Confidence Interval Calculator
Calculate a confidence interval for a mean using Student’s t (sample SD) or z (known SD), with margin of error and standard error.
Your inputs
Results
Confidence interval
45.8722 to 54.1278
- Lower bound
- 45.8722
- Upper bound
- 54.1278
- Margin of error
- 4.1278
- t critical value (df = 24)
- 2.063899
- Standard error
- 2
✓Exact calculation — rounded only for display
Notes for this result
- Assumes a random sample and an approximately normal sampling distribution (reasonable for large n by the central limit theorem).
What this calculator does
A confidence interval gives a range of plausible values for the population mean based on a sample.
Formula and method
CI = mean ± critical value × SD ÷ √n, where the critical value is t(1 − α/2, n − 1) or z(1 − α/2)
Worked example
Calculated by this tool with its default inputs:
Inputs
- Sample mean: 50
- Standard deviation: 10
- Sample size: 25
- Confidence level: 95 %
- Standard deviation is: From the sample (use Student’s t)
Results
- Confidence interval: 45.8722 to 54.1278
- Lower bound: 45.8722
- Upper bound: 54.1278
- Margin of error: 4.1278
- t critical value (df = 24): 2.063899
- Standard error: 2
How to read the results
A 95 % interval procedure captures the true mean in 95 % of repeated samples.
Assumptions and limitations
- Assumes random sampling and approximate normality of the sample mean.
Questions
Why use t instead of z?
When the SD is estimated from the sample, t accounts for the extra uncertainty, especially for small samples.
Methodology and verification
- Result type
- Exact mathematical calculation (rounded only for display)
- Formula version
- 1.0
- Automated verification
- 4 reference and validation test cases, checked against independently computed values. The site cannot be built if any of them fail.
- Editorial review
- Awaiting human editorial review. The explanation was drafted with AI assistance; the formula and examples are verified by the automated tests above.
Privacy
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