In a Binomial(n, p), what does the variance depend on?

In a Binomial(n, p), what does the variance depend on?

• A. np(1−p).
• B. Only on n.
• C. Only on p.
• D. It is constant for fixed n and p.

Answer: (A)

Variance in a Binomial distribution reflects how much the number of successes can vary across repeated trials. This variability comes from each independent trial, which is a Bernoulli experiment with variance p(1−p). When you have n independent trials, these variances add up, giving Var = n p(1−p). So the spread depends on both how many trials you conduct and the probability of success on each trial. For a fixed n, changing p changes the per-trial variability and thus the total variance; for a fixed p, increasing n linearly increases the variance. The per-trial variability p(1−p) is largest at p = 0.5 and goes to zero as p approaches 0 or 1, while the total variance scales with n. Therefore, the variance is np(1−p).

Variance in a Binomial distribution reflects how much the number of successes can vary across repeated trials. This variability comes from each independent trial, which is a Bernoulli experiment with variance p(1−p). When you have n independent trials, these variances add up, giving Var = n p(1−p). So the spread depends on both how many trials you conduct and the probability of success on each trial. For a fixed n, changing p changes the per-trial variability and thus the total variance; for a fixed p, increasing n linearly increases the variance. The per-trial variability p(1−p) is largest at p = 0.5 and goes to zero as p approaches 0 or 1, while the total variance scales with n. Therefore, the variance is np(1−p).