Hypergeometric Distribution Calculator
In probability and statistics, not all experiments involve repeated trials with replacement. Many real-world situations require calculating probabilities from finite populations where items are not replaced after selection. This is where the hypergeometric distribution becomes essential.
The Hypergeometric Distribution Calculator is designed to help students, analysts, and researchers compute probabilities, expected value, variance, and standard deviation for such scenarios quickly and accurately—without manual formulas or lengthy calculations.
What Is the Hypergeometric Distribution?
The hypergeometric distribution measures the probability of getting a specific number of successes when sampling without replacement from a finite population.
It answers questions like:
- What is the chance of drawing exactly k defective items from a batch?
- What is the probability of selecting a certain number of successful candidates from a group?
- How likely is a specific outcome when each draw affects the next?
Unlike the binomial distribution, the population size changes after each draw.
When Is Hypergeometric Distribution Used?
The hypergeometric distribution is used when:
- The population size is finite
- Sampling is done without replacement
- Each outcome is classified as success or failure
- The probability changes after each selection
This makes it ideal for real-world sampling problems.
What This Hypergeometric Calculator Computes
This calculator provides four essential statistical values:
- Probability P(X = k)
- Mean (Expected Value)
- Variance
- Standard Deviation
All results are calculated instantly based on your inputs.
Input Parameters Explained
Population Size (N)
The total number of items in the population.
Number of Successes in Population (K)
How many items in the population are classified as “success”.
Sample Size (n)
The number of items selected from the population.
Observed Successes (k)
The number of successes observed in the sample.
How to Use the Hypergeometric Distribution Calculator
Step 1: Enter Population Size (N)
Input the total number of elements in the population.
Step 2: Enter Total Successes (K)
Enter how many items in the population qualify as successes.
Step 3: Enter Sample Size (n)
Specify how many items are drawn from the population.
Step 4: Enter Observed Successes (k)
Provide the number of successes you want the probability for.
Step 5: Click Calculate
The calculator instantly displays probability and statistical measures.
Understanding the Results
Probability P(X = k)
Shows the likelihood of observing exactly k successes in the sample.
Mean (μ)
The expected number of successes in the sample over many trials.
Variance (σ²)
Measures how spread out the possible outcomes are.
Standard Deviation (σ)
Shows how much outcomes typically differ from the mean.
Hypergeometric Formula (Conceptual Overview)
The probability is calculated using combinations:
P(X = k) = [C(K, k) × C(N − K, n − k)] ÷ C(N, n)
Where:
- C(a, b) represents combinations
- The formula ensures no replacement occurs
The calculator handles this automatically.
Example Calculation
Example 1: Quality Control
- Population size (N): 50 products
- Defective products (K): 10
- Sample size (n): 5
- Observed defective items (k): 2
Results:
- Probability: Exact chance of finding 2 defective items
- Mean: Expected defective items in the sample
- Variance & Standard Deviation: Measure of spread
This is commonly used in manufacturing inspections.
Example 2: Exam Selection
- Total students: 30
- Students who studied: 12
- Students selected: 6
- Studied students selected: 3
The calculator determines the likelihood of this outcome without replacement.
Real-World Applications
Education
- Exam question sampling
- Classroom experiments
Quality Control
- Defective product sampling
- Batch testing
Healthcare
- Patient group selection
- Clinical trial sampling
Finance
- Risk analysis from finite datasets
Research & Surveys
- Sample-based inference
Hypergeometric vs Binomial Distribution
| Feature | Hypergeometric | Binomial |
|---|---|---|
| Replacement | No | Yes |
| Population | Finite | Infinite or large |
| Probability | Changes | Constant |
| Use case | Sampling | Repeated trials |
This calculator is specifically for non-replacement scenarios.
Benefits of Using This Calculator
- Eliminates manual calculations
- Prevents formula errors
- Handles large values accurately
- Instant results
- Suitable for academic and professional use
- Ideal for statistics learning
Important Notes
- k cannot exceed K or n
- Sample size must be ≤ population size
- Values must be whole numbers
- Results are exact, not approximations
Frequently Asked Questions (FAQs)
- What is hypergeometric distribution?
It calculates probability when sampling without replacement. - How is it different from binomial distribution?
Binomial assumes replacement; hypergeometric does not. - What does P(X = k) mean?
Probability of exactly k successes in the sample. - Can this calculator handle large numbers?
Yes, within standard computational limits. - Is this calculator accurate?
Yes, it uses exact combinational formulas. - What is mean in hypergeometric distribution?
Expected number of successes in the sample. - Why is variance smaller than binomial?
Because outcomes are dependent without replacement. - Is replacement ever allowed here?
No, replacement violates hypergeometric conditions. - Can k be greater than K?
No, observed successes cannot exceed total successes. - Does population size affect probability?
Yes, strongly. - Is this used in exams?
Yes, commonly in statistics and probability exams. - What does standard deviation represent?
Typical distance from the mean. - Can this be used for surveys?
Yes, when sample size is significant. - Does order matter in sampling?
No, only counts matter. - Is this calculator free to use?
Yes, completely free.
Conclusion
The Hypergeometric Distribution Calculator is an essential tool for solving probability problems involving finite populations and sampling without replacement. Whether you are studying statistics, conducting research, or analyzing real-world data, this calculator delivers fast, reliable, and accurate results for probability, mean, variance, and standard deviation—all in one place.