Variance & Standard Deviation (Discrete)
Variance and standard deviation measure spread, with \(\operatorname{Var}(X)=E(X^{2})-\big(E(X)\big)^{2}\) and standard deviation \(\sqrt{\operatorname{Var}(X)}\).
Part of the NSW Year 12 Mathematics Advanced course, in the Statistical analysis area of study (Random variables focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
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.