12.4 Mean and Variance of the Binomial Distribution

The binomial distribution is a fundamental concept in statistics used to model the number of successes in a fixed number of independent trials, where each trial has the same probability of success. Two key statistical measures that describe the binomial distribution are its mean and variance.

Mean (μ) of a Binomial Distribution

The mean, also known as the expected value, represents the average number of successes you can anticipate over a fixed series of trials. It is calculated by multiplying the total number of trials by the probability of success in any single trial.

Formula:

μ = n * p

Where:

• μ (mu) is the mean or expected value.
• n is the total number of independent trials.
• p is the probability of success in a single trial.

This formula tells us the average outcome we can expect if we were to repeat the binomial experiment many times.

Example: If a fair coin is tossed 10 times (n = 10), and the probability of getting heads (success) is 0.5 (p = 0.5), the mean number of heads expected is: μ = 10 * 0.5 = 5 On average, you would expect to get 5 heads in 10 coin tosses.

Variance (σ²) of a Binomial Distribution

The variance measures the spread or variability of the number of successes around the mean. It quantifies how much the results are likely to deviate from the expected value. The variance is calculated by multiplying the number of trials, the probability of success, and the probability of failure.

Formula:

σ² = n * p * q

Where:

• σ² (sigma squared) is the variance.
• n is the number of independent trials.
• p is the probability of success in each trial.
• q is the probability of failure in each trial, which is calculated as q = 1 - p.

The variance provides insight into the consistency of the results. A higher variance indicates that the observed outcomes are more spread out from the mean, while a lower variance suggests that the outcomes are clustered more closely around the mean.

Example: Continuing with the coin toss example: If a fair coin is tossed 10 times (n = 10), the probability of heads (success) is 0.5 (p = 0.5), and the probability of tails (failure) is q = 1 - 0.5 = 0.5. The variance is: σ² = 10 * 0.5 * 0.5 = 2.5 This means that the number of heads obtained in 10 tosses typically varies by about 2.5 around the mean of 5.

Summary Table

Interview Questions

• What is the mean of a binomial distribution, and how is it calculated?
• How do you interpret the mean in a binomial distribution?
• What does the variance of a binomial distribution represent?
• How do you calculate variance for a binomial distribution?
• Why is variance important in understanding binomial data?
• What role does the probability of failure (q) play in binomial variance?
• How do mean and variance relate to each other in the binomial distribution?
• Can you explain the effect of increasing the number of trials on the mean and variance?
• How would you use mean and variance to summarize binomial experiment results?
• What are the practical applications of mean and variance in binomial distributions?