Calculate a two-sided confidence interval for a mean, proportion, difference between independent groups or paired mean difference. Compare supported methods, inspect the critical value and margin of error, and review every substituted step.
Confidence Interval Calculator for Means and Proportions
Enter summary statistics from the correct study design. The calculator keeps unrounded values through the final interval calculation.
Confidence interval result
One mean, Student's t interval
The interval and margin of error will appear here.
Interval and point estimate
Calculated measures
| Measure | Value | Meaning or method |
|---|
Calculation steps
- Select a mode and enter valid summary statistics.
Recent calculations
A confidence interval is meaningful only when the sampling design, independence, distribution conditions and selected method fit the data. Summary inputs cannot reveal bias, dependence, outliers or data-entry errors.
How to Use This Confidence Interval Calculator
- Select one mean, one proportion, two independent means, a paired mean difference or two independent proportions.
- Enter the requested sample statistics. Keep the subtraction order consistent when comparing two groups or forming paired differences.
- Choose the statistical method. For one mean, state whether the entered deviation is a sample estimate or a known population value.
- Select a confidence level. Use Custom level for a percentage from 50 through 99.999.
- Select Calculate interval. Review the endpoints, estimate, standard error, critical value, margin, chart and substituted steps.
The fields accept integers, decimals, simple fractions and scientific notation. Success counts and sample sizes must be safe whole numbers. Standard deviations cannot be negative. The calculator rejects partial text such as 12abc instead of silently using 12.
All entries stay in the current browser page. Changing display precision changes only visible rounding. The interval uses unrounded values throughout. If a report requires a fixed convention, reproduce the result in the required statistical package and record the exact method name.
What a Confidence Interval Means
A confidence interval is an estimate plus a quantified range of sampling uncertainty. Under repeated sampling from the stated model, a 95% confidence-interval procedure is designed to cover the fixed population parameter in about 95% of samples. The procedure has the coverage property, not each completed interval.
Margin of error = critical value × standard error
After one sample is observed, its endpoints are fixed. In the usual frequentist interpretation, it is inaccurate to say there is a 95% probability that the fixed parameter lies inside those endpoints. A clear report names the parameter, confidence level, method, estimate and interval, then states the assumptions.
Confidence does not correct selection bias, measurement error, missing data, dependence, model misspecification or an unrepresentative sample. A narrow interval can be precisely centered on a biased estimate. Precision and validity are separate questions.
Choose the Correct Confidence Interval
| Target parameter | Required summary | Default method | Main condition |
|---|---|---|---|
| One population mean | Mean, SD and n | Student's t with sample SD | Independent observations and suitable mean sampling model |
| One population proportion | Successes x and total n | Wilson score | Binary outcomes from an appropriate sample |
| Difference in independent means | Mean, sample SD and n for each group | Welch t | Groups are independent |
| Paired mean difference | Mean difference, SD of differences and pairs | Paired t | Pairs are meaningful and independent of other pairs |
| Difference in independent proportions | Successes and totals for both groups | Newcombe-Wilson | Groups are independent with binary outcomes |
Do not treat repeated observations on the same person as independent. Do not use the independent-means mode for before-and-after data. Conversely, pairing unrelated observations merely because the group sizes match does not create a paired design.
Confidence Interval for One Mean
When the population standard deviation is unknown, enter the sample standard deviation and use Student's t with n minus 1 degrees of freedom. The t distribution has heavier tails than the standard normal distribution, especially with small samples.
Known population SD: x-bar ± z1-α/2 × σ / √n
The known-SD Z option is not a shortcut for a large sample. It means the population standard deviation is supplied as a known model quantity. When the entered spread was estimated from the same sample, the t method states that uncertainty directly.
For small samples, a t interval for a mean commonly relies on an approximately normal population without severe outliers. Larger samples make the sample mean more stable under many distributions, but dependence or a strongly biased sampling process still invalidates the ordinary standard error.
