Variance Of Binomial Distribution

Variance of the binomial distribution is a measure of the dispersion of the probabilities with respect to the mean value. The variance of the binomial distribution is σ²=npq, where n is the number of trials, p is the probability of success, and q i the probability of failure.

The standard deviation is the square root of the variance of the binomial distribution. Let us learn more about variance, derivation of variance of the binomial distribution, with the help of examples, FAQs.

The variance of the binomial distribution is the spread of the probability distributions with respect to the mean of the distribution. For a binomial distribution having n trails, and having the probability of success as p, and the probability of failure as q, the mean of the binomial distribution is μ = np, and the variance of the binomial distribution is σ²=npq.

The binomial distribution is called binomial, as it has two variables, P the probability of success, and q the probability of failure. Further, since p and q are the probabilities of success and failure, we have p + q = 1. The general term of the binomial distribution is B(r) = \(^nC_r.P^{n - r}.q^r\).

Variance of Binomial Distribution: σ²=npq

Variance is the square of the standard deviation, and the variance is represented as σ². Using the variance we can analyze how stretched or squeezed the data is. The binomial distribution is also normally distributed. Hence the variance of a binomial distribution is the same as the variance of the equivalent normal distribution

The formula used to derive the variance of binomial distribution is Variance \(\sigma ^2\) = E(x²) - [E(x)]². Here we first need to find E(x²), and [E(x)]² and then apply this back in the formula of variance, to find the final expression. The working for the derivation of variance of the binomial distribution is as follows.

Variance \(\sigma ^2\) = E(x²) - [E(x)]²

\(E(x^2)=\sum^n_{x=0}x^2.P(x)\)

\(E(x^2)=\sum^n_{x=0}[x + (x - 1)x].P(x)\)

\(E(x^2)=\sum x.P(x) + \sum (x - 1)x.P(x)\)

\(E(x^2) = np + \sum (x - 1)x.^nC_x.P^x.q^{n - x}\)

\(E(x^2)=np + \sum x(x - 1). \dfrac{n!}{(n - x)!. x!}.p^x.q^{n - x}\)

\(E(x^2)=np + \sum x(x - 1). \dfrac{n!}{(n - x)!. x.(x - 1).(x-2)!}.p^x.q^{n - x}\)

\(E(x^2)=np + \sum \dfrac{n.(n - 1).(n - 2)!}{[(n -2)-( x-2))!. (x - 2)!}.p^2.p^{x-2}.q^{(n-2) - (x-2)}\)

\(E(x^2)=np + n(n - 1).p^2\sum \dfrac{(n - 2)!}{[(n -2)-( x-2))!. (x - 2)!}.p^{x-2}.q^{(n-2) - (x-2)}\)

\(E(x^2)=np + n(n - 1).p^2.(p + q)^{n - 2}\)

\(E(x^2)=np + (n^2.p^2-np^2).(1)^{n - 2}\)

\(E(x^2)=np + n^2.p^2-np^2\)

Let us substitute the values \(E(x^2)=np + n^2.p^2-np^2\), and [E(x)]² = (np)², in the variance formula to find the variance of the binomial distribution.

Variance\(\sigma ^2\) = E(x²) - [E(x)]²

\(\sigma ^2 = (np + n^2.p^2 - n.p^2) - (np)^2\)

\(\sigma ^2 = np + n^2.p^2 - n.p^2 - n^2.p^2\)

\(\sigma ^2 = np - n.p^2\)

\(\sigma ^2 = np( 1 - p)\)

\(\sigma ^2 = npq\)

Thus the variance of the binomial distribution is the combined square root of the product of the number of trials, the probability of success, and the probability of failure.

What Is Variance Of Binomial Distribution?

The variance of the binomial distribution is the spread of the probability distributions with respect to the mean of the distribution. For a binomial distribution having n trails, and having the probability of success as p, and the probability of failure as q, the mean of the binomial distribution is μ = np, and the variance of the binomial distribution is σ²=npq.

What Is The Formula Of Variance Of Binomial Distribution?

The formula for the variance of the binomial distribution is σ²=npq. Here n is the number of trials, p is the probability of success, and q is the probability of failure across each of the trails.

How Do You Derive The Variance Of Binomial Distribution?

The formula of variance of binomial distribution is derived using the formula Variance \(\sigma ^2\) = E(x²) - [E(x)]².First we compute the values of E(x²)=np + n²p² - np², and (E(x))²=n²p² and substitute it in the expression Variance \(\sigma ^2\) = E(x²) - [E(x)]^2, to find the final formula of variance of binomial expression which is σ^2.=npq