Hypergeometric Distribution
Calculate hypergeometric distribution probabilities with population size, successes, and sample. Free online statistics calculator with PMF, CDF charts, mean, and variance.
About This Calculator
The Hypergeometric Distribution Calculator computes probabilities, cumulative probabilities, mean, variance, and standard deviation for sampling without replacement from a finite population. It is essential for statistics students, quality control engineers, data analysts, and researchers working with finite populations where the outcome of each draw affects subsequent probabilities.
The probability mass function is given by P(X = k) = C(K,k) × C(N-K,n-k) / C(N,n) where N is the population size, K is the number of successes in the population, n is the sample size, and k is the number of observed successes. The calculator also provides the cumulative distribution function P(X ≤ k), P(X ≥ k), and descriptive statistics including mean (n×K/N), variance, and standard deviation. The Distribution chart shows the probability for each possible k value, while the Cumulative chart displays the running total probability.
Regional Notes
Global: The hypergeometric distribution is a universal statistical concept taught worldwide in probability and statistics courses. It applies to any scenario involving sampling without replacement from a finite population — from card games to quality control in manufacturing.
India (IN): Widely used in statistical quality control in Indian manufacturing, agricultural sampling by ICAR, and in competitive exam syllabi for statistics (UPSC, IIT JAM, ISI entrance).
United States (US): Commonly used in Six Sigma quality management, pharmaceutical clinical trials (interim analyses), survey sampling by the Census Bureau, and Advanced Placement (AP) Statistics curriculum.
United Kingdom (UK): Used in A-Level Mathematics and Further Statistics, quality assurance in manufacturing, and by the Office for National Statistics (ONS) for survey sampling applications.
Frequently Asked Questions
What is the hypergeometric distribution?
The hypergeometric distribution is a discrete probability distribution that describes the probability of k successes in n draws without replacement from a finite population of size N containing exactly K successes. Unlike the binomial distribution, each draw changes the probability for subsequent draws because sampling is without replacement.
What is the formula for hypergeometric distribution?
The probability mass function is P(X = k) = C(K,k) × C(N-K,n-k) / C(N,n), where C(a,b) is the binomial coefficient. The mean is n × K/N, and the variance is n × K × (N-K) × (N-n) / (N² × (N-1)).
When should I use the hypergeometric distribution instead of the binomial distribution?
Use the hypergeometric distribution when sampling without replacement from a finite population where each draw changes the probability of success for subsequent draws. Use the binomial distribution when sampling with replacement or when the population is large enough that sampling without replacement does not significantly change probabilities.
What is an example of the hypergeometric distribution?
A common example is drawing cards from a deck. If you draw 5 cards from a standard 52-card deck containing 13 hearts (successes), the hypergeometric distribution gives the probability of drawing exactly k hearts. For k=2, P(X=2) = C(13,2) × C(39,3) / C(52,5) ≈ 0.2743.
How is the hypergeometric distribution used in quality control?
In quality control, the hypergeometric distribution is used for acceptance sampling. For example, if a batch of N items has K defects and you inspect n items without replacement, the distribution tells you the probability of finding exactly k defects. This helps determine whether to accept or reject a shipment based on sample inspection.
What are the parameters of the hypergeometric distribution?
The hypergeometric distribution has three parameters: N (population size — total number of items), K (number of success states in the population), and n (number of draws or sample size). The random variable k represents the number of successes observed in the sample.
What is the difference between hypergeometric and binomial distributions?
The hypergeometric distribution models sampling without replacement, so trials are dependent and the probability of success changes after each draw. The binomial distribution models sampling with replacement, so trials are independent and the probability of success remains constant. When the population is large relative to the sample, the two distributions give similar results.
What is Fisher's exact test and how is it related to the hypergeometric distribution?
Fisher's exact test uses the hypergeometric distribution to calculate the exact probability of observing a particular 2x2 contingency table under the null hypothesis of independence. It is used when sample sizes are too small for the chi-square test, typically when any expected cell count is less than 5. The p-value is the sum of hypergeometric probabilities for all tables at least as extreme as the observed.