Confidence Interval Calculator (Mean)
Build a confidence interval for a population mean from the sample mean, standard deviation, size, and confidence level using the t-distribution.
Result
- Sample mean100.0000
- Standard error (s/โn)3.0000
- Critical t* (df 24)2.0639
- Margin of error6.1917
- Lower bound93.8083
- Upper bound106.1917
Step-by-step
- Standard error = s รท โn = 15 รท โ25 = 3.0000.
- Critical value t* at 95% with 24 df = 2.0639.
- Margin = t* ร SE = 2.0639 ร 3.0000 = 6.1917; CI = 100.000 ยฑ 6.192 = [93.808, 106.192].
How to use this calculator
- Enter the sample mean and sample standard deviation.
- Enter the sample size and choose a confidence level (90/95/99%).
- Read the interval and the margin of error.
- Report the interval as "mean ยฑ margin," noting the confidence level.
About this calculator
A confidence interval gives a plausible range for an unknown population mean, based on a sample. This calculator builds the interval the standard way: take the sample mean and add and subtract a margin of error equal to a critical t-value times the standard error of the mean (the sample standard deviation divided by the square root of the sample size). It uses the t-distribution rather than the normal distribution because, in practice, the population standard deviation is unknown and estimated from the sample โ the t-distribution's heavier tails account for that extra uncertainty, especially at small sample sizes, and converge to the normal curve as the sample grows. A 95% confidence interval means that if you repeated the sampling many times, about 95% of the intervals constructed this way would contain the true mean.
How it works โ the formula
SE = s รท โn
Margin = t*(df = nโ1) ร SE
CI = xฬ ยฑ MarginThe standard error scales the sample spread by sample size; the t critical value sets how many standard errors wide the interval must be for the chosen confidence.
Worked examples
- Inputs:
- mean=100, sd=15, n=25, confidence=0.95
- Output:
- t*=2.064, ยฑ6.19 โ [93.81, 106.19]
- Inputs:
- mean=50, sd=8, n=100, confidence=0.95
- Output:
- ยฑ1.587 โ [48.41, 51.59]
- Inputs:
- mean=100, sd=15, n=25, confidence=0.99
- Output:
- t*=2.797, ยฑ8.39 โ [91.61, 108.39]
Limitations
- Assumes a random sample from a roughly normal population (or large n).
- For proportions or other parameters, a different interval is needed.
- Critical t-values are computed numerically (accurate to ~3 decimals).
Inferential result; validity depends on the sample being representative.