Confidence interval calculator (mean or proportion)
Your sample gives an estimate. The confidence interval shows the range where the true value probably lies.
Result
[45.8722, 54.1278]Confidence interval
- Lower bound
- 45.8722
- Upper bound
- 54.1278
- Margin of error (±)
- 4.1278
- Standard error
- 2
- Critical value t = 2.0639 (df = 24)
- Assumptions: random sample, roughly normal data or a large enough n; σ estimated by the sample standard deviation.
Inputs
- What you estimate
- Mean (Student t)
- Sample mean
- 50
- Standard deviation
- 10
- Sample size (n)
- 25
- Confidence level
- 95%
How it is calculated
Mean: mean ± q × standard deviation ÷ √n.
Result
Confidence interval
- Lower bound
- 45.8722
- Upper bound
- 54.1278
- Margin of error (±)
- 4.1278
How it is calculated
- Mean: mean ± q × standard deviation ÷ √n.
- q comes from Student’s t law with n − 1 degrees of freedom, or from the normal law when σ is known (1.96 at 95%).
- Proportion: p = yes ÷ n, Wilson interval (more accurate than p ± z√(p(1 − p)/n), shown for comparison).
Example
Mean 50, standard deviation 10, n = 25, 95%, Student: t = 2.0639 (24 df), margin ≈ 4.1278, interval [45.8722, 54.1278].
Good to know
- This is not a prediction interval: it brackets the true mean, not a single new measurement.
- 95% confidence means the method works 95 times out of 100, not that the true value has a 95% chance of lying in this exact interval.
- A biased sample gives a wrong interval, even a narrow one.
Frequently asked questions
What is this tool used for?
Your sample gives an estimate. The confidence interval shows the range where the true value probably lies.
How is the result calculated?
Mean: mean ± q × standard deviation ÷ √n.
What should I check before using the result?
This is not a prediction interval: it brackets the true mean, not a single new measurement.