How this instrument works
A point estimate is a single best-guess number for an unknown population value, calculated from sample data. For estimating a population proportion from x successes out of n trials, the most obvious point estimate is the maximum-likelihood estimate (MLE), simply x/n — but MLE has known weaknesses at the extremes, particularly when x is 0 or n (giving an estimate of exactly 0% or 100%, often too confident given limited data) or when n is small.
This calculator reports four established point-estimate formulas side by side: MLE (x/n, the raw sample proportion), Laplace's estimate ((x+1)/(n+2), from Laplace's rule of succession, which nudges extreme estimates slightly toward 0.5), Jeffrey's estimate ((x+0.5)/(n+1), a similar but gentler nudge derived from Bayesian Jeffreys-prior reasoning), and the Wilson estimate ((x + z²/2)/(n + z²), which additionally depends on a chosen confidence level via z and comes from inverting the Wilson score confidence interval).
Because no single one of these is universally correct, the calculator also returns a recommended estimate that switches formula based on where the raw MLE falls: the Wilson estimate when MLE is 0.5 or below, the MLE itself in the comfortable middle range (strictly between 0.5 and 0.9), and the smaller of Jeffrey's and Laplace's estimates when MLE is 0.9 or above — a rule of thumb leaning on Wilson's better small-sample and boundary behavior at the low end, and on the more conservative Bayesian-style estimates near the top end where MLE tends to overstate confidence.
- Enter the number of observed successes into x.
- Enter the total sample size into n (x cannot exceed n).
- Choose a confidence level for the Wilson estimate from the dropdown.
- Compare MLE, Laplace, Jeffrey, and Wilson side by side.
- Read Recommended point estimate — the calculator's suggested single best estimate, chosen by where your MLE falls.
Worked example — 45 successes out of 150 trials
You observed x = 45 successes in n = 150 trials, and want a 95% confidence level for the Wilson estimate (z = 1.96). Enter those three values.
MLE = 45/150 = 0.3 exactly. Laplace = 46/152 = 0.302632. Jeffrey = 45.5/151 = 0.301325. Wilson = (45 + 1.96²/2) / (150 + 1.96²) = 46.9208/153.8416 = 0.304994. Since MLE (0.3) is at or below 0.5, the recommended estimate is the Wilson value, 0.304994 — slightly higher than the raw MLE, reflecting Wilson's adjustment for estimation uncertainty at this sample size.
Questions
Why not just always use x/n (the MLE)?
MLE is unbiased and simple, but it behaves poorly at the extremes — with a small sample, or a result at or near 0 or 100%, x/n can overstate how confident you should be. Zero successes in 5 trials, for example, gives an MLE of exactly 0%, implying total certainty the event never happens, a strong claim from just 5 observations. The Laplace, Jeffrey, and Wilson estimates all build in a small adjustment specifically to avoid that overconfidence at small samples or extreme results.
What's the difference between the Laplace and Jeffrey estimates?
Both nudge the raw proportion slightly toward 0.5 to avoid overconfidence at the extremes, but by different amounts: Laplace's estimate, from Laplace's classic rule of succession, adds one success and one failure, (x+1)/(n+2), a stronger nudge. Jeffrey's estimate adds only half a success and half a failure, (x+0.5)/(n+1), a gentler, Bayesian-motivated adjustment. Jeffrey's estimate is generally considered to have better statistical properties across a wider range of situations, but both remain in common use.
How does the confidence level (z) affect the Wilson estimate?
A higher confidence level uses a larger z value, which pulls the Wilson estimate further from the raw MLE and generally closer to 0.5, since the z²/2 term in the numerator and z² term in the denominator both grow. A 99% confidence Wilson estimate will typically sit further from the raw sample proportion than a 90% confidence Wilson estimate computed from the exact same x and n.
How does the calculator decide which estimate to recommend?
It follows a simple rule based on the raw MLE: when MLE is 0.5 or below, it recommends the Wilson estimate, which tends to behave best in that range and at small samples. When MLE is strictly between 0.5 and 0.9, it recommends the MLE itself, since the raw sample proportion is usually trustworthy enough in that comfortable middle range. When MLE is 0.9 or above, it recommends whichever of Jeffrey's or Laplace's estimate is smaller, since MLE tends to overstate confidence near the upper extreme.
Is this the same as a confidence interval?
No — a point estimate is a single number, while a confidence interval is a range meant to contain the true population value with a stated level of confidence. The Wilson point estimate used here is actually the center of the Wilson score confidence interval, but this calculator reports only that single center value, not the full interval around it. A dedicated confidence-interval calculator for a proportion is the right tool if the interval itself is needed.