Calculate sample mean or proportion with standard error, confidence intervals, hypothesis testing, and sample size planning. Includes Wilson interval for proportions.
The Sample Mean & Proportion Calculator performs one-sample inference for means and proportions. Enter sample statistics and it returns point estimates, standard errors, confidence intervals, hypothesis tests, and sample-size guidance.
It covers both continuous data, where the parameter of interest is a mean, and binary data, where the parameter is a proportion. The calculator also includes the Wilson score interval for proportions, which is usually more reliable than the plain Wald interval for small samples or extreme rates.
That makes it useful for survey analysis, quality control, experiment summaries, and any quick inference problem where you want the estimate, its uncertainty, and a simple significance check in one place.
Sample-based inference is easiest to read when the estimate, its uncertainty, and the null-hypothesis check are shown together. This calculator keeps those pieces in one view so you can compare means and proportions without switching between separate formulas or tools.
Mean: SE = s/√n, CI = x̄ ± z*×SE, z = (x̄−μ₀)/SE. Proportion: SE = √[p̂(1−p̂)/n], CI = p̂ ± z*×SE, z = (p̂−p₀)/√[p₀(1−p₀)/n].
Result: x̄ = 78.5, SE = 2.246, 95% CI = (74.10, 82.90), z = 1.559, p = 0.119
SE = 12.3/√30 = 2.246. Margin = 1.96 × 2.246 = 4.402. CI: 78.5 ± 4.40 = (74.10, 82.90). z-test against μ₀=75: z = (78.5-75)/2.246 = 1.559, p = 0.119 (not significant).
The Central Limit Theorem (CLT) is the foundation of inferential statistics. It states that the sampling distribution of the mean approaches a normal distribution as sample size increases, regardless of the population distribution. This justifies using the z-distribution for confidence intervals and hypothesis tests when n is large enough.
Determining the right sample size before data collection prevents two problems: too few observations (underpowered, wide CIs) and too many (wasteful resources). The formulas balance four factors: desired confidence level, acceptable margin of error, estimated variability, and for proportions, the expected proportion value.
A statistically significant result (small p-value) may not be practically important. A large sample can detect tiny, meaningless differences. Always consider effect size alongside significance. The confidence interval helps — it shows both whether the effect is significant (excludes null value) and how large the effect could be.
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Use sample mean for continuous measurements (height, weight, scores). Use sample proportion for binary outcomes (pass/fail, yes/no, defective/good). The proportion is just the fraction of successes.
Standard error measures how much the sample statistic varies from sample to sample. For means, SE = s/√n. For proportions, SE = √[p̂(1−p̂)/n]. Larger samples give smaller standard errors.
The Wilson score interval is a more accurate confidence interval for proportions than the standard Wald interval. It prevents impossible values (below 0 or above 1) and has better coverage, especially for small samples or proportions near 0 or 1.
The p-value is the probability of observing a result at least as extreme as yours if the null hypothesis is true. A small p-value (< 0.05) suggests evidence against H₀. It does not give the probability that H₀ is true or false.
For means: n = (z×s/E)² where E is desired margin. For proportions: n = z²×p(1-p)/E². The sample size planning table shows requirements for various precision targets.
The z-test assumes a normal sampling distribution, which is approximately true when n ≥ 30 (Central Limit Theorem) or the population is normal. For proportions, it requires np ≥ 10 and n(1-p) ≥ 10.