Standard Deviation Calculator

Standard deviation (σ or s) is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. Core Definition: Standard deviation measures how spread out numbers are from their average (mean). It is the square root of variance. Mathematical Formulas: • Population Standard Deviation: σ = √[Σ(xi - μ)² / N] • Sample Standard Deviation: s = √[Σ(xi - x̄)² / (n-1)] Where: • xi = individual data points • μ = population mean, x̄ = sample mean • N = population size, n = sample size • Σ = summation symbol Key Characteristics: 1. Non-negative: Standard deviation is always ≥ 0 2. Zero indicates no variation: All values are identical 3. Units: Same as the original data 4. Sensitivity: Affected by outliers and extreme values Interpretation: • Low standard deviation: Data points cluster closely around the mean • High standard deviation: Data points are widely spread from the mean • Empirical Rule (68-95-99.7): In normal distribution, ~68% of data falls within ±1σ, ~95% within ±2σ, ~99.7% within ±3σ Applications: Quality control, risk assessment, scientific research, financial analysis, and any field requiring understanding of data variability.