How this calculator works
The sample mean estimates a population mean. Inverse count modes recover successes or trials from a proportion, requiring whole counts. There is no universally best point estimate without a specified criterion. For binomial counts, the maximum-likelihood estimate is successes divided by trials. Laplace and Jeffreys posterior means add prior pseudo-counts and shrink endpoint estimates away from zero and one. Wilson supplies the centre and bounds of a score-based confidence interval; its centre is not the maximum-likelihood estimate. The plug-in standard error can vanish at observed endpoints even though the Wilson interval retains uncertainty.
Formula
MLE p̂ = x/n; Laplace = (x + 1)/(n + 2); Jeffreys posterior mean = (x + 1/2)/(n + 1). Wilson centre = (p̂ + z²/(2n))/(1 + z²/n).
Worked example
Six successes in ten trials give MLE 0.6, Laplace 7/12 and Jeffreys 13/22. Observations 2, 4, 6 give sample mean 4 and estimated standard error 2/√3.
Advertisement
Frequently asked questions
Which estimate is unbiased?
For independent Bernoulli trials, x/n is unbiased for p. The pseudo-count estimates trade some bias for shrinkage and different endpoint behaviour.
Is the Wilson centre a Bayesian mean?
No prior is used to construct a Wilson score interval. The centre comes from inversion of a score test.
Why show an interval beside point estimates?
One number alone does not express sampling precision. The Wilson interval remains informative when observed counts are at an endpoint.