Variance of a Random Variable
Spread of a distribution.
Key idea
Spread of a distribution.
In depth
Var(X) = E[(X − μ)²] = E[X²] − (E[X])². It measures the spread around the mean. For independent variables, variances add.
Worked example
A die: E[X²] = 91/6 ≈ 15.17, minus 3.5² = 12.25, variance ≈ 2.92.
Formula
Real-world uses
- Measuring how risky a daily sales forecast is compared with another product
Common mistake
Adding standard deviations instead of variances.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Variance of a Random Variable?
Spread of a distribution. It is a topic at the advanced level in Probability & Statistics.
What is the formula for Variance of a Random Variable?
The key formula is Var(X) = E[X²] − (E[X])².
Where is Variance of a Random Variable used in real life?
Real-world examples: Measuring how risky a daily sales forecast is compared with another product.
What should I learn before Variance of a Random Variable?
Make sure you are comfortable with Expected Value, the previous topic in Probability & Statistics.
What is a common mistake with Variance of a Random Variable?
Adding standard deviations instead of variances.
How can I practise Variance of a Random Variable?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Variance of a Random Variable?
Move on to Linearity of Expectation next.
Related advanced topics in Probability & Statistics
- Quartiles and IQR – Quartiles split data into four parts; IQR = Q3 − Q1.
- Box Plots – Five-number summary visualised.
- Outliers – Values beyond 1.5 IQR from the quartiles.
- Variance – Average squared deviation from the mean.
- Standard Deviation – Square root of variance; typical spread.
- Sample vs Population Variance – Divide by n−1 for sample variance.