Standard Deviation Calculator
Enter a list of numbers to calculate mean, standard deviation, variance, and more descriptive statistics.
Standard Deviation Formulas
| Statistic | Formula | Notes |
|---|---|---|
| Mean | x̄ = Σxᵢ / N | Sum of all values divided by count |
| Population Variance | σ² = Σ(xᵢ − x̄)² / N | Divide by N |
| Sample Variance | s² = Σ(xᵢ − x̄)² / (N−1) | Divide by N−1 (Bessel's correction) |
| Standard Deviation | σ = √variance | Square root of variance |
Frequently Asked Questions
What is standard deviation?
Standard deviation measures how spread out a set of numbers are from their mean (average). A low standard deviation means values cluster closely around the mean; a high standard deviation means they are more spread out. It is calculated as the square root of the variance.
What is the difference between population and sample standard deviation?
Population standard deviation (σ) divides by N — used when you have data for an entire population. Sample standard deviation (s) divides by N−1 (Bessel's correction) — used when your data is a sample from a larger population. The N−1 correction makes the sample estimate less biased. In most real-world statistics, sample SD is used.
What is variance?
Variance is the average of the squared differences from the mean. Standard deviation is simply the square root of variance, which brings the result back to the same units as the original data. Variance is useful in calculations but harder to interpret directly because the units are squared.
How do I interpret standard deviation?
For data that follows a normal (bell curve) distribution: about 68% of values fall within 1 standard deviation of the mean, about 95% within 2 standard deviations, and about 99.7% within 3 standard deviations. This is known as the empirical rule (68-95-99.7 rule).
What is the coefficient of variation?
The coefficient of variation (CV) is the standard deviation divided by the mean, expressed as a percentage: CV = (σ / mean) × 100%. It normalises spread relative to the mean, making it useful for comparing variability between datasets with different scales or units. A CV below 15% generally indicates low variability.