Standard Deviation & Variance Formula

Mathematics & Education Formula • Published on July 30, 2026 • Last updated July 30, 2026

Mathematical Equation

$$\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}, \quad s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$$ me

Variable Definitions

σ / s

Population / Sample standard deviation

μ / x̄

Population / Sample mean average

N / n

Total count of data points

Detailed Explanation

In-Depth Guide

Standard deviation measures the dispersion or spread of data points relative to their mean average. A low standard deviation indicates values cluster close to the mean.

How to Calculate: Step-by-Step

1. Calculate the mean average of all numbers. 2. Subtract mean from each data point and square the difference. 3. Sum all squared differences. 4. Divide by N (for population) or n-1 (for sample) to get Variance. 5. Take square root of variance.

Worked Calculation Example

Data set [2, 4, 4, 4, 5, 5, 7, 9]: - Mean = 5. Variance = 4. - Standard Deviation = √4 = 2.0.

Common Use Cases

  • Financial stock volatility risk assessment
  • Quality control manufacturing
  • Academic test grade distribution

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