Calculate Variance Negative Binomial

The negative binomial distribution models the number of trials needed to achieve a specified number of successes. This calculator helps you determine the variance of a negative binomial distribution, which measures the spread of possible outcomes.

What is a Negative Binomial Distribution?

The negative binomial distribution is a discrete probability distribution that describes the number of trials needed to achieve a given number of successes in repeated, independent Bernoulli trials. Unlike the binomial distribution, which counts successes in a fixed number of trials, the negative binomial counts trials until a fixed number of successes occur.

Key Characteristics

Models the number of trials until r successes occur
Parameterized by success probability p and number of successes r
Right-skewed distribution
Used in quality control, reliability engineering, and other scenarios where the number of trials until success matters

The negative binomial distribution is related to the geometric distribution (special case where r=1) and the Poisson distribution (limit as p→0).

Variance Formula

The variance of a negative binomial distribution is calculated using the following formula:

Variance = (r * (1 - p)) / p²

Where:

r = number of successes
p = probability of success on each trial

Interpretation

The variance measures how spread out the distribution is. A higher variance indicates more variability in the number of trials needed to achieve r successes. The variance increases as the probability of success decreases or as the number of required successes increases.

How to Calculate Variance

To calculate the variance of a negative binomial distribution:

Determine the number of successes (r) you're interested in
Identify the probability of success (p) on each trial
Plug these values into the variance formula: (r * (1 - p)) / p²
Calculate the result

Practical Considerations

Ensure p is between 0 and 1
r must be a positive integer
The result will always be positive

Example Calculation

Suppose you're testing a new product and want to know how many trials (website visits) you might need to get 5 successful conversions, with a 10% conversion rate (p = 0.10).

Variance = (5 * (1 - 0.10)) / (0.10)² Variance = (5 * 0.90) / 0.01 Variance = 4.5 / 0.01 Variance = 450

The variance is 450, meaning the number of trials needed to achieve 5 successes has a spread of about ±√450 ≈ 21.2 trials from the mean.

Interpretation

This high variance indicates significant uncertainty in predicting exactly how many trials will be needed. The actual number of trials could vary widely from the expected value.

FAQ

What's the difference between negative binomial and binomial distributions?: The binomial distribution counts successes in a fixed number of trials, while the negative binomial counts trials until a fixed number of successes occur.
When should I use a negative binomial distribution?: Use it when you're interested in the number of trials until a certain number of successes occur, such as in quality control or reliability testing.
How does changing p affect the variance?: Lower values of p (less likely successes) increase the variance, indicating more uncertainty in the number of trials needed.
Can the variance be zero?: No, the variance is always positive for a negative binomial distribution since there's always some variability in the number of trials needed.
What's the relationship between negative binomial and Poisson distributions?: The Poisson distribution is a limiting case of the negative binomial when the number of successes r approaches infinity and the probability p approaches zero in such a way that r*p remains constant.