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Last updated on July 4th, 2025

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A Beginner’s Guide to Variance of a Binomial Distribution

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The binomial distribution is used to measure how much the probabilities differ from the expected value (mean). This value shows the difference between the sampled observations and the expected value. In this topic, we are going to learn more about the variance of binomial distribution.

A Beginner’s Guide to Variance of a Binomial Distribution for Australian Students
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What is Binomial Distribution?

First, we need to understand what the binomial distribution is before learning about the variance of a binomial distribution. A binomial distribution is a discrete probability distribution that has only two outcomes. The two outcomes are typically expressed as 1 for success and 0 for failure in a given number of trials.
 

  • This probability distribution shows how likely an event is to happen in a fixed number of independent trials.
     
  • Each trial in a binomial distribution has exactly two possible outcomes: success or failure. The probability of success stays the same, and trial are independent of one another.
     
  • As the binomial distribution is discrete, it is opposed to the continuous distribution like the normal distribution.
     
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How to Calculate the Variance of Binomial Distribution?

The variance of a binomial distribution measures how spread out the probability values are around the mean. Variance defines how much the values differ from the mean value in a data set. When we calculate the variance of a binomial distribution, we have to follow certain steps. They are: 


Step 1: The first step is to identify the total number of trials (n) and the probability of success in a single trial (p). These are the two parameters that are crucial to define a binomial distribution. 


Here, n is the total number of trials. Each trial is independent, and its outcome does not change the outcome of other trials. Also, p represents the probability of success and the value is between 0 and 1. 

Step 2: Use the variance formula. 


The symbol for variance is σ2,  and it represents the square of the standard deviation. When we find the values of n and p, we can use the formula:

Variance (σ2) = np (1 - p)  
 

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Derivation of Variance of Binomial Distribution

The binomial distribution represents the probability of getting a specific number of successes in independent trials. The possible outcomes of each trial are success and failure. In every trail the probability of success remains the same. Let X be the number of successes in the n trials. Then the variance of X can be calculated as:

σ2 = E (X2) - (E (X))2

Now, let’s find the E(X), then the mean of X, which is np, here n is the number of trials and p is the probability of success. 

Then, we have to identify the E (X2). This refers to the squared values of X. Also, we need to find the expected value of X2. X2 has a distribution where each outcome is squared because X follows a binomial distribution. 

E (X2) = k = 0n k2 . P (X = k) 
Next, using the probability mass function (PMF) of the binomial distribution, we can find the probability of getting k successes in n trials. 
P (X = k) = n   pk (1 - p) n - k 


We can substitute this formula into an equation for E (X2) and then analyze the sum. Finally, add the values of E (X) and E (X2) into the formula of variance. Then simplify it to get the variance of the binomial distribution as (σ2) = np (1 - p)

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Real-life applications of the Variance of Binomial Distribution

The variance of a binomial distribution helps us to measure how much the probabilities or the number of successes differ from the expected mean. In the fields of medical research, finance, sports analysis, and manufacturing, the role of the variance of the binomial distribution is vital. 
 

  • To predict how many patients may have side effects from new drugs, the binomial distribution is used by medical professionals such as doctors and researchers. 
     
  • Banks use the binomial distribution to model the likelihood of a specific number of credit card transactions being fraudulent.
     
  • Email providers apply the binomial distribution to estimate the chance of receiving a certain number of spam emails per day. 
     
  • Park systems use the binomial distribution to estimate the probability of the overflow of rivers in a year due to heavy rainfall. 
     
  • To calculate the probability of receiving a specific number of shopping returns each week, retail stores use the binomial distribution.  
     
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Common Mistakes and How to Avoid Them on Variance of Binomial Distribution

The variance of the binomial distribution tells us how much our actual results differ from the expected value on average. However, some mistakes can lead to incorrect calculations and interpretations. By understanding the common mistakes of the variance of the binomial distribution, students can improve their statistical skills and practical knowledge. 

Mistake 1

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Using the incorrect formula

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 Students should ensure that they are using the correct formula for calculating the variance. The correct formula is:

Variance (σ2) = np (1 - p)

Where p is the probability of success and n is the number of trails. If students use the wrong formula, it can lead to errors and wrong conclusions. 
 

Mistake 2

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Confusing the values of success and failure

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 Carefully find the values of the probability of success (p) and the probability of failure (q). Sometimes, kids mistakenly interpret the values of p and q. Before applying the formula, make sure the probability of failure is identified correctly as q = 1- p.

Mistake 3

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Ignoring the conditions of a binomial distribution

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Remember that while applying the formula for the variance of the binomial distribution, the situation should meet the required conditions. They are trials of a binomial distribution are independent, with only two possible outcomes. If we use the formula for situations that do not meet the conditions, it will affect the final results.

