Last updated on July 4th, 2025
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.
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.
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)
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 k ) 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)
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.
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.
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.
Find the variance of the binomial distribution having 15 trials and a probability of success of 0.6.
3.6
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
A factory produces 10 bulbs daily. The probability of a defective bulb is 0.2. Find the variance of the defective bulbs per day.
1.6
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
Felix takes 20 quizzes. The probability of passing each quiz is 0.8. Find the variance of the number of quizzes passed.
3.2
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
A basketball player takes 30 free throws. The probability of making a basket is 0.6. Find the variance of successful shots.
7.2
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
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.
9.6
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