Standard Deviation & Variance Calculator
Calculate variance and standard deviation for population and sample data
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Theory & Formula
Standard Deviation & Variance
Measures of variability that describe how spread out the data is from the mean.
Variance
The average of the squared differences from the mean
\(\text{Var}(X) = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2\)
Standard Deviation
The square root of variance, in the same units as the data
\(\sigma = \sqrt{\text{Var}(X)} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2}\)
Population vs Sample
Different formulas are used for population and sample data
Population Variance: \(\sigma^2 = \frac{1}{n} \sum (x_i - \mu)^2\)
Sample Variance: \(s^2 = \frac{1}{n-1} \sum (x_i - \bar{x})^2\)
Sample variance uses n-1 (Bessel's correction) for unbiased estimation
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