Mistake 4

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Thinking the value of variance is a whole number

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Understand the fact that the value of variance can be a fraction or decimal. Kids expect the value of variance as a whole number and end up with incorrect conclusions. There is no rule that the variance will be a whole number. For example, if n = 9, p = 0.5, then the formula is:

Variance (σ2) = np (1 - p)

 σ2 = 9 (0.5) (1 - 0.5)

 σ2 = 9 (0.5) (0.5)

 σ2 = 9 × 0.25

 σ2 = 2.25

Here, the variance of the binomial distribution is 2.25, which is a decimal. 

Mistake 5

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Forgetting the relationship between variance, n, and p 

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 Before using the formula for the variance of a binomial distribution, kids should understand that variance depends on n and p. The value of variance increases with n. For example, if n = 12, p = 0.4, the variance will be 12 (0.4) (1 - 0.4) = 12 × 0.4 × 0.6 

σ2 = 12 × 0.24 = 2.88 

Next, if n = 30, p = 0.4, the variance will be 30 (0.4) (1 - 0.4) = 30 × 0.4 × 0.6 

σ2 = 30 × 0.24 = 7.2

The value of variance increases with the value of n. 
 

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Tips and Tricks of Variance of Binomial Distribution

The variance of the binomial distribution helps us to understand how much the results fluctuate around the mean. Understanding the concepts of variance of binomial distribution is useful in working with statistics, probability, and risk assessment. Here are some of the tricks and tips that help us to effectively work with the fundamental concept. 

  • Use the correct formula for the variance of the binomial distribution. Before concluding, double-check the structure of the formula. 
     
  • The probability of success (p) must be between 0 and 1. Here, p must satisfy 0 ≤ p ≤ 1. 
     
  • The value of variance will decrease if p is close to 0 or close to 1. 
     
  • The value of variance depends on the value of n. But in these cases, the value of p should be constant. The variance increases with a larger n. 
     
  • Mean and variance are related to each other. The variance denotes how much the values in a probability distribution differ from the mean. The mean in a binomial distribution is denoted as (μ) = np. Also, variance (σ2) = np (1 - p).
     
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Solved examples of Variance of Binomial Distribution

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Problem 1

Find the variance of the binomial distribution having 15 trials and a probability of success of 0.6.

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3.6

Explanation

We can use the formula for the variance of a binomial distribution:
Variance (σ2) = np (1 - p)


Here, n is the number of trials = 15


p is the probability of success = 0.6


Hence, the prob failure = 1 - p = 1 - 0.6 = 0.4


Now, we can substitute the values to the formula:  (σ2) = np (1 - p)


15 × 0.6 × 0.4 


15 × 0.24 = 3.6


The variance is 3.6.

It means that the number of successes will fluctuate around the mean with a variance of 3.6
 

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Problem 2

A factory produces 10 bulbs daily. The probability of a defective bulb is 0.2. Find the variance of the defective bulbs per day.

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1.6

Explanation

To find the variance of the defective bulbs per day, we can apply the binomial variance formula. Here,

n = 10

p = 0.2 

1 - p = 1 - 0.2 = 0.8

The variance formula is:

Variance (σ2) = np (1 - p)

σ2 = 10 × 0.2 × 0.8 = 1.6

Hence, the variance of defective bulbs per day is 1.6
 

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Problem 3

Felix takes 20 quizzes. The probability of passing each quiz is 0.8. Find the variance of the number of quizzes passed.

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3.2

Explanation

Variance (σ2) = np (1 - p) is the formula for the variance of the binomial distribution.

Here, n = 20

p = 0.8 

1 - p = 1 - 0.8 = 0.2

Now, we can substitute the values.
 
 σ2 = 20 × 0.8 × 0.2 

 σ2 = 20 × 0.16 = 3.2

The number of quizzes Felix passes fluctuates around the mean with a variance of 3.2
 

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Problem 4

A basketball player takes 30 free throws. The probability of making a basket is 0.6. Find the variance of successful shots.

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7.2

Explanation

To find the answer, we can use the formula, Variance (σ2) = np (1 - p)

Where, n = 30 

p = 0.6

1 - p = 1 - 0.6 = 0.4

So the formula will be:
σ2 = 30 × 0.6 × 0.4 

σ2 = 30 × 0.24 = 7.2

The variance of successful shots is 7.2
 

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Problem 5

In a shop, 40 customers visit daily. The probability that a customer makes a purchase is 0.6. Find the variance of the number of customers making a purchase.

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9.6

Explanation

To find the variance of the number of customers making a purchase, we can use the formula.
 Here, n = 40

p = 0.6 

1 - p = 1 - 0.6 = 0.4

The binomial variance formula is:

 Variance (σ2) = np (1 - p)

σ2 = 40 × 0.6 × 0.4 

σ2 = 40 × 0.24 = 9.6

The variance of the number of customers making a purchase is 9.6
 

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FAQs on Variance of Binomial Distribution

1.Define the variance of the binomial distribution.

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2.Explain the formula of variance of the binomial distribution.

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3. What is variance in a binomial distribution?

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4.Is it possible for the variable to be negative?

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5.How are n and variance related?

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