Calculate Standard Deviation Step by Step
Find population and sample standard deviation with detailed step-by-step solutions. Includes the 68-95-99.7 empirical rule and coefficient of variation.
How It Works
σ = √(Σ(x-μ)²/n), s = √(Σ(x-μ)²/(n-1))Population standard deviation (σ) divides by n — use when you have the entire population. Sample standard deviation (s) divides by n-1 (Bessel's correction) — use when you have a sample of a larger population.
Quick Tips
Population vs Sample
Use σ (divide by n) when data IS the entire population. Use s (divide by n-1) when data is a SAMPLE.
Bessel's Correction
Dividing by n-1 instead of n gives an unbiased estimate of population variance from a sample.
Empirical Rule
68% within 1σ, 95% within 2σ, 99.7% within 3σ of the mean (for normal distributions).
CV
Coefficient of Variation = (σ/μ)×100%. Compares variability between datasets with different units or scales.
Step-by-Step Instructions
- 1Enter your data as comma-separated numbers.
- 2Click Calculate to see both population and sample standard deviation.
- 3Review the step-by-step solution showing each calculation.
Frequently Asked Questions
Should I use population or sample standard deviation?▼
Use population (σ) if your data includes every member of the population (e.g., all students in a class). Use sample (s) if your data is a subset of a larger population (e.g., 50 students from a school of 500).
Why divide by n-1 instead of n?▼
Bessel's correction (n-1) compensates for the fact that a sample tends to underestimate population variance. Without it, sample variance would be biased low.
What is a good standard deviation?▼
It depends on context. For exam scores out of 100, σ=10 means moderate spread. σ=5 means tight clustering. σ=20 means wide variation. Compare CV (coefficient of variation) across different datasets.