How to Use This Calculator
- Paste or enter numbers: Separate values with commas, spaces, or line breaks.
- Compare sample vs population: Use Sample Standard Deviation (s) for experiments and surveys; use Population (σ) when analyzing the complete universal set.
- Review central tendency: Check mean, median, mode, and variance alongside standard deviation.
Formula and Calculation Steps
Mean (x̄) = ( Σ xᵢ ) ÷ n
Sample Variance (s²) = Σ (xᵢ − x̄)² ÷ (n − 1)
Sample Standard Deviation (s) = √[ Σ (xᵢ − x̄)² ÷ (n − 1) ]
Population Standard Deviation (σ) = √[ Σ (xᵢ − μ)² ÷ N ]Worked Examples and Reference Table
| Dataset | Mean | Sample SD (s) | Spread Interpretation |
|---|---|---|---|
| 10, 10, 10, 10 | 10.0 | 0.0000 | Zero variance (all identical) |
| 9, 10, 10, 11 | 10.0 | 0.8165 | Tightly clustered around average |
| 1, 5, 15, 19 | 10.0 | 8.5635 | Wide dispersion across scale |
| 10, 12, 16, 23 | 15.25 | 5.7373 | Moderate natural variance |
Frequently Asked Questions
What is the difference between sample and population standard deviation?
Population standard deviation (σ) divides by N and is used when your data represents every single member of the group. Sample standard deviation (s) uses Bessel's correction (dividing by N − 1) to provide an unbiased estimate when using a representative sample subset.
What does standard deviation measure?
Standard deviation measures the dispersion or spread of values around the arithmetic mean. A small standard deviation indicates that data points are clustered closely to the average, while a large standard deviation indicates wide variability.
How do variance and standard deviation relate?
Standard deviation is the square root of variance: s = √s². Variance is expressed in squared units, while standard deviation is expressed in the original units of measurement.
What is the 68-95-99.7 Empirical Rule?
In a normal bell-curve distribution, approximately 68% of all values fall within 1 standard deviation of the mean, 95% fall within 2 standard deviations, and 99.7% fall within 3 standard deviations.
Can standard deviation ever be negative?
No. Because differences from the mean are squared before averaging, standard deviation is always greater than or equal to zero. It equals zero only if every single number in the dataset is identical.