Calculator Input
Example Data Table
| Scenario | Model | Input Data | Sufficient Statistic | Statistic Result |
|---|---|---|---|---|
| Call arrivals per interval | Poisson | 2, 3, 5, 4, 6 | Σxᵢ | 20 |
| Binary test outcomes | Bernoulli | 1, 0, 1, 1, 0, 1 | Σxᵢ | 4 |
| Normal observations | Normal unknown μ, σ² | 12, 15, 11, 14, 13 | (Σxᵢ, Σxᵢ²) | (65, 855) |
Formula Used
The calculator applies the Neyman–Fisher factorization idea. A statistic is sufficient when the likelihood can be written as a product where all parameter dependence passes through that statistic.
- Bernoulli: T(X) = Σxᵢ
- Binomial with fixed trials: T(X) = Σxᵢ
- Poisson: T(X) = Σxᵢ
- Exponential: T(X) = Σxᵢ
- Normal with known variance: T(X) = Σxᵢ or x̄
- Normal with unknown mean and variance: T(X) = (Σxᵢ, Σxᵢ²)
The page also reports supporting summaries such as sample mean, variance, median, range, and sum of squares for clearer interpretation.
How to Use This Calculator
- Select the probability model matching your sample data.
- Enter the observations using commas or spaces.
- Provide known variance or trials if the chosen model needs them.
- Set a significance level and optional dataset label.
- Press Calculate to display the sufficient statistic above the form.
- Use the CSV or PDF buttons to save the output.
Frequently Asked Questions
1. What does a sufficient statistic do?
It compresses the sample without losing information about the target parameter under the chosen model. Inference can then rely on that statistic alone.
2. Why do different models share the same statistic form?
Several exponential family models depend on parameters through totals or sums. That structure makes the same statistic sufficient across different data-generating settings.
3. Why does the normal model sometimes need two statistics?
If both mean and variance are unknown, the likelihood depends on both the sample sum and sum of squares. One value alone is not enough.
4. Can I use decimal values for every model?
No. Bernoulli, binomial, and Poisson inputs must follow count rules. Exponential and normal models can use positive or continuous measurements as appropriate.
5. Does this calculator prove sufficiency formally?
It applies standard model results based on factorization theory. For custom likelihoods, a separate symbolic proof may still be required.
6. What is the benefit of the CSV export?
CSV export makes it easy to store results, compare runs, or import outputs into spreadsheets, reports, dashboards, or classroom materials.
7. When should I use the PDF export?
Use it when you want a printable snapshot of the result section for reports, study notes, submission records, or quick sharing.