Variance of Binomial Distribution

Variance of Binomial Distribution is a measure of the dispersion of probabilities with respect to the mean value (expected value). This value tells us the typical extent to which sampled observations tend to differ from the expected value.

In this article, we will explore the variance of the binomial distribution, the formula for variance in the binomial distribution, and the derivation of the variance formula for the binomial distribution. We will also solve some examples related to the variance of the binomial distribution. Let’s begin our learning journey on the topic of the Variance of Binomial Distribution.”

What is Binomial Distribution?

Binomial Distribution is a probability distribution that describes the number of successes (events with a specified outcome) in a fixed number of independent Bernoulli trials. Each trial has only two possible outcomes, typically labelled “success” and “failure,” and the probability of success (denoted by p) remains constant across all trials.

Variance of Binomial Distribution

In statistics, variance is a measure of how much the values in a data set differ from the mean (average) value. It quantifies the dispersion or spread of the data points around the mean.

Mathematically, the variance of a data set X, denoted by σ² or Var(X) (or s² for a sample variance), is calculated as the average of the squared differences between each data point and the mean.

Variance of Binomial Distribution measures the spread or dispersion of the distribution. It’s a statistical measure that quantifies how much the values in a dataset differ from the mean (average) value.

Read more about Statistics in Maths.

Variance of Binomial Distribution Formula

For a binomial distribution, where you have a fixed number of independent trials, each with the same probability of success (denoted by p) and failure (denoted by q = 1 − p), the variance (σ²) is calculated using the formula:

σ² = npq

or

σ² = n × p × (1 – p)

Derviation Of Variance Of Binomial Distribution

The binomial distribution describes the probability of obtaining a certain number of successes in a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes (success or failure) and the probability of success (denoted by p ) remains constant across all trials.

Let X be a random variable representing the number of successes in n trials. The variance of X (denoted by \sigma^2 ) can be calculated using the formula:

σ² = E[X²] – (E[X])²

First, let’s find E[X] , the mean of X, which is np , where n is the number of trials and p is the probability of success.

Next, we need to find E[X²] , the mean of the squared values of X . This involves finding the expected value of X² .

Since X follows a binomial distribution, X² follows a distribution where each outcome is squared. Therefore, we have:

E[X²] =

Using the probability mass function (PMF) of the binomial distribution, P(X=k) = , we can substitute this into the expression for E[X²] and evaluate the sum.

Finally, substitute the values of E[X] and E[X²] into the formula for variance σ², and simplify to obtain the variance of the binomial distribution, .

Which is the required formula.

How to Find the Variance of Binomial Distribution?

The steps to find the variance of binomial distribution are given below:

• First, identify the number of trials and probability of success
• Then, identify the probability of failure is given or not. If the probability of failure is not present, then determine it by 1 – probability of success.
• Then use the variance of binomial distribution formula i.e., σ² = npq
• The resultant value gives the variance of the binomial distribution.

Read More,

• Variance and Standard Deviation
• Standard Deviation
• Difference Between Variance and Standard Deviation

Solved Examples on Variance of Binomial Distribution

Example 1: Find the variance of binomial distribution given that the number of trials is 100 and probability of success is 0.7.

Solution:

To find the variance of binomial distribution we use given formula:

σ² = n × p × (1 – p)

Here, n = 100, p = 0.7 and (1 – p) = 0.3

σ² = 100 × 0.7 × (1 – 0.7)

σ² = 100 × 0.7 × 0.3

σ² = 21

Example 2: If in a binomial distribution the number of trials is 200, the probability of success is 0.9 and the probability of failure is 0.1 then, calculate the variance of the binomial distribution.

Solution:

To find the variance of binomial distribution we use formula:

σ² = npq

Here, n = 200, p = 0.9 and q = 0.1

σ² = 200 × 0.9 × 0.1

σ² = 18

Example 3: Find the variance of binomial distribution given that the number of trials is 150 and probability of failure is 0.4.

Solution:

To find the variance of binomial distribution we use formula.

σ² = npq

Here, n = 150, q = 0.4 and p = 1 – q = 1 – 0.4 = 0.6

σ² = 150 × 0.6 × 0.4

σ² = 36

Practice Problems on Variance of Binomial Distribution

Q1: Find the variance of binomial distribution given that the number of trials is 500 and probability of success is 0.6.

Q2: If in a binomial distribution the number of trials is 350, the probability of success is 0.5 and the probability of failure is 0.5 then, calculate the variance of the binomial distribution.

Q3: Find the variance of binomial distribution given that the number of trials is 430 and probability of failure is 0.2.

FAQs on Variance of Binomial Distribution

What is the Variance of Binomial Distribution?

The variance of the binomial distribution is the disperse the probabilities with respect to its expected value i.e., mean.

What is the Formula of Variance of Binomial Distribution?

The formula for the variance of binomial distribution is given by:

σ² = npq

where,

• n is number of trials
• p is probability of success
• q is probability of failure

What is the Variance of Binomial Distribution Always Equals to?

The variance of binomial distribution is always less than its mean i.e., expected value.

How Do You Find the Variance of a Binomial Distribution?

To find the variance of a binomial distribution we use the variance of binomial distribution formula i.e., σ² = npq.