Variance (Random Variable) Equation, Derivation & Rearrangements

Core idea

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.

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.

Symbols

Variables

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

V a r (X)

Variance

V a r iab l e

E (X^{2})

E(X^2)

V a r iab l e

μ

Mean E(X)

V a r iab l e

Walkthrough

Derivation

Derivation of Variance of a Random Variable

Variance measures expected squared deviation from the mean. Expanding the definition gives the useful identity Var(X) = E( $X^{2}$ ) − [E(X)]^2.

X is a discrete random variable.
E(X) exists.

1

Start from the Definition:

Variance is the expected value of squared deviations from the mean $μ$ .

Var (X) = E [(X - μ)^{2}]

2

Expand the Square:

Expand $(X - μ)^{2}$ before applying expectation.

E [X^{2} - 2 μ X + μ^{2}]

3

Use Linearity of Expectation:

Expectation distributes over sums. Since $μ$ is constant, $E (μ^{2}) = μ^{2}$ .

E (X^{2}) - 2 μ E (X) + E (μ^{2})

4

Substitute E(X)=μ and Simplify:

Replace $E (X)$ with $μ$ and simplify.

E (X^{2}) - 2 μ^{2} + μ^{2} = E (X^{2}) - μ^{2}

5

Final Identity:

Variance equals the mean of squares minus the square of the mean.

Var (X) = E (X^{2}) - [E (X)]^{2}

Result

Var (X) = E (X^{2}) - [E (X)]^{2}

Source: Edexcel A-Level Mathematics — Statistics (Random Variables)

Free formulas

Rearrangements

Solve for $V a r (X)$

Variance (Random Variable)

V a r (X) = E (X^{2}) - μ^{2}

This formula defines the variance of a random variable X in terms of its expected value, often expressed using the mean symbol $μ$ .

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

The graph of variance against the expected value of a random variable forms a downward-opening parabola. This parabolic shape arises because the variance is defined by the difference between the mean of the squares and the square of the mean, resulting in a quadratic relationship with respect to the expected value.

Graph type: quadratic

Why it behaves this way

Intuition

Imagine a histogram or probability distribution curve for a random variable; the variance quantifies the 'width' of this distribution, with a larger variance indicating a flatter, more spread-out shape, and a smaller

Var(X)

The average squared deviation of a random variable's values from its mean.

A quantitative measure of how 'spread out' or dispersed the values of a random variable are around its central tendency.

E(X)

The expected value or mean of the random variable X.

Represents the central tendency or the long-run average value that the random variable X is expected to take.

E (X^{2})

The expected value of the square of the random variable X.

Reflects the average of the squared magnitudes of the variable, which includes contributions from both its mean and its spread.

(E (X))^{2}

The square of the expected value (mean) of the random variable X.

This term accounts for the contribution of the mean's position to E(

X^{2}

); it is subtracted to isolate the true spread around the mean.

Signs and relationships

-(E(X))^2: The subtraction of (E(X))^2 from E( $X^{2}$ ) is essential because E( $X^{2}$ ) measures the average of the squared values from the origin. By subtracting the square of the mean, we effectively shift the reference point from the

Free study cues

Insight

Canonical usage

The variance of a random variable X will have units that are the square of the units of X, ensuring dimensional consistency within the equation.

Common confusion

Confusing the units of variance (squared units of the random variable) with the units of standard deviation (original units of the random variable).

Dimension note

If the random variable X is dimensionless (e.g., a count, a probability, a ratio, or an index), then its variance Var(X) will also be dimensionless.

Unit systems

$V a r (X)$ [unit of X]^2 · The unit of variance is the square of the unit of the random variable X.

$E (X)$ [unit of X] · The expected value has the same unit as the random variable X.

$E (X^{2})$ [unit of X]^2 · The expected value of X squared has units that are the square of the unit of X.

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: $V$

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

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Variance of dice rolls.

Study smarter

Tips

Subtract the square of the mean, not just the mean itself.
Variance must be zero or positive; a negative result indicates a calculation error.
Remember that the units of variance are the square of the original units.

Avoid these traps

Common Mistakes

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

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Variance measures expected squared deviation from the mean. Expanding the definition gives the useful identity Var(X) = E(X^2) − [E(X)]^2.

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.

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.

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