Confidence Interval Calculator
A confidence interval is the estimate plus or minus a margin of error: a sample of 25 with mean 50 and SD 8 gives a 95 % interval of 46.6978 to 53.3022.
46.6978 to 53.3022
- Margin of error
- ±3.30224
- Standard error
- 1.6
- Critical value t (df = 24)
- 2.0639
Step-by-step solution
- SE = s ÷ √n = 8 ÷ √25 = 1.6
- Margin of error = t × SE = 2.0639 × 1.6 = 3.30224
- CI = 50 ± 3.30224
Confidence Interval Formula
x̄ ± t(α/2, n−1) × s ÷ √n
x̄ sample mean, s sample SD, n sample size; z replaces t when σ is known; p̂ ± z√[p̂(1−p̂)/n] for a proportion
Worked Example
A sample of 25 with mean 50 and standard deviation 8, at 95 %:
- SE = s ÷ √n = 8 ÷ √25 = 1.6
- Margin of error = t × SE = 2.0639 × 1.6 = 3.30224
- CI = 50 ± 3.30224
Standard error: 1.6 · Critical value t (df = 24): 2.0639
What a confidence interval means
A 95 % confidence interval means that if you repeated the sampling many times, about 95 % of the intervals built this way would contain the true value. It does not mean there is a 95 % chance that this particular interval contains it. The interval narrows with a larger sample and widens with higher confidence. Use the t-distribution for a mean when the population standard deviation is unknown, especially for samples under about 30.
Frequently asked questions
When do I use t instead of z?
Use t when the population standard deviation is unknown and you use the sample SD, which is almost always. z is for a known σ or for proportions.
What does a 95 % confidence interval mean?
If you repeated the sampling many times, about 95 % of the intervals built this way would contain the true value. It is not a 95 % chance for this particular interval.
How do I make the interval narrower?
Increase the sample size (the margin falls with √n) or accept a lower confidence level.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology .
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
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APA
Scientific Converters. (2026, October 15). Confidence Interval Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/confidence-interval-calculator/
IEEE
Scientific Converters, "Confidence Interval Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/confidence-interval-calculator/