A point estimate is a single best guess for an unknown population value. For a probability – the chance that a coin lands heads, that a user clicks a button, that a part is defective – the obvious estimate is the observed share of successes, S/T. This is the maximum likelihood estimate (MLE), and with large samples it is hard to beat.
With small samples, however, the MLE can be misleading. Two successes in two trials give an MLE of 100%, which is almost certainly too high; zero successes give 0%. The adjusted estimators pull the estimate towards 50% by adding pseudo-observations: Laplace’s rule of succession adds one success and one failure, Jeffreys adds half of each, and the Wilson estimate adds z²/2 of each, where z depends on the confidence level (about 1.92 successes and 1.92 failures at 95%).
Which one is “best” depends on where the MLE lies. A common rule of thumb from usability research – based on simulation studies of small samples – is: Wilson when the MLE is 50% or less, the MLE itself between 50% and 90%, and the smaller of Laplace and Jeffreys when the MLE is 90% or more. The calculator applies this rule and shows all four values so you can compare them.
How to use the Point Estimate Calculator
- Number of successes (S)
- Number of trials (T)
- Confidence level
Results update as you type, and everything is calculated in your browser.
How it works
The four estimates are computed from S and T; z = Φ⁻¹(1 − α/2) for the chosen confidence level enters the Wilson estimate and interval. The recommended estimate is Wilson if MLE ≤ 0.5, MLE if 0.5 < MLE < 0.9, otherwise the smaller of Laplace and Jeffreys.
Worked example
In a usability test 27 of 40 participants complete a task. MLE = 27/40 = 0.675, Laplace = 28/42 = 0.6667, Jeffreys = 27.5/41 = 0.6707 and, at 95% confidence (z = 1.96), Wilson = (27 + 1.921)/(40 + 3.841) = 0.6597. Because the MLE lies between 0.5 and 0.9 the rule picks the MLE, so the best point estimate is 0.675 (67.5%). For 8 successes in 30 trials (MLE 0.267) it would pick Wilson instead: 0.2932.
Which estimate the rule picks
| Observed MLE | Recommended estimate |
|---|---|
| 0 to 50% | Wilson |
| 50% to 90% | Maximum likelihood (S/T) |
| 90% to 100% | Smaller of Laplace and Jeffreys |
Inputs explained
| Field | What to enter |
|---|---|
| Number of successes (S) | A number |
| Number of trials (T) | A number |
| Confidence level | Choose one: 80%, 90%, 95%, 98% … |
What each result means
- z for the confidence level = ninv(1 – (1 – [Confidence level])÷2)
- Maximum likelihood estimate (MLE) = [Number of successes (S)]÷[Number of trials (T)]
- Laplace estimate = ([Number of successes (S)] + 1)÷([Number of trials (T)] + 2)
- Jeffreys estimate = ([Number of successes (S)] + 0.5)÷([Number of trials (T)] + 1)
- Wilson estimate = ([Number of successes (S)] + [z for the confidence level]^2÷2)÷([Number of trials (T)] + [z for the confidence level]^2)
- Best point estimate as a percentage = [Best point estimate]×100
- Standard error of the MLE = sqrt([Maximum likelihood estimate (MLE)]×(1 – [Maximum likelihood estimate (MLE)])÷[Number of trials (T)])
- Wilson interval lower bound = ([Wilson estimate] – [z for the confidence level]×sqrt([Maximum likelihood estimate (MLE)]×(1 – [Maximum likelihood estimate (MLE)])÷[Number of trials (T)] + [z for the confidence level]^2÷(4×[Number of trials (T)]^2))÷(1 + [z for the confidence level]^2÷[Number of trials (T)]))×100
- Wilson interval upper bound = ([Wilson estimate] + [z for the confidence level]×sqrt([Maximum likelihood estimate (MLE)]×(1 – [Maximum likelihood estimate (MLE)])÷[Number of trials (T)] + [z for the confidence level]^2÷(4×[Number of trials (T)]^2))÷(1 + [z for the confidence level]^2÷[Number of trials (T)]))×100
Formula
MLE = S ÷ T · Laplace = (S + 1) ÷ (T + 2) · Jeffreys = (S + 0.5) ÷ (T + 1) · Wilson = (S + z²/2) ÷ (T + z²)
Worked example
Inputs — Number of successes (S): 27; Number of trials (T): 40; Confidence level: 95%.
Results — z for the confidence level = 1.96; Maximum likelihood estimate (MLE) = 0.675; Laplace estimate = 0.6667; Jeffreys estimate = 0.6707; Wilson estimate = 0.6597; Best point estimate = 0.675; Best point estimate as a percentage = 67.5 %; Method chosen = Maximum likelihood (MLE between 50% and 90%); Standard error of the MLE = 0.0741; Wilson interval lower bound = 52.02 %; Wilson interval upper bound = 79.92 %.
Key terms
- Point estimate
- Single best guess of a population parameter.
- Maximum likelihood estimate
- The parameter value that makes the observed data most probable; here S/T.
- Rule of succession
- Laplace's estimate (S + 1)/(T + 2).
- Pseudo-observations
- Imaginary successes and failures added to stabilise a small-sample estimate.
Frequently asked questions
What is a point estimate?
A single number used as the best guess of an unknown population value, such as the sample proportion S/T for a probability.
How do you calculate a point estimate of a proportion?
The basic estimate is MLE = S ÷ T; 27 successes in 40 trials give 0.675.
What is the Laplace estimate?
(S + 1) ÷ (T + 2), Laplace's rule of succession. It never gives exactly 0 or 1.
When is the Wilson estimate preferred?
For small samples with low observed proportions (MLE ≤ 50%), where it reduces the error of the plain MLE.
Why does the best estimate change with the confidence level?
Only the Wilson estimate depends on z; the confidence level does not affect the MLE, Laplace or Jeffreys values.
What is the difference between a point estimate and an interval estimate?
A point estimate is one number; an interval estimate (confidence interval) gives a range that reflects the uncertainty around it.
Which estimate should I use for 92 successes in 100 trials?
MLE = 0.92 ≥ 0.9, so the smaller of Laplace (93/102 = 0.9118) and Jeffreys (92.5/101 = 0.9158): Laplace, 0.9118.