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Variance (Random Variable) Calculator

Measure of spread for a random variable.

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Variance

Formula first

Overview

Variance characterizes the spread of a random variable by calculating the average of the squared differences from the mean. This specific formula, known as the computational formula for variance, simplifies calculations by using the first and second moments of the distribution.

Symbols

Variables

Var(X) = Variance, E(X^2) = E(X^2), \mu = Mean E(X)

Variance
E(X^2)
Mean E(X)

Apply it well

When To Use

When to use: Apply this formula when analyzing the volatility of a dataset or when the raw moments of a distribution are provided. It is the standard method for determining the dispersion of discrete or continuous random variables in statistical modeling.

Why it matters: Variance is foundational in portfolio theory to measure financial risk and in scientific research to determine the reliability of experimental results. It allows for the derivation of the standard deviation, which translates spread back into the original units of the data.

Avoid these traps

Common Mistakes

  • Squaring E(X) first then subtracting.
  • Negative variance (impossible).

One free problem

Practice Problem

A discrete random variable has an expected value of 5 and the expected value of its square is 34. Calculate the variance of this distribution.

Mean E(X)5
E(X^2)34

Solve for:

Hint: Subtract the square of the mean from the expected value of X².

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Wikipedia: Variance
  2. A First Course in Probability by Sheldon Ross
  3. Probability and Statistics for Engineers and Scientists by Walpole, Myers, Myers, and Ye
  4. A First Course in Probability (Sheldon Ross)
  5. Ross, Sheldon M. A First Course in Probability. 8th ed. Pearson Prentice Hall, 2010.
  6. Edexcel A-Level Mathematics — Statistics (Random Variables)