How Standard Deviation Works
Population: σ = √(Σ(xᵢ − μ)² / N)
Sample: s = √(Σ(xᵢ − x̄)² / (N − 1))
Standard deviation measures the spread of data around the mean. A low SD means data points are close to the mean; a high SD means they're spread out.
Population vs. Sample
- Population (σ): Use when you have ALL data points in the group (entire class, all employees)
- Sample (s): Use when you have a subset (a survey, an experiment). Divides by N−1 (Bessel's correction) for an unbiased estimate.
Frequently Asked Questions
What does standard deviation tell you? About 68% of data falls within 1 SD of the mean, 95% within 2 SDs, and 99.7% within 3 SDs (for normal distributions).
Is a high standard deviation bad? Not necessarily. It depends on context. Investment returns with high SD mean more risk. Manufacturing with high SD means poor quality control.