Compute confidence intervals for means, proportions, mean differences, proportion differences, odds ratios, and risk ratios. Choose 80, 90, 95, or 99 percent. The calculator uses the exact t-distribution for means, the Wilson score interval for single proportions, the Newcombe hybrid method for proportion differences, and the log-transformed Wald approach for odds and risk ratios. Each tab exports a reproducible R script.
Confidence interval for the population mean from a single sample, using the exact t-distribution.
SE = SD / sqrt(n); CI = mean +/- t crit x SE
Mean, proportion, mean difference, proportion difference, OR or RR from a 2x2 table, or a generic estimate plus standard error.
Choose 80, 90, 95, or 99 percent. Match the level your reporting guideline or journal expects.
Plug in mean and SD, successes and trials, or the four cells of a 2x2 table. Sample data are one click away.
The result panel shows the point estimate, the lower and upper bounds, and the precise method used (Wilson, Newcombe, log Wald).
Copy a reproducible R snippet for the manuscript supplement or to drop into your analysis script.
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Point estimates without intervals are uninterpretable. The interval shows precision and lets readers judge whether the estimate is consistent with no effect.
The Wilson score interval has correct coverage even at small samples and extreme proportions. The Wald interval is only acceptable for moderate p and large n.
Odds ratios and risk ratios should be log-transformed before constructing a CI, then exponentiated back. The resulting interval is asymmetric on the original scale.
A 95 percent CI does not mean a 95 percent probability that the true value is in this specific interval. It means the procedure produces an interval that captures the truth in 95 percent of repeated samples.
A confidence interval is the inferential complement to a point estimate. Where the point estimate gives a single best guess for the population parameter, the confidence interval communicates how much that guess could plausibly vary from sample to sample. Neyman (1937) formalised the construction by defining intervals that, if computed across an infinite sequence of samples, would contain the true parameter at the chosen long-run frequency. Modern reporting guidelines, including the EQUATOR network, the CONSORT statement for clinical trials, and the PRISMA 2020 update for systematic reviews (Page et al., 2021), treat the confidence interval as a mandatory companion to any reported estimate.
The mathematics behind the interval depends on the parameter being estimated. For a sample mean, the standard error follows from the sample standard deviation and the sample size, and the t-distribution provides the correct critical value because it accounts for the extra uncertainty in estimating the variance from the same sample. For a single proportion, the most defensible approach is the Wilson score interval (Wilson 1927), which has accurate coverage even with sparse data and stays inside the natural bounds of zero and one. Agresti and Coull (1998) showed that the Wald interval has dramatically poor coverage for proportions near zero or one, which is why the Wilson method is now the default in most reputable software and why this calculator surfaces it as the primary choice.
Comparisons between groups bring additional complexity. Welch's t-interval handles two independent means without assuming equal variances, an assumption that is rarely defensible in practice. For the difference between two proportions the Newcombe hybrid score interval (Newcombe 1998) combines two Wilson intervals in a way that preserves the bounds of minus one to plus one and outperforms simple Wald differences when sample sizes are uneven. For ratio measures such as the odds ratio and risk ratio, the standard approach is to construct the interval on the log scale, where the sampling distribution is approximately normal, and then exponentiate the bounds. The result is an asymmetric interval on the original scale, with the lower bound closer to one than the upper bound.
Common misinterpretations should be avoided. A 95 percent confidence interval does not mean there is a 95 percent probability that the true value lies inside this specific calculated interval; under the frequentist framework, the parameter is fixed and the interval is the random object. The 95 percent refers to the long-run reliability of the construction procedure. Bayesian credible intervals use a different definition that does support the direct probabilistic statement, but they require a prior and produce different numbers from frequentist intervals. The Cochrane Handbook for Systematic Reviews of Interventions (Higgins et al., 2023) recommends reporting the procedure used and the precise interpretation alongside the numbers themselves.
