Standard Deviation Calculator
Adjust the inputs below. Results update as you type.
How it works
Population SD: σ = √(Σ(xᵢ−μ)²/N). Sample SD: s = √(Σ(xᵢ−x̄)²/(N−1)). Divide by N−1 (Bessel's correction) for samples to get an unbiased variance estimate. SD measures spread around the mean—useful alongside mean for comparing datasets. For normal distributions: 68% of data within ±1SD, 95% within ±2SD, 99.7% within ±3SD (empirical rule). Outliers inflate SD significantly.
Input guidance
- Check units and decimal placement before running calculations.
- For percentage and ratio tasks, define the baseline value clearly.
- Use rounded output carefully and keep full precision for intermediate steps.
The formula
Population SD: σ = √(Σ(xᵢ − μ)² ÷ N). Sample SD: s = √(Σ(xᵢ − x̄)² ÷ (N − 1)). The N−1 (Bessel's correction) gives an unbiased estimate for samples.
Worked example
For 2, 4, 6, 8 (mean 5), the squared deviations sum to 20; population SD = √(20÷4) = √5 ≈ 2.24.
More examples to test
- Baseline example: solve with clean numbers first to verify formula direction.
- Edge-case example: test near-zero or boundary values to avoid interpretation errors.
How to interpret results
Use outputs as quick checks, then verify with step-by-step work for graded or high-stakes use.
When this can be inaccurate
Errors usually come from unit mismatch, wrong baseline assumptions, or rounding too early in multistep problems.
Change history
- July 2026: Published sample vs population standard deviation guidance.
Site-wide corrections also appear on the corrections log.
Build math and logic skills with guided lessons
Educational platform recommendation.
Frequently asked questions
Population or sample standard deviation?+
Use population SD when you have every data point; use sample SD (dividing by N−1) when your data is a sample of a larger group.
What does standard deviation tell me?+
How spread out values are around the mean. A small SD means data clusters tightly; a large SD means it is widely dispersed.
What is the empirical rule?+
For normal data, about 68% falls within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD.
Why do my classroom and calculator results differ?+
Differences usually come from rounding conventions, order-of-operations handling, or different baseline assumptions.
What is the safest way to validate an answer?+
Recompute with units shown and check against a second method or manual arithmetic for critical problems.