Ci Interval Calculator
Confidence Interval Calculator works out Lower limit and Upper limit from Interval for, Sample size (n) and Confidence level. The default figures on this page give margin of error of 1.959964; change any entry and the result, the working and the sources update together.
Your figures
Updates as you typeHow this is worked out
4 steps
- Standard error = 10 / sqrt(100) = 1.
- The 97.5th percentile of the standard normal distribution is 1.9599639845.
- Margin of error = 1.9599639845 x 1 = 1.9599639845.
- Interval = 50 +/- 1.9599639845 = (48.04003602, 51.95996398).
Its 5 worked examples are re-run on every release, last on 15 September 2026. What changed
Sources and references
5 sources
What the Confidence Interval Calculator calculates
Computes a confidence interval for a population mean (Student t when sigma is unknown, z when it is known) or for a proportion (Wald or Wilson score), with the critical value, standard error, margin of error and an optional finite-population correction, plus the Bayesian credible interval from a Beta posterior.
How to use this Confidence Interval Calculator
Enter Interval for, Sample size (n), Confidence level and Sample mean. Lower limit updates as you type, and “Use a worked example” loads checked figures you can adjust. Change one figure at a time to see what drives the result.
How the result is worked out
Open “How this is worked out” beside the result for the step-by-step calculation behind Lower limit, Upper limit and Point estimate.
The method follows NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.5.2 Confidence Limits for the Mean, Wilson, E.B. (1927), "Probable Inference, the Law of Succession, and Statistical Inference", JASA 22(158), 209-212, Brown, L.D., Cai, T.T. & DasGupta, A. (2001), "Interval Estimation for a Binomial Proportion", Statistical Science 16(2), 101-133, Cochran, W.G. (1977), Sampling Techniques, 3rd ed., section 2.8 and Gelman et al. (2013), Bayesian Data Analysis, 3rd ed., chapter 2.
Mean with a known sigma: n = 100, mean 50, sigma 10, 95% example
With Interval for: mean-z, Sample size (n): 100, Sample mean: 50, Population standard deviation (sigma): 10, the calculator returns Standard error: 1, Critical value: 1.95996398, Margin of error: 1.95996398. This example is checked against an independent source in the test suite.
What you can compare
Change one input at a time to see how it affects Lower limit, Upper limit and Point estimate. The inputs you can vary are Interval for, Sample size (n), Confidence level, Sample mean, Sample standard deviation (s) and Population standard deviation (sigma).
Understanding Lower limit, Upper limit and Point estimate
A confidence interval only describes sampling error under its stated model. It says nothing about bias, non-response or measurement error, and the Bayesian mode is only as defensible as the prior you enter.
Results are general information, not financial, tax, legal, medical or engineering advice. Check any decision that matters with a qualified professional.
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Frequently asked questions
How to calculate a 95% confidence level?
Computes a confidence interval for a population mean (Student t when sigma is unknown, z when it is known) or for a proportion (Wald or Wilson score), with the critical value, standard error. Open “How this is worked out” beside the result to see each step for your own figures.
How many SD is 95% confidence?
On the figures this page opens with — confidence level 95%, sample mean 50 and population standard deviation (sigma) 10 — the margin of error is 1.959964. No single number fits everyone: change any figure above and the answer updates as you type, with each step of the arithmetic beside it.
How is confidence calculated?
It needs Interval for, Sample size (n), Confidence level, Sample mean, Sample standard deviation (s) and Population standard deviation (sigma). Nothing else is required, and the figures stay in your browser.
How to use TI-84 for confidence interval?
The result gives Lower limit, Upper limit, Point estimate, Margin of error, Standard error and Critical value, each labelled with its unit so the figure can be read on its own.
What information do I need for the Confidence Interval Calculator?
You need Interval for, Sample size (n), Confidence level, Sample mean, Sample standard deviation (s), Population standard deviation (sigma), Number of successes (x) and Sample proportion. The page opens with default figures that give margin of error of 1.959964, so you can see the scale before entering your own.
What does this Confidence Interval Calculator show?
It shows Lower limit, Upper limit, Point estimate, Margin of error, Standard error, Critical value, Degrees of freedom and Confidence level, with the working and any warnings for your entries.