Calculator Inputs
Enter one probability and one count for each category. Use commas between values. The observed counts must add up to the total trials.
Example Data Table
| Category | Probability | Observed Count | Expected Count |
|---|---|---|---|
| Category A | 0.20 | 3 | 2.00 |
| Category B | 0.50 | 4 | 5.00 |
| Category C | 0.30 | 3 | 3.00 |
This example uses 10 trials over 3 categories. The observed vector is (3, 4, 3), and the probability vector is (0.2, 0.5, 0.3).
Formula Used
Multinomial probability mass function
P(X1=x1, ..., Xk=xk) = n! / (x1! x2! ... xk!) × ∏ pixi
Subject to: Σxi = n and Σpi = 1
Expected count: E[Xi] = n pi
Variance: Var(Xi) = n pi(1 - pi)
Covariance: Cov(Xi, Xj) = -n pi pj for i ≠ j
The calculator evaluates the PMF using logarithms first, which improves numerical stability for larger trial counts or small category probabilities.
How to Use This Calculator
- Enter the total number of trials.
- Provide one probability for each category, separated by commas.
- Enter the observed counts in the same category order.
- Optionally add labels for easier reading in the table and chart.
- Enable normalization when your probabilities do not sum exactly to one.
- Submit the form to view the PMF, expected counts, residuals, and covariance matrix.
- Use the CSV and PDF buttons to export the displayed output.
Frequently Asked Questions
1. What does this calculator measure?
It computes the probability of one exact outcome vector from a multinomial experiment. It also reports expected counts, per-category variance, covariance, standardized residuals, and a quick chi-square style comparison against the expected profile.
2. When should I use a multinomial model?
Use it when each trial ends in exactly one of several categories, the number of trials is fixed, and the category probabilities remain constant across trials. Survey responses, dice groupings, and product choice models are common examples.
3. Why must the counts add up to total trials?
The multinomial distribution partitions all trials across categories. Every trial must be counted once, so the category counts must add to n. If they do not, the specified outcome cannot represent a valid multinomial event.
4. What happens if probabilities do not sum to one?
A valid probability vector must total one. This page can automatically normalize your inputs, which rescales them proportionally. Keep that option off when you want the tool to strictly validate the original values instead.
5. Why are off-diagonal covariance values negative?
When one category receives more trials, some other category must receive fewer because the total is fixed. That tradeoff creates negative covariance between different category counts in a multinomial experiment.
6. What is the benefit of the log PMF?
For large n or tiny probabilities, the exact PMF can underflow toward zero in floating-point arithmetic. The log PMF stays stable and still lets you compare outcomes, rank scenarios, and understand relative likelihood.
7. Are the residuals a formal test result?
No. They are diagnostic indicators showing where observed counts differ from expected counts, scaled by category variability. They are useful for interpretation, but a formal goodness-of-fit conclusion needs the right statistical test and assumptions.
8. Can I use decimal probabilities with many categories?
Yes. The calculator accepts decimal probabilities and any practical number of categories, provided the probability, count, and label lists match in length. Very large category sets may simply create wider tables and more detailed charts.