The variance of the binomial distribution is alwaysReview LaterEqual to meanless than meangreater than meanequal to standard deviation

The variance of a binomial distribution can be expressed mathematically as:

$\sigma^2 = n p (1 - p)$

where:

• $n$ is the number of trials,
• $p$ is the probability of success, and
• $1 - p$ is the probability of failure.

The mean of a binomial distribution is given by:

$\mu = n p$

Key Analysis

1. Relationship Between Variance and Mean: The variance ($\sigma^2$) involves the product of $n$, $p$, and $1 - p$, while the mean ($\mu$) is simply $n p$. This means that the variance is dependent on both the number of trials and the probabilities involved.

2. Standard Deviation Relationship: The standard deviation ($\sigma$) is the square root of the variance:

$\sigma = \sqrt{n p (1 - p)}$

Conclusion

The variance of a binomial distribution is typically calculated as $n p (1 - p)$, which is not necessarily equal to the mean but is rather a function of it. Thus, the variance can be:

• Equal to the mean only in specific cases,
• Less than the mean for certain values of $p$,
• Greater than the mean when $p$ is close to 0 or 1.

Based on this assessment, the best answer is "less than mean" for many values of $p$ within the probability constraints.