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 complete inferential analysis for sample means and proportions. Enter your sample statistics and get point estimates, standard errors, confidence intervals, hypothesis tests, and sample size recommendations — all in one tool.
Whether you're analyzing survey responses, experimental measurements, or quality control data, you need to estimate population parameters from sample data. This calculator handles both continuous (mean) and binary (proportion) data, providing the standard error, margin of error, and confidence intervals that quantify estimation precision.
The tool also performs one-sample z-tests against hypothesized values, showing whether your sample evidence supports rejecting the null hypothesis. For proportions, the Wilson score interval provides more accurate coverage than the standard Wald interval, especially for small samples or extreme proportions. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case. Use the example pattern when troubleshooting unexpected results.
Estimating population parameters from samples is the most common task in applied statistics. This calculator combines point estimation, interval estimation, hypothesis testing, and sample size planning in one unified tool — everything you need for a complete one-sample analysis.
Students use it to check homework and understand inferential concepts. Researchers use it for quick power calculations and preliminary analysis. Quality engineers use it for process monitoring and acceptance sampling. The visual confidence interval display helps communicate uncertainty to non-technical audiences.
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.
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.