A Beginner’s Guide to Variance of a 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.
   

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 ) p^k (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. 
 

• 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.  
   

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).