Math
Standard Deviation Calculator
Mean, variance, and standard deviation for any dataset.
Population vs sample standard deviation
Population SD (σ): use when you have data for the entire group. Sample SD (s): use when your data is a sample from a larger population. Sample SD divides by n−1 (Bessel's correction) to avoid underestimating variability.
The formula
σ = √[ Σ(xᵢ − μ)² ÷ N ] (population)
s = √[ Σ(xᵢ − x̄)² ÷ (N−1) ] (sample)
The 68-95-99.7 rule
In a normal distribution: 68% of values fall within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD. This is the foundation of statistical quality control.
⚡ CalcWolf Insight
The empirical rule: 68% of data falls within 1 standard deviation, 95% within 2, and 99.7% within 3.
Frequently asked questions
When should I use sample vs population standard deviation?▾
Use sample (s) when analyzing a subset of a larger group — which is most real-world cases. Use population (σ) when you have data for every single member of the group.
What is a good standard deviation?▾
It depends on context. A small SD relative to the mean indicates consistent data. In manufacturing, low SD means quality. In investing, higher SD means more risk.
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