Standard Deviation Calculator
Provide numbers separated by commas to calculate the standard deviation, variance, mean, sum, and margin of error.
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About Standard Deviation
Standard deviation, typically denoted by σ, is a measure of variation or dispersion between values in a set of data. The lower the standard deviation, the closer the data points tend to be to the mean μ. Conversely, a higher standard deviation indicates a wider range of values. The calculator above computes population standard deviation and sample standard deviation, as well as confidence interval approximations. When used in this manner, standard deviation is often called the standard error of the mean, or standard deviation of the estimate with regard to a mean.
Population Standard Deviation
The population standard deviation, the standard definition of σ, is used when an entire population can be measured. The formula divides the sum of squared differences by N (total population count).
σ = √[ Σ(xᵢ − μ)² / N ]
Where μ is the population mean and N is the population size.
Sample Standard Deviation
When only a sample of the full population is available, the sample standard deviation s uses N−1 (Bessel's correction) to give an unbiased estimate.
s = √[ Σ(xᵢ − x̄)² / (N−1) ]
Margin of Error (95% Confidence)
The margin of error uses a z-score of 1.96 for a 95% confidence level. It is computed as: ME = 1.96 × (σ / √N)
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