Confidence Interval for One Proportion
The sample proportion is p-hat = x/n. The simple Wald interval uses p-hat plus or minus a normal critical value times the estimated standard error. It is familiar, but it can have poor coverage, become too narrow and extend outside 0 through 1 when n is small or p-hat is near a boundary.
Wilson score is the default because it adjusts both the center and width. Agresti-Coull adds a critical-value-based adjustment to the counts before applying a Wald-style calculation. Clopper-Pearson inverts exact binomial tails. Its coverage is at least the nominal level under the binomial model, which often makes it conservative and wider.
| Method | Strength | Limitation |
|---|---|---|
| Wilson score | Good general coverage without an invalid probability range | Not centered exactly on p-hat |
| Agresti-Coull | Simple adjusted approximation | Still an approximation and may be conservative |
| Clopper-Pearson | Exact binomial-tail construction | Often wider than needed |
| Wald | Simple textbook formula | Unreliable near boundaries or with limited counts |
Two Means and Paired Differences
For two independent means, the estimate is mean 1 minus mean 2. Welch's method combines the two standard-error contributions without assuming equal population variances. It uses Welch-Satterthwaite degrees of freedom, which can be fractional. This is the safer default for independent groups.
Welch SE = √(s12/n1 + s22/n2)
The pooled method estimates one common variance and uses n1 + n2 minus 2 degrees of freedom. Select it only when the analysis plan supports an equal-variance model. Similar sample standard deviations do not by themselves prove the population variances are equal.
For paired measurements, first calculate one difference per pair in the same direction. The paired interval is then a one-sample t interval for the mean difference. Enter the standard deviation of the differences. Separate before and after standard deviations do not contain the within-pair correlation needed to reconstruct it.
Difference Between Two Proportions
The point estimate is p-hat 1 minus p-hat 2. The Newcombe-Wilson option combines separate Wilson score limits into a confidence interval for that difference. It respects the possible difference range from -1 through 1 and is the recommended default here.
The unpooled Wald option uses the square root of p1(1-p1)/n1 plus p2(1-p2)/n2 as its standard error. It is included for comparison and for procedures that explicitly require it. Like the one-proportion Wald method, it can behave poorly with sparse successes or failures and may extend outside the feasible parameter range.
Keep group order explicit. Switching sample 1 and sample 2 reverses the sign and swaps the endpoint signs. The substantive conclusion may be equivalent, but the reported direction changes.
Worked Confidence Interval Examples
One mean with unknown population deviation
A sample has mean 72.4, sample standard deviation 8.5 and n = 25. At 95% confidence, df = 24 and the two-sided t critical value is about 2.0639. The standard error is 8.5 / 5 = 1.7. The margin is about 2.0639 × 1.7 = 3.5087, producing an interval from about 68.8913 to 75.9087.
One proportion with Wilson score
Suppose 56 successes occur in 100 independent trials. The estimate is 0.56. A 95% Wilson score interval is approximately 0.4623 to 0.6533. The interval is slightly asymmetric around 0.56 because Wilson adjusts its center as well as its width.
Paired mean difference
For 20 complete pairs, a mean difference of 2.4 and sample SD of differences 4.8 give SE = 4.8 / square root of 20, about 1.0733. With df = 19, the 95% t critical value is about 2.0930, giving a margin near 2.2465 and an interval near 0.1535 to 4.6465.
Confidence Level, Width and Sample Size
Holding the data and method fixed, higher confidence requires a larger critical value and a wider interval. A 99% interval is wider than a 95% interval, while a 90% interval is narrower. Higher confidence does not improve the sample or remove bias.
Larger independent samples usually reduce standard error at roughly the square-root rate. Quadrupling n often halves the standard error when spread stays similar. The relationship is not a promise for clustered, paired, weighted or complex-survey data because those designs need specialized variance estimates.
Do not choose a confidence level after inspecting which interval supports a preferred conclusion. State the level and analysis method in advance when possible. Sample-size planning should use the desired precision, expected variability or proportion, design effect and anticipated nonresponse rather than reverse-engineering the observed interval.
How to Interpret the Result
- Name the parameter. A mean interval does not describe where 95% of individual observations fall.
