Hypergeometric Distribution Calculator

Category: Statistics
- May 14, 2025
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Calculate probabilities for sampling without replacement from a finite population of successes and failures. Useful for quality control, card games, lottery analysis, and population sampling.

Population Parameters

Sample Parameters

Calculation Type

Display Options

About Hypergeometric Distribution

The hypergeometric distribution describes the probability of k successes in n draws, without replacement, from a finite population of size N that contains exactly K successes.

Key Properties
  • Mean: μ = n × (K/N)
  • Variance: σ² = n × (K/N) × (N-K)/N × (N-n)/(N-1)
  • Sampling: Without replacement from a finite population
  • Difference from Binomial: Probabilities change after each draw
Applications
  • Quality Control: Probability of finding defects in a sample from a batch
  • Card Games: Calculating probabilities in poker, bridge, etc.
  • Lottery Analysis: Probability of matching a certain number of winning numbers
  • Audit Sampling: Determining likelihood of finding errors in financial records
  • Population Studies: Analyzing traits in samples from a known population
Formula:
P(X = k) = [ (K choose k) × (N-K choose n-k) ] / (N choose n)

What is the Hypergeometric Distribution Calculator?

The Hypergeometric Distribution Calculator is a powerful Statistics tool that helps you find probabilities when sampling without replacement from a fixed population. Whether you are doing statistical analysis for quality control, card games, lottery predictions, or audit sampling, this statistical computation resource simplifies your work by calculating exact, cumulative, or range probabilities easily.

It focuses on scenarios where each selection changes the remaining possibilities, which differs from sampling with replacement, commonly found in binomial distributions.

How to Use the Calculator Effectively

  • Step 1: Enter the Population Size (N) — the total number of items.
  • Step 2: Enter the number of Successes in Population (K) — items that qualify as a "success."
  • Step 3: Enter the Sample Size (n) — how many items you plan to draw.
  • Step 4: Enter the desired Successes in Sample (k).
  • Step 5: Choose your calculation type: Exact probability, Cumulative Lower, Cumulative Upper, or a Range.
  • Step 6: Customize your output: Choose decimal places, display calculation steps, and show the full probability distribution.
  • Step 7: Click "Calculate" to see results including the probability, mean, variance, and standard deviation.

Why Use the Hypergeometric Distribution Calculator?

This calculator serves as a hypergeometric distribution solver and a probability and stats helper, offering clear answers for real-world applications:

  • Quality Control: Assess the likelihood of finding defects in random samples.
  • Card Games: Analyze hands and predict probabilities in games like poker or bridge.
  • Lotteries: Estimate your chances of matching a set number of winning numbers.
  • Audit Sampling: Evaluate error rates in Financial or compliance reviews.
  • Population Studies: Understand traits in biological or demographic studies.

It also doubles as a data analysis helper when working with finite population statistics and as a probability distribution solver for academic or professional projects.

Key Features of the Calculator

  • Supports Exact, Cumulative, and Range probability calculations.
  • Displays the probability mass function (PMF) and cumulative distribution function (CDF).
  • Calculates the expected value (mean), variance, and standard deviation easily.
  • Includes a full chart and distribution table for better visualization of results.
  • Interactive settings to control precision and output visibility.

FAQs About the Hypergeometric Distribution Calculator

What is the Hypergeometric Distribution used for?

The hypergeometric distribution is used for calculating the probability of a certain number of successes in a sample drawn without replacement from a finite population. It's essential for sampling where each draw affects the next, such as quality inspection or card games.

Can this tool help in analyzing data variance and standard deviation?

Yes, alongside calculating probabilities, it shows the mean and median, data variance, and standard deviation, making it a complete statistical analysis tool for finite population studies.

Is this calculator similar to a Permutation and Combination Calculator?

While a Permutation and Combination Calculator focuses on counting possibilities, this calculator uses combination formulas within its probability calculations, similar to what you find when you solve combinations problems.

How accurate are the results?

You can choose up to 10 decimal places, ensuring highly accurate outputs for any statistical computations you require.

Can I use this for educational purposes?

Absolutely. It's an excellent descriptive statistics guide and probability distribution reference tool for both students and professionals learning about hypergeometric probability and related topics like mean and median or standard deviation.

Conclusion

The Hypergeometric Distribution Calculator is a highly useful Statistics Calculator that provides quick, accurate, and insightful analysis when sampling without replacement. Whether you are doing a school project, a professional audit, or simply exploring statistical analysis, this data distribution solver is an indispensable part of your toolkit.