Standard Deviation
Calculate standard deviation, variance, and coefficient of variation
Enter Data
Separate values with commas
Theory & Formula
Standard Deviation & Variance
Standard deviation measures how spread out data points are from the mean. Variance is the square of standard deviation.
Variance
The average of squared differences from the mean. It measures the overall spread of data.
\(\text{Var}(X) = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2\)
Standard Deviation
The square root of variance, expressed in the same units as the original data. It shows typical deviation from the mean.
\(\sigma = \sqrt{\text{Var}(X)} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2}\)
Population vs Sample
Use population formulas when analyzing an entire population. Use sample formulas when analyzing a subset of a population.
Population: divide by n: \(\sigma^2 = \frac{1}{n} \sum (x_i - \mu)^2\)
Sample: divide by (n-1) for Bessel's correction: \(s^2 = \frac{1}{n-1} \sum (x_i - \bar{x})^2\)
Bessel's correction provides an unbiased estimate of population variance from sample data
Interpretation
- Low standard deviation: Data points are close to the mean
- High standard deviation: Data points are spread out from the mean
- Standard deviation has the same units as the original data