Variance of Binomial Distribution

A binomial distribution is a type of probability model that counts the number of “successes” in a given set of independent trials. It is used only when there are only two possible outcomes per trial (e.g., heads vs. tails or pass vs. fail) and the probability of success is constant. Essentially, it calculates the likelihood of receiving a specific result in a repeated “yes or no” experiment.
 

\(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\)
 

Consider a student guessing randomly on a 10-question multiple-choice test, where each question has four answer options.
This scenario fits a binomial distribution because:
 

• Fixed Trials: There are exactly 10 questions.
   
• Binary Outcome: Each guess is either “Correct” or “Incorrect.”
   
• Constant Probability: Since the student is guessing blindly, the chance of getting it right is always 25% (1 in 4) for every question.
   
• Independence: Guessing on question #1 doesn't change the odds for question #2.
   

You could use the binomial formula to calculate the exact odds of the student passing (getting 6 or more correct) by pure luck.

The variance of a binomial distribution measures how much the number of successes in your trials is likely to differ from the average (mean) number of successes.

In simpler terms, it tells you how “spread out” your results are.

Low Variance: Results are tightly clustered around the average. You can be confident the outcome will be close to the mean.

High Variance: Results are widely scattered.3 predicting the exact outcome is harder because the possibilities are more spread out.

If you were to graph this:
 

• Large Variance makes the graph look flatter and wider (like a low hill).
   
• Small Variance makes the graph look taller and thinner (like a spike).

The formula for the variance (\(\sigma^2\)) of a binomial distribution is:

\(Var(X) = \sigma^2 = n \cdot p \cdot (1 - p)\)

Sometimes written as:

\(Var(X) = npq\)

Where,

n is the Total number of trials,

p is the Probability of success in a single trial, and

q is the Probability of failure (1-p).
 

Example Calculation:

Imagine you flip a fair coin (p=0.5) 100 times (n=100).
 

• Find the Mean (\(\mu\)): n \times p = \(100 \times 0.5 = 50\ heads.\)
   
• Calculate Variance (\(\sigma^2\)): \(100 \times 0.5 \times (1 - 0.5) = 25\)
   
• Standard Deviation (\(\sigma\)): \(\sqrt{25} = 5.\)
   

This means that while you expect 50 heads, it is very normal for the result to vary by about 5 heads (getting 45 to 55 heads).
 

Calculating the variance of a binomial distribution is a straightforward process once you have your variables defined. It essentially measures the “spread” or volatility of your data around the average.
Here is the step-by-step guide to performing this calculation manually.

Step 1: Identify the Total Trials (n)

Determine how many times the experiment or trial is being performed.
 

• Example: You are flipping a coin 50 times. (n = 50)
   

Step 2: Identify the Probability of Success (p)

Determine the likelihood of getting the “successful” outcome in a single, individual trial.
 

• Example: The probability of getting “Heads” in one flip is 0.5. (p = 0.5)
   

Step 3: Calculate the Probability of Failure (q)

Subtract the probability of success from 1. This gives you q, or (1 - p).
 

• Example: 1 - 0.5 = 0.5. (q = 0.5)
   

(Note: If p was 0.2, then q would be 0.8)

Step 4: Multiply Them All Together

Multiply n by p by q.
 

• Calculation: \(50 \times 0.5 \times 0.5\)
   
• \(50 \times 0.5 = 25\)
   
• \(25 \times 0.5 = 12.5\)
   

Answer: The variance is 12.5.

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. 

 \(E (X^2) = \displaystyle\sum_{k=0}^{n}k^2\). 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) = \(\binom{n}{k}\)  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 measures how much the number of successes deviates 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. 

• Medical: 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. 
   
• Banking: Banks use the binomial distribution to model the likelihood of a specific number of credit card transactions being fraudulent.
   
• Emails: Email providers apply the binomial distribution to estimate the chance of receiving a certain number of spam emails per day. 
   
• Park Systems: Park systems use the binomial distribution to estimate the probability of the overflow of rivers in a year due to heavy rainfall. 
   
• Retail: 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 variance decreases when p is near 0 or 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. 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).