Standard Deviation Calculator
Enter numbers separated by commas to calculate standard deviation, variance, mean, sum, and margin of error.
Formulas
σ = √[Σ(x−μ)² / n]
s = √[Σ(x−μ)² / (n−1)]
s = √[Σ(x−μ)² / (n−1)]
What is standard deviation?
Standard deviation (σ or s) measures how spread out data points are from the mean (μ). A low standard deviation means values cluster near the mean; a high standard deviation means a wider range. It is one of the most important statistics in science, finance, and quality control.
Population vs sample standard deviation
Population (σ): Divide by n when your data is the entire population.
σ = √[ Σ(xi − μ)² / n ]
Sample (s): Divide by n − 1 (Bessel's correction) when data is a sample from a larger population.
s = √[ Σ(xi − x̄)² / (n − 1) ]
Margin of error
For sample data, the 95% margin of error is approximated as:
MOE ≈ 1.96 × (s / √n)
This estimates how far the sample mean might differ from the true population mean.
Frequently Asked Questions
Use population when you have every value in the group. Use sample when you measured a subset and want to estimate the whole population.
At least 2 values are required. More data gives a more reliable standard deviation estimate.