📊 Confidence Interval Calculator
Build a confidence interval for a population mean or proportion, with the margin of error, critical value and every step of the calculation.
Quick answer: For a sample mean of 50 with s = 8 and n = 25, the 95% confidence interval is 50 ± 2.0639 × 8 ÷ √25 = 50 ± 3.3022, i.e. 46.6978 to 53.3022.
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Confidence Interval Calculator inputs
Result
95% confidence interval for the mean
46.6978 to 53.3022
50 ± 3.3022
- Margin of error
- ±3.3022
- Critical value t* (df = 24)
- 2.0639
- Standard error
- 1.6
Step by step
- Critical value for 95% confidence: t* (df = 24) = 2.0639
- Standard error: s ÷ √n = 8 ÷ √25 = 1.6
- Margin of error: 2.0639 × 1.6 = 3.302238
- Interval: 50 ± 3.3022 = 46.6978 to 53.3022
We are 95% confident that the true population mean lies between 46.6978 and 53.3022.
The same data at other confidence levels
| Confidence | Critical value | Margin of error | Interval |
|---|---|---|---|
| 80% | 1.3178 | ±2.1085 | 47.8915 to 52.1085 |
| 85% | 1.4871 | ±2.3794 | 47.6206 to 52.3794 |
| 90% | 1.7109 | ±2.7374 | 47.2626 to 52.7374 |
| 95% | 2.0639 | ±3.3022 | 46.6978 to 53.3022 |
| 98% | 2.4922 | ±3.9875 | 46.0125 to 53.9875 |
| 99% | 2.7969 | ±4.4751 | 45.5249 to 54.4751 |
| 99.9% | 3.7454 | ±5.9926 | 44.0074 to 55.9926 |
Confidence interval formulas
Mean: x̄ ± t* × s ÷ √n · x̄ ± z* × σ ÷ √n · Proportion: p̂ ± z* × √(p̂(1 − p̂) ÷ n)
The part after ± is the margin of error: the critical value times the standard error.
Worked example
A sample of 25 has a mean of 50 and a standard deviation of 8. Find the 95% interval for the population mean.
- σ is unknown, so use t with 25 − 1 = 24 degrees of freedom: t* = 2.0639.
- Standard error = 8 ÷ √25 = 1.6.
- Margin of error = 2.0639 × 1.6 = 3.3022.
- Interval = 50 ± 3.3022 = 46.6978 to 53.3022.
With z instead of t (1.96 × 1.6 = 3.136) the interval would be slightly narrower – which is why using z with a small sample overstates your precision.
Critical values
| Confidence | z* | t* (df = 10) | t* (df = 30) |
|---|---|---|---|
| 80% | 1.2816 | 1.3722 | 1.3104 |
| 90% | 1.6449 | 1.8125 | 1.6973 |
| 95% | 1.9600 | 2.2281 | 2.0423 |
| 98% | 2.3263 | 2.7638 | 2.4573 |
| 99% | 2.5758 | 3.1693 | 2.7500 |
How to use the calculator
Choose whether you are estimating a mean or a proportion. For a mean, enter the sample mean, the standard deviation and the sample size; for a proportion, enter the number of successes and the sample size. Pick the confidence level and the interval, margin of error and working appear instantly.
Assumptions
- The sample is random and the observations are independent.
- For a mean: the data is roughly normal, or n is large enough (about 30+) for the central limit theorem.
- For a proportion: the Wald interval needs at least about 10 successes and 10 failures; otherwise rely on the Wilson interval.
Frequently asked questions
Should I use z or t?
Use t whenever you estimate the spread from the sample itself (the usual case) – it widens the interval to allow for the extra uncertainty, with n − 1 degrees of freedom. Use z only when the population standard deviation σ is genuinely known. For large samples the two are almost identical.
What does a 95% confidence interval mean?
If you repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true population value. It does not mean there is a 95% probability that this particular interval contains it.
Wald or Wilson for a proportion?
The Wald interval p̂ ± z√(p̂(1−p̂)/n) is the textbook formula but performs poorly for small samples or proportions near 0% or 100% – it can even go below 0. The Wilson score interval has much better coverage and is recommended in those cases.
How can I make the interval narrower?
Increase the sample size (the margin shrinks with √n, so quadrupling n halves it), accept a lower confidence level, or reduce variability in the measurements.