Variance of a Binomial Distribution Equation, Derivation & Rearrangements

Core idea

Overview

The variance represents the spread of the probability distribution around the mean. It is derived from the properties of independent Bernoulli trials, specifically the sum of variances of n independent Bernoulli variables. A higher variance indicates greater uncertainty in the outcome of the binomial experiment.

When to use: Use this when you need to quantify the dispersion of outcomes in a binomial experiment with a fixed number of trials and a constant probability of success.

Why it matters: It is essential in quality control and risk management for predicting the volatility or deviation from the expected average in binary outcomes.

Symbols

Variables

n = Number of trials, p = Probability of success

n

Number of trials

Variable

p

Probability of success

Variable

Walkthrough

Derivation

Derivation of Variance of a Binomial Distribution

This derivation uses the variance formula Var(X) = E(X²) - [E(X)]² applied to a sum of independent Bernoulli trials.

X follows a binomial distribution B(n, p)
X can be expressed as the sum of n independent Bernoulli random variables Y₁, Y₂, ..., Yn, where each Yᵢ ~ B(1, p)

1

Define Variance for Bernoulli Trials

Since Yᵢ is a Bernoulli trial, E(Yᵢ) = p. Because Yᵢ takes values 0 or 1, Yᵢ² = Yᵢ, thus E(Yᵢ²) = E(Yᵢ) = p.

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

Note: Remember that for a Bernoulli variable, the variance is p(1-p).

2

Calculate Variance of a Single Trial

Substituting the expected values into the variance formula gives the variance for a single success-failure trial.

Var (Y_{i}) = p - p^{2} = p (1 - p)

Note: Let q = 1-p, then Var(Yᵢ) = pq.

3

Sum of Independent Variables

Because the trials in a binomial distribution are independent, the variance of the sum is the sum of the variances.

Var (X) = Var (i = 1 \sum n Y_{i}) = i = 1 \sum n Var (Y_{i})

Note: Independence is a strictly required condition for this step.

4

Final Result

Summing the variance of n identical and independent Bernoulli trials yields the total variance for the binomial distribution.

Var (X) = n \times p (1 - p) = n p (1 - p)

Note: This is often written as npq where q = 1-p.

Result

Var (X) = n \times p (1 - p) = n p (1 - p)

Source: A-Level Mathematics Specification: Statistics and Probability

Free formulas

Rearrangements

Solve for $n$

Make n the subject

\frac{Var ( X )}{p ( 1 - p )} = n

Isolate n by dividing the variance by the product of the probability of success and the probability of failure.

Difficulty: 2/5

Solve for $p$

Make p the subject

p = \frac{n \pm ( - n ) ^{2} - 4 n Var ( X )}{2 n}

Rearrange the equation into a quadratic form in terms of p and solve using the quadratic formula.

Difficulty: 4/5

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

Why it behaves this way

Intuition

Imagine a path of n 'decision points' (like a Galton board). Each step is a choice between success (p) and failure (1-p). The variance is the 'spread' or 'volatility' of the final count, which is maximized when p=0.5 (maximum uncertainty at each step) and scales linearly with the number of opportunities (n).

Var(X)

Variance

A measure of how much the number of successes deviates from the average expected value over many repetitions.

n

Number of trials

The 'scale' of the experiment; more trials provide more opportunities for outcomes to scatter, increasing total variance.

p

Probability of success

The 'tuning dial' for outcomes; if p is close to 0 or 1, outcomes are very predictable, but at p=0.5, uncertainty is highest.

(1-p)

Probability of failure

The 'mirror' probability; variance depends on the interplay between the chance of winning and the chance of losing.

Signs and relationships

Multiplication (np(1-p)): Variance is additive for independent trials; since each trial has variance p(1-p), summing this n times results in the product.

One free problem

Practice Problem

A coin is tossed 20 times. Given the probability of landing heads is 0.5, calculate the variance of the number of heads.

Number of trials0.5

Solve for: Var(X)

Hint: Use the formula np(1-p) where n=20 and p=0.5.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In a mathematical model involving Variance of a Binomial Distribution, Variance of a Binomial Distribution is used to calculate Number of trials from the measured values. The result matters because it helps estimate likelihood and make a risk or decision statement rather than treating the number as certainty.

Study smarter

Tips

Remember that q is often used as a shorthand for (1-p).
Variance is always positive, so check your inputs if you get a negative result.
Take the square root of this value if you need to find the standard deviation.

Avoid these traps

Common Mistakes

Confusing the variance with the standard deviation (which requires a square root).
Using the wrong probability value, such as using the probability of failure instead of success.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

This derivation uses the variance formula Var(X) = E(X²) - [E(X)]² applied to a sum of independent Bernoulli trials.

Use this when you need to quantify the dispersion of outcomes in a binomial experiment with a fixed number of trials and a constant probability of success.

It is essential in quality control and risk management for predicting the volatility or deviation from the expected average in binary outcomes.

Confusing the variance with the standard deviation (which requires a square root). Using the wrong probability value, such as using the probability of failure instead of success.