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Mean, variance, standard deviation and parameter recovery of a binomial variable

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Binomial moments

For X with a binomial distribution, n is the number of trials and p is the probability of success on each trial. These two parameters determine the centre and spread of the distribution

\[X\sim\operatorname{Bi}(n,p),\quad \operatorname{E}(X)=np,\quad \operatorname{Var}(X)=np(1-p)\]
  1. 1Identify n and p Use n for the fixed number of trials and p for the success probability on one trial \(\displaystyle X\sim\operatorname{Bi}\left(10,\frac{3}{5}\right)\)
  2. 2Find the mean Multiply the trial count by the success probability \(\displaystyle \operatorname{E}(X)=np=10\times\frac{3}{5}=6\)
  3. 3Find the variance Multiply n, p and the failure probability 1-p. \(\displaystyle \operatorname{Var}(X)=np(1-p)=10\times\frac{3}{5}\times\frac{2}{5}=\frac{12}{5}\)
  4. 4Take the principal root The standard deviation is the non-negative square root of the variance \(\displaystyle \operatorname{SD}(X)=\sqrt{\operatorname{Var}(X)}\)
Year 12, VCE Units 3 & 4, Chapter 11, 4 difficulty levels, Calculator allowedBuild a worksheet

Binomial moments

For X with a binomial distribution, n is the number of trials and p is the probability of success on each trial. These two parameters determine the centre and spread of the distribution

\[X\sim\operatorname{Bi}(n,p),\quad \operatorname{E}(X)=np,\quad \operatorname{Var}(X)=np(1-p)\]
  1. Identify n and p Use n for the fixed number of trials and p for the success probability on one trial \(\displaystyle X\sim\operatorname{Bi}\left(10,\frac{3}{5}\right)\)
  2. Find the mean Multiply the trial count by the success probability \(\displaystyle \operatorname{E}(X)=np=10\times\frac{3}{5}=6\)
  3. Find the variance Multiply n, p and the failure probability 1-p. \(\displaystyle \operatorname{Var}(X)=np(1-p)=10\times\frac{3}{5}\times\frac{2}{5}=\frac{12}{5}\)
  4. Take the principal root The standard deviation is the non-negative square root of the variance \(\displaystyle \operatorname{SD}(X)=\sqrt{\operatorname{Var}(X)}\)

Worked example

Find the mean, variance and standard deviation of the binomial variable shown

\[X\sim\operatorname{Bi}\left(8,\frac{3}{4}\right)\]
  1. \(\displaystyle 1-p=1-\frac{3}{4}=\frac{1}{4}\)Find the failure probability before using the variance rule
  2. \(\displaystyle \operatorname{E}(X)=np=8\times\frac{3}{4}=6\)Multiply n by p to find the mean
  3. \(\displaystyle \operatorname{Var}(X)=np(1-p)=8\times\frac{3}{4}\times\frac{1}{4}=\frac{3}{2}\)Multiply n, p and 1-p to find the variance
  4. \(\displaystyle \operatorname{SD}(X)=\sqrt{\operatorname{Var}(X)}=\sqrt{\frac{3}{2}}\)Take the principal square root of the variance

Common mistakes

Omitting the complement in the variance
✕\(\displaystyle \operatorname{Var}(X)=np=10\times\frac{3}{5}=6\)✓\(\displaystyle \operatorname{Var}(X)=np(1-p)=10\times\frac{3}{5}\times\frac{2}{5}=\frac{12}{5}\)
The variance must include the failure probability 1-p as well as n and p.
Squaring p in the mean
✕\(\displaystyle \operatorname{E}(X)=np^2=10\times\left(\frac{3}{5}\right)^2=\frac{18}{5}=3.6\)✓\(\displaystyle \operatorname{E}(X)=np=10\times\frac{3}{5}=6\)
The mean is n multiplied by p once, with no squared parameter
Using variance as standard deviation
✕\(\displaystyle \operatorname{SD}(X)=\frac{3}{2}\)✓\(\displaystyle \operatorname{SD}(X)=\sqrt{\frac{3}{2}}\)
Standard deviation is the principal square root of the variance

Difficulty levels

introCalculate the mean and variance from a stated binomial n and p.
standardCalculate a stated-precision standard deviation using the variance first
challengeRecover the success probability and trial count from a stated mean and variance
examChoose a correct moment rule and calculate the requested exact value

Questions about this topic

What do n and p mean in a binomial distribution?+

n is the fixed number of trials and p is the constant probability of success on each trial

Why does the variance include 1-p?+

The variance rule includes the success and failure probabilities

Why is standard deviation the square root of variance?+

Standard deviation measures spread in the original units, so it is the principal square root of the variance

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 11: Binomial distribution

In this chapter

Bernoulli trials and the Bernoulli distribution11A Bernoulli trials and the Bernoulli distribution: Finite events, probability assignments and complementsSolve a quadratic exactly with the quadratic formula11A Bernoulli trials and the Bernoulli distribution: Round to decimal places and significant figures11B Binomial distribution and parameters: Binomial conditions, the distribution table and graph shapeMean, variance, standard deviation and parameter recovery of a binomial variableExact probability of repeated independent successesEvaluate factorial and combination expressionsSolve a linear equation11B Binomial distribution and parameters: Round to decimal places and significant figuresCumulative, at-least-one and conditional probabilities of a binomial variableSmallest trial count for an at-least-one threshold11C Binomial distribution events: Finite events, probability assignments and complementsConditional probability from exact representationsSolve and interpret a linear inequation11D Binomial shape and specified order: Binomial conditions, the distribution table and graph shapeProbabilities of sequences of independent eventsDescribe the shape of a distribution