Point Estimate Calculator (MLE, Wilson, Laplace & Jeffreys for a proportion)

Understanding the point estimate calculator — concept and principle

This calculator gives four point estimates of an unknown proportion from x successes in n trials: the maximum likelihood estimate x ÷ n, the Wilson (Agresti–Coull) midpoint, Laplace's rule of succession and the Jeffreys estimate. It also shows the standard error and margin of error. Use it for survey results, conversion rates, pass rates, defect rates and A/B tests, especially with small samples.

How it is calculated

• Maximum likelihood (MLE): p̂ = x ÷ n
• Wilson midpoint: (x + z²/2) ÷ (n + z²)
• Laplace: (x + 1) ÷ (n + 2)
• Jeffreys: (x + 0.5) ÷ (n + 1)
• Standard error: SE = √(p̂(1 − p̂) ÷ n)
• Margin of error: z × SE, with z from the confidence level (1.96 for 95%)

The principle

The sample proportion is the value of p that makes the observed data most likely, which is why it is called the maximum likelihood estimate. With very small samples, or with 0 or n successes, it can be extreme: 3 out of 3 gives 100%, which is hard to believe. The other estimates add a few imaginary observations, half successes and half failures, which pulls the estimate slightly toward 50%. Wilson adds z²/2 of each (about 2 at 95%), Laplace adds 1 of each, and Jeffreys adds 0.5 of each, matching the mean of a Bayesian Beta(x + ½, n − x + ½) posterior. As n grows, all four converge.

Worked example

Default inputs: x = 27, n = 40, 95% confidence.

1. MLE: 27 ÷ 40 = 0.675.
2. z = 1.96, z² = 3.8415. Wilson: (27 + 1.9207) ÷ (40 + 3.8415) ≈ 0.6597.
3. Laplace: 28 ÷ 42 ≈ 0.6667. Jeffreys: 27.5 ÷ 41 ≈ 0.6707.
4. SE = √(0.675 × 0.325 ÷ 40) ≈ 0.07406.
5. Margin = 1.96 × 0.07406 ≈ 0.1451, so a rough 95% range is 0.53 to 0.82. All four estimates are close, so x ÷ n is the usual report.

How to read the result

When the four estimates agree to a couple of decimals, report x ÷ n with its margin of error. When they differ noticeably, the sample is small or the proportion is near 0 or 1, and an adjusted estimate (Wilson or Jeffreys) with a proper interval is more trustworthy. The margin of error shows how far the true proportion may plausibly be from the estimate.

Common mistakes

• Reporting 0% or 100% from a tiny sample as if it were certain.
• Treating the margin of error as covering all sources of error; it covers sampling error only.
• Using the simple p̂ ± z·SE interval with very small n or extreme proportions.
• Entering a percentage instead of a count for x.

Frequently asked questions

Which estimate is best?

For moderate or large samples they barely differ. For small samples or extreme results, Wilson or Jeffreys usually behave better than x ÷ n.

What is the rule of succession?

Laplace's (x + 1) ÷ (n + 2), which estimates the chance of success on the next trial.

Why is it called a point estimate?

It gives a single value, in contrast to an interval estimate that gives a range.

Sources: Agresti, A. & Coull, B. A. (2001), Statistical Science 16(2), 117–120