Class 12 Statistics Notes · GSEB

Mean and Variance

Mean and Variance — compute expected value, variance, and standard deviation of discrete random variables using the computational formula. GSEB Class 12 Commerce Statistics notes with step-by-step calculations.

Last updated: 22 Sep 2026

Notes

Mean (Expected Value)

Mean / Expected Value E(X)

The mean (or expected value) of a discrete random variable is the weighted average of all possible values, weighted by their probabilities. It represents the long-run average if the experiment were repeated infinitely.

Mean / Expected Value

E(X) = \sum x \cdot P(x)

Think of it as a long-run average

If you toss 2 coins 1000 times, the average number of heads will be close to E(X) = 1.0. The mean is what you "expect" on average — not what you get in any single trial.

Real-life analogy

A taxi driver earns ₹500 on 60% of days and ₹800 on 40% of days. His expected daily earning = 500 × 0.6 + 800 × 0.4 = ₹620. Some days he earns more, some less — but on average, ₹620 per day.

Variance and Standard Deviation

Variance V(X)

Variance measures how spread out the values of X are from the mean. A high variance means the values are widely scattered; a low variance means they cluster near the mean.

Computational Formula for Variance

V(X) = E(X^2) - [E(X)]^2

Standard Deviation

\sigma = \sqrt{V(X)}

Why E(X²) − [E(X)]²?

The formula V(X) = Σ(x − μ)²P(x) is conceptually clear but tedious. The computational form E(X²) − [E(X)]² gives the same answer with less arithmetic — just compute two sums and subtract.

V(X) = 0 means constant

If V(X) = 0, the random variable takes only one value (no variability). Example: a rigged die that always shows 3.

Try It: Interactive Calculator

Select a preset distribution and watch the mean, variance, and standard deviation compute step by step.

Mean & Variance Calculator

x	P(x)	x · P(x)	x² · P(x)
0	0.2500	0.0000	0.0000
1	0.5000	0.5000	0.5000
2	0.2500	0.5000	1.0000
Σ	1.0000	1.0000	1.5000

Mean E(X)

1.0000

Σx · P(x)

E(X²)

1.5000

Σx² · P(x)

Variance V(X)

0.5000

E(X²) − [E(X)]²

Standard Deviation

σ = √V(X) = 0.7071

Solved Examples

Solved Example

Problem

A random variable X has distribution: P(X=1) = 0.2, P(X=2) = 0.5, P(X=3) = 0.3. Find E(X) and V(X).

Solution

E(X) = 2.1, V(X) = 0.49, SD = 0.7

Solved Example

Problem

A fair die is rolled. Find E(X) and V(X).

Solution

E(X) = 3.5, V(X) = 2.917, SD = 1.708

Key Takeaways

E(X) = Σx · P(x) — the weighted average of all values, weighted by probability.
V(X) = E(X²) − [E(X)]² — the computational formula for variance (easier than Σ(x−μ)²P(x)).
Standard deviation σ = √V(X) — in the same units as X.
V(X) measures spread; V(X) = 0 means no variability (constant).
Mean gives the centre of the distribution; variance gives the spread.

Practice

X has distribution: P(0) = 0.1, P(1) = 0.4, P(2) = 0.3, P(3) = 0.2. Find E(X) and V(X).
If E(X) = 5 and V(X) = 2, what is E(X²)?
A random variable has E(X) = 4 and E(X²) = 20. Find V(X) and SD.