Enter a list of numbers to compute sample standard deviation, population standard deviation, both variances, standard error of the mean, and coefficient of variation.
How to use this calculator
Enter your data values separated by commas or spaces.
Read both sample SD (s) and population SD (σ), plus variance for each.
Standard error and coefficient of variation are also computed.
Formula & method
Standard deviation formulas
σ² = Σ(xᵢ − μ)² / N · s² = Σ(xᵢ − x̄)² / (N−1) · SE = s / √N
σ = population SD, s = sample SD, μ = population mean, x̄ = sample mean.
Example
[2,4,4,4,5,5,7,9]: mean = 5, Σ(xᵢ−μ)² = 32, σ² = 4, σ = 2, s² ≈ 4.57, s ≈ 2.14.
Key insights
Sample SD (s) divides by N−1 (Bessel's correction) to correct for estimation bias when working with a sample.
Population SD (σ) divides by N — use this only when you have the entire population.
For dataset [2,4,4,4,5,5,7,9]: mean = 5, population σ = 2, sample s ≈ 2.14.
Standard error SE = s/√N estimates how far the sample mean is from the population mean.
How to interpret your result
Sample vs population SD
Use sample SD (s) when your data is a sample from a larger population. Use population SD (σ) only when you have every member of the population.
Coefficient of variation
CV = (s / |mean|) × 100. Expresses dispersion as a percentage of the mean — useful for comparing variability across datasets with different units or scales.
How this differs from the Statistics Calculator
This calculator specialises in variance and standard deviation outputs. The Statistics Calculator adds quartiles, IQR, skewness, and a five-number summary.
Frequently asked questions
What is standard deviation?
Standard deviation measures how spread out values are from the mean. A low SD means data cluster tightly; a high SD means high variability.
Why does sample SD divide by N−1?
N−1 (Bessel's correction) corrects for the fact that sample variance underestimates population variance when dividing by N.
What is the coefficient of variation?
CV = (SD / |mean|) × 100. It normalises variability as a percentage of the mean, enabling comparison across datasets with different units.
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