Point estimate calculator

The formulas

With x successes in n trials and z the critical value for your confidence level:

MLE = x ÷ n Laplace = (x + 1) ÷ (n + 2) Jeffrey = (x + 0.5) ÷ (n + 1) Wilson = (x + z²/2) ÷ (n + z²)

The maximum likelihood estimate (MLE) is the observed proportion. The other three each add a small number of imaginary successes and failures before dividing. Laplace adds one of each, Jeffrey adds half of each, and Wilson adds z²/2 of each, which at 95% confidence is about 1.92. That is why they are sometimes called shrinkage estimates: they shrink x/n towards 0.5.

A worked example

The default: 15 successes in 40 trials, with 95% confidence (z = 1.95996, so z² = 3.84146).

MLE = 15 ÷ 40 = 0.375 Laplace = 16 ÷ 42 = 0.380952 Jeffrey = 15.5 ÷ 41 = 0.378049 Wilson = (15 + 1.92073) ÷ (40 + 3.84146) = 16.9207 ÷ 43.8415 = 0.385953

The MLE is 0.375, which is at most 0.5, so the rule of thumb picks Wilson: about 0.386, or 38.6%. All four estimates sit between 37.5% and 38.6%, a spread of about one percentage point. That spread is small next to the sampling uncertainty with only 40 trials, where a 95% interval is roughly ±15 points wide.

For a contrast, try 45 successes in 50 trials. The MLE is 0.9, so the rule uses the smaller of Jeffrey (0.892157) and Laplace (0.884615), giving 0.8846. With 30 of 50 the MLE of 0.6 falls in the middle band and is used as it is.

The rule for choosing

The calculator applies a rule of thumb that circulates in textbooks and online calculators: Wilson when x/n ≤ 0.5, the MLE when 0.5 < x/n < 0.9, and the smaller of Jeffrey and Laplace when x/n ≥ 0.9. Treat this as a convention, not a law. The estimators trade a little bias for less variance, and which trade is better depends on the true p and on the loss you care about. If your field has a standard (for example, reporting x/n with a Wilson interval), follow it.

From a confidence interval

Switch the input to a confidence interval when a report gives only the bounds. For a symmetric interval the point estimate is the midpoint and the margin of error is half the width. The default interval 42.1 to 47.9 gives an estimate of (42.1 + 47.9) ÷ 2 = 45 and a margin of (47.9 − 42.1) ÷ 2 = 2.9, so the interval reads 45 ± 2.9.

How to interpret the result

A point estimate is your single best guess, and it will almost never equal the true value exactly. Always report it with some measure of uncertainty: a standard error, a margin of error or a confidence interval. When the four estimates here disagree noticeably, that is itself a signal: the sample is small or the result is near 0 or 1, and the interval matters more than the point.

Common mistakes

• Taking the midpoint of an asymmetric interval. The Wilson and exact intervals for a proportion are not centred on x/n.
• Entering a percentage as x. x is a count of successes, a whole number no bigger than n.
• Reporting 0% or 100% from a small sample. With 0 successes in 12, the MLE is 0 but Laplace gives 1/14 ≈ 0.071, which is a more honest summary.
• Treating the choice of estimator as more important than the sample size. Doubling n narrows the uncertainty far more than switching formulas.