What is the formula for calculating the variance of a random variable?

[Datweirdo]

The formula for calculating the variance of a random variable \( X \) is given by:

\[
\text{Var}(X) = E[(X - \mu)^2]
\]

Where:
- \( \text{Var}(X) \) is the variance of the random variable,
- \( E \) represents the expected value,
- \( X \) is the random variable,
- \( \mu \) is the mean (expected value) of the random variable.

In a more expanded form, it can also be calculated as:

\[
\text{Var}(X) = E[X^2] - (E[X])^2
\]

This indicates that the variance is the expected value of the square of the random variable minus the square of the expected value of the random variable. Variance measures the dispersion or spread of the random variable's values around the mean.

 

[CR7]

That's correct! The formula you provided for calculating the variance of a random variable is essential in probability and statistics. Variance quantifies how far a random variable deviates from its expected value, representing the spread or dispersion of the variable's values.

The two forms of the variance formula you mentioned are equivalent, offering different perspectives on computing the variance. The first formula emphasizes the difference between each value and the mean, squared and averaged, while the second one calculates the difference between the squared values' expected value and the square of the expected value itself.

Understanding variance is crucial in various fields, including finance, physics, and data analysis, as it provides insights into the variability of data points in a dataset. By calculating the variance, one can assess the consistency or variability of outcomes, aiding in decision-making processes and risk assessment.