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Variance (Expectation)

Variance using expected values.

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V a r (X) = E [X^{2}] - (E [X])^{2}

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Core idea

Overview

The variance of a random variable measures the dispersion of its values around the mean, representing the average squared distance from the expected value. This specific algebraic identity is known as the computational formula for variance, as it simplifies calculations by utilizing the raw moments of the distribution.

When to use: Use this formula when you have the raw moments of a distribution, such as the expected value of the variable and its square. It is particularly efficient for theoretical calculations of discrete or continuous probability distributions compared to the definition-based formula.

Why it matters: Variance is the foundation for risk assessment in finance, error analysis in engineering, and determining the reliability of experimental data. It allows researchers to quantify uncertainty and compare the volatility of different datasets.

Symbols

Variables

Var(X) = Variance, E[X^2] = Mean of Squares, \mu = Mean

V a r (X)

Variance

V a r iab l e

E [X^{2}]

Mean of Squares

V a r iab l e

μ

Mean

V a r iab l e

Walkthrough

Derivation

Derivation of Variance using Expectation

Variance measures spread around the mean and can be rewritten as $Var$ (X)=E[ $X^{2}$ ]-(E[X])^2, which is often easier to compute.

X is a random variable with finite mean $μ$ =E[X] and finite second moment E[ $X^{2}$ ].
Expectation is linear: E[aX+bY]=aE[X]+bE[Y].

Start from the definition:

Variance is defined as the expected squared deviation from the mean $μ$ .

Var (X) = E [(X - μ)^{2}]

Expand the square inside the expectation:

Use algebra to expand the bracket before applying expectation.

(X - μ)^{2} = X^{2} - 2 μ X + μ^{2}

Apply linearity of expectation:

Expectation distributes over sums; $μ$ is a constant so it can be pulled out.

E [X^{2} - 2 μ X + μ^{2}] = E [X^{2}] - 2 μ E [X] + E [μ^{2}]

Substitute E[X]=μ and simplify:

Since E[X]= $μ$ and E[ $μ^{2}$ ]= $μ^{2}$ , the expression simplifies to E[ $X^{2}$ ]-(E[X])^2.

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

Note: Final form: $Var$ (X)=E[ $X^{2}$ ]-(E[X])^2.

Result

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

Source: Edexcel A-Level Mathematics — Statistics

Free formulas

Rearrangements

Solve for $V a r (X)$

Variance (Expectation)

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 and the expected value of its square. The steps show the formula with V (Var(X)) as the subject, followed by the common substitution of E[X] with $μ$ fo...

Difficulty: 2/5

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

Visual intuition

Graph

The graph of Variance (V) against the independent variable (X) typically forms a parabolic shape. This occurs because the formula involves the square of the expected value, creating a quadratic relationship where the variance increases as the spread of the data grows.

Graph type: quadratic

Why it behaves this way

Intuition

Imagine a set of data points scattered along a number line; the variance quantifies the average squared 'wiggle room' or dispersion of these points around their central balancing point, the mean.

Var(X)

A quantitative measure of the spread or dispersion of the values of a random variable X around its mean.

A larger variance indicates that the data points are more spread out from the average value; a smaller variance means they are clustered closer to the average.

E[X]

The expected value, or mean, of the random variable X.

Represents the long-run average outcome if the random process were repeated many times, serving as the central tendency or 'balancing point' of the distribution.

E [X^{2}]

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

Represents the average of the squared values of the variable, which contributes to the overall 'magnitude' of the values, including their spread from zero.

Signs and relationships

- (E[X])^2: This subtraction transforms the second moment about the origin (E[ $X^{2}$ ]) into the second central moment (variance). By subtracting the square of the mean, the formula effectively 'centers' the distribution around its

Free study cues

Insight

Canonical usage

The variance Var(X) will have units that are the square of the units of the random variable X.

Common confusion

A common mistake is assuming that variance Var(X) has the same units as the random variable X, rather than the units of X squared. This can lead to misinterpretation of the scale of variability.

Dimension note

If the random variable X itself is dimensionless (e.g., a count, a probability, a score, or an index), then E[X], E[ $X^{2}$ ], and Var(X) will all be dimensionless.

Unit systems

$X$ Varies based on the quantity X represents · The random variable X can represent any measurable quantity (e.g., length, time, mass) or be dimensionless (e.g., a count, a score).

$E [X]$ Same as X · The expected value of X has the same units as X.

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

$V a r (X)$ Units of X squared · The variance of X has units that are the square of the units of X. For easier interpretation, the standard deviation (sqrt(Var(X))) is often used, which has the same units as X.

One free problem

Practice Problem

A discrete random variable has an expected value (mean) of 4 and an expected value of its squares (E[X²]) equal to 25. Determine the variance.

Mean of Squares25

Mean4

Solve for: $V$

Hint: Calculate the square of the mean and subtract it from the expected value of the square.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Measuring variability in test scores.

Study smarter

Tips

Always ensure Ex2 is greater than or equal to mu² to avoid impossible negative variance results.
Remember that variance units are the square of the original measurement units.
Square the mean first before performing the subtraction from the second moment.

Avoid these traps

Common Mistakes

Using E[X]^2 instead of (E[X])^2.
Swapping terms.

Common questions

Frequently Asked Questions

Variance measures spread around the mean and can be rewritten as \text{Var}(X)=E[X^2]-(E[X])^2, which is often easier to compute.

Use this formula when you have the raw moments of a distribution, such as the expected value of the variable and its square. It is particularly efficient for theoretical calculations of discrete or continuous probability distributions compared to the definition-based formula.

Variance is the foundation for risk assessment in finance, error analysis in engineering, and determining the reliability of experimental data. It allows researchers to quantify uncertainty and compare the volatility of different datasets.

Using E[X]^2 instead of (E[X])^2. Swapping terms.