How does variance measure the spread of a probability distribution around its mean?

In probability theory, the variance measures the spread of a distribution by calculating the weighted average of the squared deviations of a random variable from its mean ($\mu$). By squaring the difference between each value ($x$) and the mean, the variance ensures that deviations are treated as positive values, preventing them from canceling each other out.1

The calculation depends on the nature of the random variable:

• Discrete variables: The variance is the sum of the squared deviations weighted by the probability mass function, expressed as:
  $$\text{Var}(x)={\sigma }^2=\sum (x-\mu {)}^{2}f(x)$$1

• Continuous variables: The variance is the integral of the squared deviations weighted by the probability density function:
  $$\text{Var}(x)={\sigma }^2=\int (x-\mu {)}^{2}f(x)dx$$1

Because variance is measured in squared units, the standard deviation—the positive square root of the variance—is often preferred as a measure of dispersion because it shares the same units as the original data.12