Variance Calculator

Variance measures the average squared deviation from the mean:

Population variance:

$$\sigma^2 = \frac{\sum_{i=1}^{N}(x_i - \mu)^2}{N}$$

Sample variance:

$$s^2 = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n - 1}$$

Step-by-step process:

1. Enter your data and set the count of values.
2. Choose Population or Sample based on your data context.
3. The calculator finds the mean \(\bar{x}\) of your values.
4. For each value, it computes the deviation from the mean \((x_i - \bar{x})\).
5. Each deviation is squared to eliminate negative signs: \((x_i - \bar{x})^2\).
6. The squared deviations are summed to get the sum of squares (SS).
7. SS is divided by \(N\) (population) or \(N-1\) (sample) to yield variance.

Why square the deviations? Because the sum of raw deviations from the mean always equals zero (positive and negative deviations cancel out). Squaring ensures all contributions are positive and gives extra weight to large deviations.

The sum of squared deviations (SS) is a critical intermediate quantity used throughout statistics. It appears in ANOVA, regression analysis, and the computation of many test statistics.

Variance is measured in squared units of the original data. For instance, if your data is in meters, the variance is in square meters. This can make direct interpretation less intuitive than standard deviation, which is in the same units as the data.

• Variance = 0: All values are identical; there is no variability.
• Small variance: Values are tightly clustered around the mean.
• Large variance: Values are widely dispersed from the mean.

Variance is additive for independent variables: \(\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)\) when X and Y are independent. This property makes variance especially useful in theoretical statistics and portfolio risk analysis.