Calculator Inputs
Example Data Table
| Category | Probability | Observed Count | Meaning |
|---|---|---|---|
| A | 0.25 | 12 | First possible outcome |
| B | 0.30 | 15 | Second possible outcome |
| C | 0.20 | 9 | Third possible outcome |
| D | 0.25 | 14 | Fourth possible outcome |
Formula Used
The calculator uses the multinomial form of a polynomial distribution.
P(X1=x1,...,Xk=xk) = n! / (x1! x2! ... xk!) × p1^x1 × p2^x2 × ... × pk^xk
Here, n is total trials, xi is each observed count,
and pi is each category probability.
Expected count uses E(Xi)=npi.
Variance uses Var(Xi)=npi(1-pi).
Standard deviation is the square root of variance.
How to Use This Calculator
- Enter category labels separated by commas.
- Enter matching probabilities in the same order.
- Enter observed counts in the same order.
- Enter total trials. It must equal the count total.
- Enable normalization when rough probabilities do not add to one.
- Press calculate to view probability, coefficient, moments, and graph.
- Use CSV or PDF buttons to export the calculated report.
Understanding Polynomial Distribution
A polynomial distribution describes many possible outcome counts from one repeated experiment. It is often called a multinomial distribution in statistics. Each trial can fall into one category. Every category has its own probability. The probabilities should add to one. The calculator checks that rule and can also normalize values when needed.
Why It Matters
This model is useful when two outcomes are not enough. A binomial model handles success and failure. A polynomial model handles several choices at once. Examples include survey answers, dice faces, product defects, traffic sources, and exam options. The method also connects directly with polynomial expansion. The term from (p1x1 + p2x2 + ... + pkxk)^n gives the probability pattern for counts across categories.
What The Calculator Measures
The tool computes the multinomial coefficient first. This coefficient counts how many arrangements can create the same observed counts. It then multiplies that coefficient by each category probability raised to its count. The final value is the probability of that exact count pattern. It also shows log probability. This helps when probabilities are extremely small.
Advanced Output
The result table compares observed counts with expected counts. Expected count equals total trials times category probability. The table also shows variance and standard deviation. These values help judge whether a category is close to its expected level. A z score is included for quick comparison. Large positive or negative z scores may deserve closer review.
Graph And Exports
The Plotly chart compares observed counts, expected counts, and probabilities. This makes imbalances easier to see. CSV export is useful for spreadsheets. PDF export is useful for reports, assignments, and client notes. Both exports use the same calculated values shown on the page.
Good Input Practice
Use nonnegative integer counts. Keep labels in the same order as probabilities and counts. Make sure the count total equals the number of trials. Use decimal probabilities such as 0.25 or percentages converted to decimals. If rough probabilities are entered, normalization can scale them. Still, the final model should match the real experiment.
For learning tasks, compare several scenarios. Change counts or probabilities slowly. Watch how small edits move the final probability and graph.
FAQs
1. What is a polynomial distribution?
It models counts across several possible outcomes after repeated trials. It is commonly known as the multinomial distribution.
2. How is it different from a binomial distribution?
A binomial distribution uses two outcomes. A polynomial distribution supports three or more outcomes in one repeated experiment.
3. Should probabilities add to one?
Yes. For a standard model, all category probabilities should add to one. The calculator can normalize them when needed.
4. What does the coefficient mean?
The coefficient counts how many ordered arrangements can produce the same category counts. Larger coefficients mean more matching arrangements.
5. Why does the calculator show log probability?
Exact probabilities can become extremely small. Log probability keeps the value readable and helps avoid numeric underflow.
6. What are expected counts?
Expected count is the average count predicted for a category. It equals total trials multiplied by that category probability.
7. What does the z score show?
The z score compares observed count with expected count. It uses standard deviation to show relative distance.
8. Can I export the result?
Yes. Use the CSV button for spreadsheet work. Use the PDF button for printable reports and study notes.