Calculator Inputs
Example Data Table
| Scenario | Distribution | Sample Values | Main Estimate |
|---|---|---|---|
| Measurement Errors | Normal | 5, 7, 6, 8, 5, 9, 7, 6 | μ̂ = 6.625 |
| Waiting Times | Exponential | 2.1, 1.6, 3.4, 2.8, 1.9 | λ̂ = n / Σx |
| Daily Defects | Poisson | 2, 0, 1, 3, 2, 4, 1 | λ̂ = x̄ |
| Binary Outcomes | Bernoulli | 1, 0, 1, 1, 0, 1 | p̂ = successes / n |
Formula Used
Maximum likelihood estimation chooses parameter values that make the observed sample most probable under a selected model. The calculator evaluates closed-form estimators for common distributions.
Normal: μ̂ = Σx / n, and σ²̂ = Σ(x − μ̂)² / n.
Exponential: λ̂ = n / Σx.
Poisson: λ̂ = Σx / n.
Bernoulli: p̂ = successes / n.
The log-likelihood summarizes model fit. AIC = 2k − 2ln(L̂), and BIC = kln(n) − 2ln(L̂), where k is parameter count.
How to Use This Calculator
- Select the probability distribution that matches your data pattern.
- Enter sample values using commas, spaces, or semicolons.
- Choose a confidence level for the approximate interval display.
- Press Estimate Parameters to calculate MLE values.
- Review the result block shown above the form.
- Export the result table as CSV or PDF if needed.
Use Normal for continuous symmetric data, Exponential for waiting times, Poisson for counts, and Bernoulli for binary outcomes.
FAQs
1. What does maximum likelihood estimation do?
It finds parameter values that maximize the probability of observing your sample under a chosen statistical distribution.
2. Which distribution should I choose?
Choose Normal for continuous symmetric data, Exponential for waiting times, Poisson for event counts, and Bernoulli for binary observations.
3. Why is the Poisson option restricted to whole numbers?
Poisson models count data. Counts must be non-negative integers, so decimals or negative values do not fit that likelihood function.
4. Why does Bernoulli accept only zeros and ones?
Bernoulli trials represent two possible outcomes, usually coded as success and failure. That makes 0 and 1 the correct sample format.
5. What is log-likelihood?
Log-likelihood is the natural logarithm of the likelihood. It makes large probability products easier to compute and compare.
6. What do AIC and BIC mean?
AIC and BIC balance fit with model complexity. Lower values usually indicate a more efficient model among comparable candidates.
7. Are the confidence intervals exact?
No. These intervals are approximate and intended for quick interpretation. Formal inference may require distribution-specific exact methods.
8. Can I export the results?
Yes. Use the CSV button for spreadsheet-friendly output and the PDF button for printable documentation of the calculated results.