Confidence intervals are the engine of meta-analytic synthesis. Random-effects pooling weights each study by the inverse of the sum of within-study and between-study variances, both of which are derived from the standard errors that produce confidence intervals. When primary studies fail to report intervals, you can back-calculate them from p-values using our p-value to confidence interval converter, and you can convert standard errors to standard deviations using the SE to SD converter. For full statistical analysis with adjusted models, robust standard errors, multilevel structures, or Bayesian intervals, our statistical analysis service delivers PhD-level support and APA-ready manuscript text.
A confidence interval is a range of values, computed from sample data, that is constructed to contain the true population parameter with a specified long-run frequency. A 95 percent interval, for example, is built using a procedure that captures the true value 95 times out of 100 if the study were repeated under identical conditions. The interval communicates both the point estimate and its precision in a single expression.
For a sample mean, multiply the standard error by the t critical value with n minus one degrees of freedom and add and subtract from the mean. For a proportion, the recommended approach is the Wilson score interval, which performs well even with small samples or proportions near zero or one. This calculator chooses the appropriate formula automatically based on the input type you select.
Always prefer Wilson, especially when the sample is small or the proportion is close to zero or one. The Wald interval relies on a normal approximation that breaks down at extreme proportions, sometimes producing intervals that extend below zero or above one. Wilson is bounded inside zero and one and has more accurate coverage even at sample sizes as small as 10.
Newcombe (1998) developed a hybrid score interval for the difference between two independent proportions. It uses Wilson intervals on each proportion and combines them in a way that respects the natural bounds of minus one to plus one. It outperforms simple Wald intervals when sample sizes are unequal or proportions are extreme, and it is the method used in this calculator.
Ratios have skewed sampling distributions because they are bounded below by zero. Taking the natural log produces an approximately normal distribution, so a symmetric Wald interval can be constructed on the log scale. The interval is then exponentiated back to the original scale, yielding asymmetric bounds that respect the lower bound of zero.
Yes. When any cell is zero, the calculator applies a 0.5 continuity correction to all four cells before computing the odds ratio and standard error. This is the standard fix recommended by Sweeting et al. (2004) for sparse 2x2 tables and avoids divide-by-zero failures while keeping the overall estimate close to the true parameter.
The p-value to CI converter starts from a reported p-value and effect estimate and back-calculates the standard error and interval. This calculator starts from raw summary statistics such as means, SDs, and counts and computes the interval directly. Use this tool when you have summary data and the converter when a paper only reports a p-value with the effect.
The Mean tab handles paired differences if you compute the within-subject difference scores yourself and pass the mean and SD of those differences. For more complex designs such as cluster sampling, multilevel data, or repeated measures with time-varying covariates, our statistical analysis service produces correctly modelled intervals using mixed-effects regression.
No. All inputs and results stay inside your browser tab. Nothing is transmitted to a server, so the calculator is safe to use with proprietary or confidential clinical data.
Yes. Each tab provides a Copy R Code button that outputs a reproducible script using the appropriate base R functions or packages such as PropCIs and epitools. Drop the code into a manuscript supplement to make your analysis fully reproducible.
If you only have a p-value and an effect estimate, the p-value to confidence interval converter back-calculates the standard error and interval. To run regression with full coefficient intervals, use the linear regression calculator. For meta-analytic effect sizes such as Cohen's d, see the effect size calculator. If you only have standard error rather than standard deviation, the SE to SD converter handles the back-calculation. To plan a study large enough to produce a precise CI, use the sample size calculator.
Reviewed by
Dr. Sarah Mitchell holds a PhD in Biostatistics from Johns Hopkins Bloomberg School of Public Health and has over 15 years of experience in systematic review methodology and meta-analysis. She has authored or co-authored 40+ peer-reviewed publications in journals including the Journal of Clinical Epidemiology, BMC Medical Research Methodology, and Research Synthesis Methods. A former Cochrane Review Group statistician and current editorial board member of Systematic Reviews, Dr. Mitchell has supervised 200+ evidence synthesis projects across clinical medicine, public health, and social sciences. She reviews all Research Gold tools to ensure statistical accuracy and compliance with Cochrane Handbook and PRISMA 2020 standards.
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