Point Estimate Calculator

A point estimate is a single value computed from sample data that serves as the best guess for an unknown population parameter. For the population mean μ, the point estimate is the sample mean x̄. For a population proportion p, it is the sample proportion p̂ = x/n. Point estimates are simple but give no information about uncertainty - that is why they are paired with confidence intervals (which give a range of plausible values).

Maximum Likelihood Estimation is a general method for finding the parameter value that makes the observed data most probable. For the population mean under normality, MLE gives x̄ (sample mean). For a proportion, MLE gives p̂ = x/n. MLE is asymptotically unbiased and efficient (minimum variance among consistent estimators for large n), making it the default choice in most situations.

The Wilson Score CI is a confidence interval for a proportion p that adjusts for the known inaccuracy of the Wald interval (p̂ ± z × SE) when p̂ is near 0 or 1, or n is small. The center of the Wilson CI is pulled slightly toward 0.5: p̃ = (p̂ + z²/(2n)) / (1 + z²/n), with half-width adjusted accordingly. The Wilson CI always falls within [0, 1] and has better actual coverage than the Wald CI, especially for extreme proportions.

Laplace smoothing (add-one smoothing) is a simple Bayesian estimate for proportions: p̂_Laplace = (x + 1) / (n + 2). It assumes a uniform prior on p ∈ [0,1]. Adding a pseudocount of 1 to both numerator and denominator prevents estimates of exactly 0 or 1, which can be problematic in subsequent calculations (e.g., log-likelihood). It is commonly used in natural language processing and naive Bayes classifiers.

The Jeffreys estimate uses a Jeffreys prior (Beta(0.5, 0.5), the non-informative prior for proportions): p̂_Jeffreys = (x + 0.5) / (n + 1). It adds half a pseudocount rather than a full one, giving less shrinkage toward 0.5 than Laplace. The Jeffreys estimate is theoretically motivated - the Jeffreys prior is invariant under reparameterisation. It produces Wilson-like CIs and is generally preferred over Laplace for proportion estimation.

A point estimate is a single best guess, useful for prediction or reporting a summary statistic. A confidence interval (CI) quantifies the uncertainty around that estimate - it says 'with 95% confidence, the true parameter lies within this range.' In practice, always report both: the point estimate tells you the central value, and the CI tells you how precise that estimate is. A narrow CI indicates high precision (large n or small variance); a wide CI indicates high uncertainty.

The standard deviation (s) measures the spread of individual observations around the sample mean. The standard error of the mean (SE = s/√n) measures the precision of the sample mean as an estimator of the population mean - it decreases as n increases. For proportions, SE = √(p̂(1−p̂)/n). The SE is the standard deviation of the sampling distribution of the estimator, not of the data itself.

The margin of error (MoE) is the half-width of a confidence interval: MoE = z × SE. For a 95% CI, z = 1.96. For example, if a poll shows 54% support with n = 1000, SE = √(0.54×0.46/1000) = 0.01575, and MoE = 1.96 × 0.01575 ≈ ±3.1%. The full 95% CI is (50.9%, 57.1%). Margin of error decreases with larger n - to halve it, you need to quadruple the sample size.