with the entire population or just a sample.
Population Standard Deviation (σ): Used when the dataset contains every member of the group you are studying. The sum of squared deviations is divided by the total count N.
σ = √(SS / N)
Sample Standard Deviation (s): Used when the dataset is a sample representing a larger population. The sum of squared deviations is divided by N - 1 to correct for sample bias (Bessel's correction).
s = √(SS / (N - 1))
FAQ
What is the Sum of Squares (SS)?
The Sum of Squares represents the sum of squared differences between each data point and the mean. It is a key intermediate value used to compute variance and standard deviation.
Why do we divide by N - 1 for sample standard deviation?
Dividing by N - 1 instead of N corrects the bias in the estimation of the population variance. It accounts for the fact that a sample is likely to have slightly less spread than the full population.