Standard Deviation & Variance Calculator
Use our free online Standard Deviation Calculator to perform fast, accurate calculations with instant formulas, step-by-step arithmetic, and verified reference benchmarks.
Standard Deviation Calculator
Instant Live Computation • Zero Waiting
Adjust inputs on the left for instantaneous live computation.
How to Calculate: The Standard Deviation Calculator Formula
Measures the dispersion or spread of data points around the arithmetic mean. Sample formula uses Bessel's correction (n - 1).
Step-by-Step Calculation Guide
- Input comma-separated numbers.
- Calculate the arithmetic mean (average).
- Calculate squared differences from the mean for each point.
- Divide sum of squared deviations by n - 1 (sample) or N (population) and take the square root.
Practical Standard Deviation Calculator Examples
Dataset: 10, 12, 23, 23, 16, 23, 21, 16.
Dataset: 50, 51, 49, 50, 50.
Empirical Rule (68–95–99.7) for Normal
Official reference values and benchmark classifications based on standard institutional guidelines.
| Standard Deviation Band | Percentage of Data Included | Odds Outside Range | Z-Score Range |
|---|---|---|---|
| ± 1 Standard Deviation (μ ± 1σ) | 68.27% | 1 in 3 points | -1.0 to +1.0 |
| ± 2 Standard Deviations (μ ± 2σ) | 95.45% | 1 in 22 points | -2.0 to +2.0 |
| ± 3 Standard Deviations (μ ± 3σ) | 99.73% | 1 in 370 points | -3.0 to +3.0 |
| ± 6 Standard Deviations (Six Sigma) | 99.99966% | 3.4 parts per million | -6.0 to +6.0 |
Frequently Asked Questions about Standard Deviation Calculator
When should you use Sample vs Population standard deviation?
Use Population (N) when you have measured every single member of the group. Use Sample (n - 1) when inferring conclusions about a broader population from a smaller sample.
What is Bessel's correction?
Dividing by n - 1 instead of n corrects the inherent bias in sample variance, preventing underestimation of the true population spread.
What does a high standard deviation indicate?
A high standard deviation indicates that data points are widely dispersed across a broad range of values away from the mean.
How is variance related to standard deviation?
Variance is simply the standard deviation squared (Variance = SD²). Standard deviation is in the original units of measurement, making it more interpretable.