Mean Confidence Interval Calculator
Calculate a two-sided confidence interval for a population mean from summary statistics. Choose Student t with sample SD or z with a known population SD.
YOUR CALCULATION
Result
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Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Divide the selected standard deviation by the square root of sample size for the standard error. Multiply by a two-sided critical value, using t with n−1 degrees of freedom when population SD is estimated. Add and subtract the margin from the sample mean.
Interval=x̄±critical×SD/√n
Worked example
With mean100, sample SD10 andn=25, the95% Student-t critical value is about2.063899. Standard error is2 and the interval is approximately[95.872203,104.127797].
Assumptions and input details
Summary values refer to independent, randomly sampled observations. Exact t coverage assumes a normal population; a large-sample approximation does not fix selection bias or dependence. Use2–100000 observations for t and1–100000 for known-sigma z. SD is nonnegative and at most1e12; mean magnitude≤1e12. Confidence levels are90%,95%,99%. A zero SD produces a point interval under the entered model. These limits concern a mean, not individual observations or a prediction interval.
References: NIST confidence limits for the mean
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Results use floating-point arithmetic and displayed rounding. Read the formula, assumptions and any stated data precision before interpreting the result.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
Enter a list of numbers to calculate mean, median, modes, quartiles, population and sample standard deviation, with a frequency histogram.
Estimate the sample count needed for a target margin of error for a mean or a proportion. Optionally apply a finite-population correction and round upward.
Project a sampled proportion to a known population. Compare the estimate with the logically possible count range and, for small sample fractions, a Wilson sampling interval.
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