What a point estimate is
A point estimate is a single number used to approximate an unknown population parameter from sample data. This calculator estimates a population proportion — the fraction of a population that has some attribute (for example, the share of voters who favor a candidate, or the defect rate of a production line) — from two numbers: the count of successes S and the number of trials T.
The most obvious estimate is the observed sample proportion, S/T, called the maximum likelihood estimate (MLE). It is unbiased and works well most of the time, but it behaves badly at the edges: when S/T is near 0 or 1, or when T is small, it can report an implausibly extreme value (such as exactly 0 or exactly 1) and understate the true uncertainty. To handle those cases, statisticians use adjusted estimators. This tool computes four of them and picks the most appropriate.
The four formulas
Let S be the number of successes, T the number of trials, and z the critical value for your confidence level (z = 1.95996 for 95%). The calculator evaluates:
- MLE (maximum likelihood): p̂ = S / T. The raw sample proportion.
- Wilson: p̂ = (S + z²/2) / (T + z²). Shrinks the estimate toward 0.5 and never leaves the 0–1 range; excellent for small samples and extreme proportions.
- Laplace (add-one / rule of succession): p̂ = (S + 1) / (T + 2). Adds one imaginary success and one imaginary failure.
- Jeffrey (Jeffreys prior): p̂ = (S + 0.5) / (T + 1). Adds half a success and half a failure, from the Beta(½,½) prior.
How the “best” estimate is chosen
The calculator uses the standard selection rule based on the MLE and sample size:
- If MLE ≤ 0.5 → use the Wilson estimate.
- If 0.5 < MLE < 0.9 → use the plain MLE.
- If MLE ≥ 0.9 → use Laplace when T ≤ 100, otherwise Jeffrey.
The idea: near the boundaries (low or very high proportions) the raw S/T is least trustworthy, so a shrinkage estimator is substituted; in the comfortable middle range the MLE is left alone.
A worked example
Suppose 95 of 100 sampled items pass inspection, so S = 95 and T = 100 at 95% confidence (z = 1.95996, z² ≈ 3.8415). Then MLE = 95/100 = 0.95. Because MLE ≥ 0.9 and T ≤ 100, the best estimate is Laplace: (95 + 1)/(100 + 2) = 96/102 ≈ 0.9412. For comparison, Wilson = (95 + 1.9207)/(100 + 3.8415) = 96.9207/103.8415 ≈ 0.9334, and Jeffrey = 95.5/101 ≈ 0.9455. Notice all three adjusted values sit just below the raw 0.95, reflecting that a small sample of 100 does not justify claiming a 95% rate exactly.
Common reference points
- When S = T (every trial a success), MLE = 1 but Laplace, Jeffrey, and Wilson stay below 1 — a more honest estimate, since one clean sample rarely means a perfect population.
- When S = 0, MLE = 0 but the adjusted estimators return a small positive value, the “rule of three” intuition that a rate can be low but is almost never truly zero.
- At S/T = 0.5 the four estimators nearly coincide; the corrections matter most as you move toward 0 or 1.