Point Estimate Calculator

Quick answer

From x successes in n trials, the maximum-likelihood estimate of the proportion is x ÷ n, Laplace's is (x + 1) ÷ (n + 2) and the Wilson/Agresti–Coull estimate is (x + 2) ÷ (n + 4). For 8 of 20: 0.4, 0.4091 and 0.4167.

Enter the number of successes and trials. The calculator returns three point estimates.

Related tools: Sample Size Calculator: Sample size for estimating a mean or a proportion. Confidence Interval Calculator: Confidence interval for a mean (t or z). Standard Error Calculator: Standard error of a mean or a proportion.

How to use the Point Estimate Calculator

1. Enter successes (a whole number, 0 or more).
2. Enter trials (a whole number, 1 or more).
3. Press Calculate to see the result, and use Reset to start again.

Formula and method

Successes cannot exceed trials.

Worked examples

This example is verified: each one is an automated test case that the live calculator must reproduce exactly.

The first line is the headline result (sample proportion). Below it the point Estimate Calculator lists laplace estimate and wilson estimate, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.

Why adjust

With few trials or extreme results the plain proportion can be misleading: 0 of 5 gives 0%, whereas Laplace gives 14%, accounting for uncertainty. Laplace's rule effectively adds one success and one failure.

Choosing

For large samples the three agree closely. Agresti–Coull is recommended for confidence intervals on small samples. x cannot exceed n.

How the result changes with successes

Holding every other input at the first worked example, this table changes only successes. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.

What happens with invalid input

These messages are produced by the calculator itself, so you know what to fix.

• Successes: entering "9" shows "Successes cannot exceed the number of trials." beside that field and no result.

Choosing between the Point Estimate Calculator and similar calculators

Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.

Background: hypothesis testing and inference

A hypothesis test compares a test statistic with what the null hypothesis predicts. The decision rule is to reject the null hypothesis when the p-value is at or below the significance level α, which you should choose before seeing the data (0.05 is conventional, not special). For a two-tailed z test the p-value is 2 × (1 − Φ(|z|)); for a one-sample t test the statistic is (x̄ − μ₀) ÷ (s ÷ √n) with n − 1 degrees of freedom.

Failing to reject the null hypothesis does not prove it true, and a statistically significant result can still be too small to matter in practice.

Assumptions and limitations

• x ≤ n

If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for successes to confirm this.

Related calculators

• Sample Size Calculator — Sample size for estimating a mean or a proportion.
• Confidence Interval Calculator — Confidence interval for a mean (t or z).
• Standard Error Calculator — Standard error of a mean or a proportion.
• Margin of Error Calculator — Margin of error for a mean or a proportion.
• Odds Ratio Calculator — Odds ratio from a 2×2 table.
• Relative Risk Calculator — Relative risk from a 2×2 table.

Sources

• NIST/SEMATECH e-Handbook of Statistical Methods (itl.nist.gov/div898/handbook)
• Standard textbook definitions (e.g. Moore, McCabe & Craig, Introduction to the Practice of Statistics)

Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.