1. What is Binomial Distribution Variance?

The variance of a binomial distribution measures the spread or dispersion of the distribution. It quantifies how much the number of successes in n independent Bernoulli trials is expected to vary from the mean.

2. How Does the Calculator Work?

The calculator uses the binomial variance formula:

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 and reaches maximum when p = 0.5, decreasing as p approaches 0 or 1.

3. Importance of Variance Calculation

Details: Understanding variance helps assess the reliability of expected outcomes, determine confidence intervals, and evaluate risk in binomial processes across statistics, quality control, and probability modeling.

4. Using the Calculator

Tips: Enter number of trials (n ≥ 1) and probability of success (0 ≤ p ≤ 1). Both values must be valid for calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is the range of possible variance values?
A: Variance ranges from 0 (when p=0 or p=1) to n/4 (when p=0.5), with maximum variance occurring at p=0.5.

Q2: How is variance related to standard deviation?
A: Standard deviation is the square root of variance (σ = √σ²), providing the spread in the same units as the original measurement.

Q3: When is binomial distribution appropriate?
A: When trials are independent, each has only two outcomes (success/failure), and probability of success remains constant across trials.

Q4: What if p is exactly 0 or 1?
A: Variance becomes 0 since there's no variability - outcomes are completely predictable (all failures or all successes).

Q5: How does variance change with n and p?
A: Variance increases linearly with n and follows a parabolic relationship with p, peaking at p=0.5.