1. What is Binomial Variance?

The variance of a binomial distribution measures how spread out the possible outcomes are from the expected value. It quantifies the variability in the number of successes in a fixed number of independent trials.

2. How Does the Formula Work?

The formula for binomial variance is:

Where:

• \( n \) — Number of independent trials
• \( p \) — Probability of success on each trial
• \( (1 - p) \) — Probability of failure on each trial

Explanation: The variance increases with more trials (n) and is maximized when p = 0.5 (equal chance of success and failure).

3. Importance of Binomial Variance

Details: Understanding the variance helps assess the reliability of the expected number of successes. It's crucial for statistical inference, hypothesis testing, and confidence interval construction in binomial scenarios.

4. Using the Calculator

Tips: Enter the number of trials (positive integer) and probability of success (between 0 and 1). The calculator will compute the variance of the binomial distribution.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between variance and standard deviation in binomial distribution?
A: Variance is the square of the standard deviation. While variance is in squared units, standard deviation is in the same units as the original data.

Q2: When is the variance maximized for a given n?
A: Variance reaches its maximum when p = 0.5 (equal probability of success and failure).

Q3: What happens to variance when n increases?
A: Variance increases linearly with n, as it's directly proportional to the number of trials.

Q4: Can variance be zero in a binomial distribution?
A: Yes, when p = 0 or p = 1 (certainty of all failures or all successes), the variance becomes zero.

Q5: How does this relate to the normal approximation?
A: When n is large and p isn't too close to 0 or 1, the binomial distribution can be approximated by a normal distribution with the same mean and variance.