Editverse Free Calculator
Confidence Interval Calculator for a Mean
Estimate the plausible range for a population mean from one sample with the one-sample t interval at 90%, 95% or 99% confidence, alongside the mean, SD and standard error. See the reproducible R command and read a full interpretation guide below. Validated against R 4.3.3.
In Simple Words
A confidence interval (CI) gives a range of plausible values for something you cannot measure directly — usually a population mean — based on the sample you collected. You enter your sample values and a confidence level, and the calculator returns a lower and upper bound, for example “mean 24.15, 95% CI 22.76 to 25.54”.
The width of the interval reflects your uncertainty: larger samples and less variability give narrower, more precise intervals. One subtle but important point: a 95% CI does not mean there is a 95% probability the true value sits inside this particular interval. It means that if you repeated the study many times, about 95% of the intervals built this way would contain the true value.
When to Use a Confidence Interval
Report a confidence interval whenever you estimate a population parameter — a mean, a proportion, a difference between groups, or an effect size — from sample data. A CI communicates both the estimate and its precision, which a p-value alone cannot.
Use the t-based interval (used here) when estimating a mean from a continuous, roughly normal sample, especially when the population standard deviation is unknown and the sample is small. For large samples, the t and z intervals nearly coincide. Major reporting guidelines now recommend presenting confidence intervals alongside, or instead of, bare p-values, because they show the range of effects compatible with the data rather than a single significant/not-significant verdict. Pair this with a sample size calculation at the design stage to target the precision you need. Not sure which estimate fits your design? Our interactive Test Selector can help.
Expected format — Sample values
Drop a CSV here or click to choose. Parsed in your browser — nothing is uploaded.
Expected format — Sample values
x <- c({data})
t.test(x, conf.level = {conf})$conf.intUser guide
When to use. Estimate the plausible range for a population mean from a sample.
Assumptions. Sample is random; for small n the underlying distribution is approximately normal.
Common mistakes. A 95% CI does not mean a 95% probability the true value lies in this one interval.
Worked example
Click Load example data above, then Run Analysis to see a fully worked result with figure, formula and interpretation for this test.
References
- Neyman J. Outline of a theory of statistical estimation. Philos Trans R Soc A. 1937;236:333-380.
Worked Examples
Example 1 — Lab measurements
Six replicate measurements (23.1, 25.4, 22.8, 24.0, 26.1, 23.5) have a mean of 24.15. The 95% confidence interval runs from 22.76 to 25.54, the range of population means compatible with this small sample. Report the mean with its interval, not the mean alone, so readers can judge precision.
Example 2 — Why the level changes the width
Re-running the same data at 99% confidence produces a wider interval, and at 90% a narrower one. Higher confidence buys a wider net; it does not make the estimate more accurate. Choose the level before analysis — 95% is the convention — rather than picking whichever looks best.
Assumptions to Check
Random sampling. The interval is only meaningful if the sample is representative of the population you want to describe.
Approximate normality (small n). The t-based interval assumes the underlying data are roughly normal. With larger samples the central limit theorem makes this less critical.
Independent observations. Each value should be independent. Clustered or repeated measurements need methods that account for the correlation.
Common Mistakes
Misreading the probability statement
A 95% CI does not mean a 95% chance the true value is in this one interval. The 95% refers to the long-run performance of the procedure across many samples.
Treating “overlapping CIs” as a formal test
Two intervals can overlap and still represent a significant difference. To compare groups, test the difference directly rather than eyeballing overlap.
Reporting only a p-value
A p-value says whether an effect is detectable; the CI shows how large it plausibly is. Report both.
Choosing the confidence level after seeing the data
Decide on 90%, 95%, or 99% in advance. Picking the level to obtain a desired conclusion is a form of bias.
Frequently Asked Questions
What does a 95% confidence interval actually mean?
If you repeated the study many times and built an interval each time, about 95% of those intervals would contain the true population value. It is a statement about the method, not about this single interval.
Why report a CI instead of just a p-value?
A confidence interval shows the range of effects compatible with your data and its precision, which is more informative than a single significant/not-significant verdict.
How do I get a narrower interval?
Increase the sample size or reduce measurement variability. A larger n shrinks the margin of error.
Does the result match SPSS, R, and Stata?
Yes. The calculator uses the same t-distribution and standard-error formulas as standard software and shows the exact R command so you can reproduce it.
Related Calculators
References
- Neyman J. Outline of a theory of statistical estimation based on the classical theory of probability. Philos Trans R Soc Lond A. 1937;236(767):333–380.
- Rothman KJ. Six persistent research misconceptions. J Gen Intern Med. 2014;29(7):1060–1064. PMID 24452418.
- Greenland S, Senn SJ, Rothman KJ, et al. Statistical tests, P values, confidence intervals, and power: a guide to misinterpretations. Eur J Epidemiol. 2016;31(4):337–350. PMID 27209009.
Need This Analysis Done for Your Manuscript?
Our PhD biostatisticians compute and interpret your confidence intervals, and deliver publication-ready tables with reproducible R output.
Get Expert Statistical Help