Standard Deviation Calculator
Calculate standard deviation and variance for population and sample data. Get step-by-step calculations with detailed explanations of the formulas and statistical concepts.
Population Standard Deviation
Formula: σ = √[(Σ(x - μ)²) / N]
Sample Standard Deviation
Formula: s = √[(Σ(x - x̄)²) / (n - 1)]
Understanding Standard Deviation
Population vs Sample
- Population Standard Deviation (σ): Used when you have data for the entire population
- Sample Standard Deviation (s): Used when you have data from a sample of the population
- Key Difference: Sample standard deviation uses (n-1) in the denominator (Bessel's correction)
What Standard Deviation Tells Us
- Low Standard Deviation: Data points are close to the mean
- High Standard Deviation: Data points are spread out from the mean
- Zero Standard Deviation: All data points are identical
Standard Deviation Formulas
Population Standard Deviation
σ = √[(Σ(x - μ)²) / N]
Where: σ = population standard deviation, μ = population mean, N = population size
Where: σ = population standard deviation, μ = population mean, N = population size
Sample Standard Deviation
s = √[(Σ(x - x̄)²) / (n - 1)]
Where: s = sample standard deviation, x̄ = sample mean, n = sample size
Where: s = sample standard deviation, x̄ = sample mean, n = sample size
Related Calculators
- Mean Calculator - Calculate average and other statistics
- Percentage Calculator - Calculate percentages and ratios
- Math Calculators - More mathematical tools