- Report the estimate and both endpoints with sensible units and precision.
- State the confidence level and method, such as 95% Welch t or 95% Wilson score.
- For a difference, say which group was subtracted from which.
- Compare the full interval with a meaningful effect size, not only with zero.
- Review whether the design supports independence and the stated sampling model.
A confidence interval that excludes zero corresponds to rejection by a matching two-sided hypothesis test at alpha = 1 minus the confidence level, provided both calculations use the same model and method. That link does not turn statistical significance into practical importance.
Common Confidence Interval Mistakes
- Using sample SD as known population SD: This makes the interval too confident for the stated information.
- Choosing Wald for sparse proportions: Prefer Wilson or an appropriate exact method.
- Mixing paired and independent designs: The standard errors use different information.
- Reversing subtraction order: Keep group and difference definitions consistent.
- Entering percentages as whole counts: Proportion modes require successes and total trials.
- Ignoring clusters or repeated observations: Ordinary formulas assume the required independence.
- Rounding intermediate values: Keep full precision until the endpoints are calculated.
- Calling the interval a prediction range: A parameter interval is not a range for future individuals.
Calculation Method and Review Sources
The general t interval for a mean was reviewed against the NIST confidence-limits-for-a-mean reference. One-proportion options follow the formulas and method cautions in the NIST confidence-interval guide and NIST proportion confidence-limit reference.
Independent-mean structure was checked against the NIST two-sample t reference. Paired-difference structure was reviewed against Penn State's paired t-interval lesson. The Newcombe-Wilson difference method follows the interval approach described in the Newcombe independent-proportions paper record.
The engine evaluates normal and Student's t critical values numerically, uses stable scaling for Welch and pooled standard errors, and inverts regularized beta probabilities for exact binomial endpoints. Browser floating-point arithmetic has finite precision, so reproduce regulated or high-stakes results with approved software.
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Frequently Asked Questions
What is a confidence interval?
A confidence interval is a range produced by a statistical procedure to estimate an unknown population parameter. Its confidence level describes the procedure's long-run coverage under its assumptions.
How do I calculate a 95% confidence interval?
Calculate the point estimate and standard error, find the two-sided critical value for 95% confidence, then subtract and add the margin of error. The correct critical distribution and standard error depend on the parameter and design.
Should I use Z or Student's t for a mean?
Use Student's t when the population standard deviation is unknown and you enter a sample standard deviation. Use Z when the population standard deviation is treated as known for the stated procedure.
What is the margin of error?
The margin of error is the critical value multiplied by the estimated standard error. For a symmetric interval, it is the distance from the point estimate to either endpoint.
Is a 99% confidence interval wider than a 95% interval?
With the same data and method, yes. A higher confidence level uses a larger critical value, which increases the margin of error and widens the interval.
Why is Wilson recommended for one proportion?
The Wilson score interval generally has more reliable coverage than the simple Wald interval, especially with small samples or proportions near zero or one.
What is the difference between Welch and pooled intervals?
Welch's interval allows different population variances and estimates fractional degrees of freedom. The pooled interval assumes equal variances and combines the two sample variances.
How do I calculate a paired confidence interval?
Calculate one difference for each matched pair, then enter the mean and sample standard deviation of those differences with the number of complete pairs. The interval uses a one-sample t procedure on the differences.
Does a 95% confidence interval mean a 95% chance the parameter is inside?
Not under the usual frequentist interpretation. After the data are observed, the interval is fixed. The 95% label refers to the long-run success rate of the procedure under repeated sampling and valid assumptions.
Can a confidence interval prove there is no effect?
No. An interval shows estimates compatible with the selected procedure and data. An interval containing zero may still include meaningful positive or negative effects, while a narrow interval requires a design-specific interpretation.
Educational, statistical and financial disclaimer: This calculator provides mathematical estimates for learning and general analysis. It does not provide statistical, academic, medical, legal, employment or financial advice. Verify important data, assumptions, methods and results with approved software or a qualified professional.