Variance & Standard Deviation (Discrete)
Learn variance and standard deviation for discrete random variables in NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-07). Two distributions can share the same mean yet differ in spread, and these measures show how far outcomes typically fall from the centre.
You will learn to calculate variance from a probability table using \(E(X^2)\) minus the square of the mean, take its square root for the standard deviation, and compare the spread of different distributions — useful for weighing up risk.
Theory
Variance measures spread about the mean; the standard deviation is its square root. This Year 12 Mathematics Advanced topic (MAV-12-07) computes both for a discrete variable.
\(\mathrm{Var}(X)=\sum x^{2}P(x)-\mu^{2}=E(X^{2})-\mu^{2}\), and \(\sigma=\sqrt{\mathrm{Var}(X)}\). Find \(\mu\) first, then \(E(X^{2})=\sum x^{2}P(x)\).
Method
- Find \(\mu=E(X)\).
- Find \(E(X^{2})=\sum x^{2}P(x)\).
- Subtract \(\mu^{2}\); square-root for \(\sigma\).
| \(\mathrm{Var}(X)\) | \(=\) | \(10-9=1\) |
| \(\mu\) | \(=\) | \(0.5\) |
| \(\mathrm{Var}(X)\) | \(=\) | \(0.5-0.25=0.25\) |
| \(\sigma\) | \(=\) | \(\sqrt9=3\) |
| \(E(X^{2})\) | \(=\) | \(5\) |
| \(\mathrm{Var}(X)\) | \(=\) | \(5-4=1\) |
Common pitfalls
Frequently asked questions
How do you find the variance of a discrete random variable?
Compute E of X squared as the sum of x squared times P of x, then subtract the mean squared.
What is the standard deviation?
The square root of the variance; it has the same units as the variable.
Why subtract mu squared?
Because the variance is the mean of the squares minus the square of the mean.
What does a larger standard deviation mean?
The values are more spread out from the mean.