What is Standard Deviation?
Standard deviation (σ or s) measures how spread out the values in a dataset are relative to the mean. A low standard deviation means that values are clustered close to the mean; a high value means they are spread far apart.
Population vs. Sample Standard Deviation
- Population (σ): Used when your dataset contains the entire population. Divides by N.
- Sample (s): Used when your dataset is a sample from a larger population. Divides by N−1 (Bessel's correction) to give an unbiased estimate.
Formulas
Population σ = √[Σ(xᵢ − μ)² / N]
Sample s = √[Σ(xᵢ − x̄)² / (N−1)]
Where μ / x̄ = mean, N = count
Sample s = √[Σ(xᵢ − x̄)² / (N−1)]
Where μ / x̄ = mean, N = count
How to Interpret Standard Deviation
- In a normal distribution, ~68% of values fall within 1σ of the mean
- ~95% fall within 2σ of the mean
- ~99.7% fall within 3σ of the mean (the 68-95-99.7 rule)