Standard Deviation Calculator
Calculate the standard deviation, variance, and coefficient of variation for a data set. See also Variance Calculator and Mean Calculator.
How to Calculate Standard Deviation
Standard deviation measures how spread out values are from the mean. To calculate: (1) find the mean, (2) subtract the mean from each value and square the result, (3) find the average of those squared differences (divide by n for population, n−1 for sample), (4) take the square root. A low standard deviation means values are close to the mean; a high one means they are spread out.
Standard Deviation Formulas
Population: σ = √[Σ(xᵢ − μ)² / N]
Sample: s = √[Σ(xᵢ − x̄)² / (n − 1)]
Variance = σ² or s²
Coefficient of Variation = (σ / |μ|) × 100%
Example Calculation
Data: 10, 12, 23, 23, 16, 23, 21, 16
Mean = 144 / 8 = 18
Σ(xᵢ − x̄)² = 64 + 36 + 25 + 25 + 4 + 25 + 9 + 4 = 192
Sample Variance = 192 / 7 ≈ 27.428571
Sample Std Dev = √27.428571 ≈ 5.237229
Population vs Sample Standard Deviation
| Feature | Population (σ) | Sample (s) |
|---|---|---|
| Divisor | N | n − 1 |
| Use when | You have all data points | Data is a subset |
| Symbol | σ (sigma) | s |
| Bias | Exact | Corrected (Bessel's) |
Frequently Asked Questions
What is the difference between population and sample standard deviation?
Population standard deviation divides by N (total count) and is used when you have data for the entire population. Sample standard deviation divides by n−1 (Bessel's correction) and is used when your data is a sample from a larger population.
What does a high standard deviation mean?
A high standard deviation means the data points are spread far from the mean. A low standard deviation means they are clustered close to the mean.
What is the coefficient of variation?
The coefficient of variation (CV) is the standard deviation divided by the mean, expressed as a percentage. It allows comparison of variability between data sets with different units or scales.
Can standard deviation be negative?
No. Standard deviation is always zero or positive because it is the square root of variance, which is a sum of squared values. A standard deviation of zero means all values are identical.