Variance (Expectation) Calculator
Variance using expected values.
Formula first
Overview
The variance of a random variable measures the dispersion of its values around the mean, representing the average squared distance from the expected value. This specific algebraic identity is known as the computational formula for variance, as it simplifies calculations by utilizing the raw moments of the distribution.
Symbols
Variables
Var(X) = Variance, E[X^2] = Mean of Squares, \mu = Mean
Apply it well
When To Use
When to use: Use this formula when you have the raw moments of a distribution, such as the expected value of the variable and its square. It is particularly efficient for theoretical calculations of discrete or continuous probability distributions compared to the definition-based formula.
Why it matters: Variance is the foundation for risk assessment in finance, error analysis in engineering, and determining the reliability of experimental data. It allows researchers to quantify uncertainty and compare the volatility of different datasets.
Avoid these traps
Common Mistakes
- Using E[X]^2 instead of (E[X])^2.
- Swapping terms.
One free problem
Practice Problem
A discrete random variable has an expected value (mean) of 4 and an expected value of its squares (E[X²]) equal to 25. Determine the variance.
Solve for:
Hint: Calculate the square of the mean and subtract it from the expected value of the square.
The full worked solution stays in the interactive walkthrough.
References
Sources
- Probability and Statistics for Engineering and the Sciences, Jay L. Devore
- Wikipedia: Variance
- Britannica: Variance (statistics)
- Ross, Sheldon M. A First Course in Probability. 9th ed., Pearson, 2014.
- Edexcel A-Level Mathematics — Statistics