How is the variance of a random variable calculated?

In probability and statistics, the variance of a random variable is a measure of the weighted average of the squared deviations from the mean ($\mu$). It is denoted as $\text{Var}(X)$ or ${\sigma }^2$ and is defined as $\text{Var}(X)=E\{[X-E(X){]}^2\}$.12

The calculation method depends on whether the random variable is discrete or continuous:

• Discrete Random Variables: The variance is calculated using the probability mass function, $f(x)$:
  $$\text{Var}(X)=\sum (x-\mu {)}^{2}f(x)$$
• Continuous Random Variables: The variance is calculated using the probability density function, $f(x)$:
  $$\text{Var}(X)=\int (x-\mu {)}^{2}f(x)dx$$

Alternatively, the variance can be calculated using the formula $\text{Var}(X)=E(X^2)-\{E(X){\}}^2$. The standard deviation is the positive square root of the variance, providing a measure of dispersion in the same units as the original variable.231