Confidence Interval Calculator
Build a confidence interval for a mean or a proportion.
What the Confidence Interval Calculator does
A confidence interval gives a range of plausible values for a population parameter, based on your sample. A 95% interval does not mean a 95% chance the true value is inside — it means 95% of intervals built this way would capture it.
Formula
Mean: CI = x̄ ± t × (s ÷ √n)Proportion: CI = p ± z × √(p(1−p) ÷ n)Finite population correction = √((N − n) ÷ (N − 1))
Inputs explained
| Input | Unit | Required | Notes |
|---|---|---|---|
| Interval for | one of 2 options | Yes | — |
| Sample mean | number | In some modes | Shown Interval for is A mean. |
| Standard deviation | number | In some modes | Accepts more than 0. Shown Interval for is A mean. |
| Sample proportion | % | In some modes | Shown Interval for is A proportion. |
| Sample size (n) | number | Yes | Accepts 2 or more. |
| Confidence level | one of 6 options | Yes | — |
| Population size | number | Optional | Optional — applies a finite population correction. Accepts 0 or more. |
How to use it
- Choose Interval for and Confidence level.
- Enter Sample size (n).
- Fill in the remaining inputs the form shows for your choice.
- Optionally add Population size.
- Select Calculate.
Worked example
A sample of 50 with a mean of 72.4 and a standard deviation of 8.6, at 95% confidence.
- Mean
- 72.4
- SD
- 8.6
- Sample size
- 50
- Confidence
- 95%
SE = 1.216, t = 2.010, giving 69.96 to 74.84.
Reading the result
- The "interval at different confidence levels" table makes the notes' trade-off visible directly — a wider interval at 99% versus 95% is the concrete price of more certainty.
- For a mean, the gap between the shown t critical value and the "large-sample" z value indicates how much the small-sample correction matters here — the gap shrinks toward zero past roughly 30 observations.
Assumptions and limitations
- For a proportion, the normal approximation needs at least 5 successes and 5 failures, per the notes — below that threshold the interval becomes unreliable, and no warning is raised if the entered numbers fall short.
- The finite population correction only applies when a population size is entered — leaving it blank assumes an effectively infinite population, appropriate for most surveys but not for sampling a small, fully-enumerable group.
Common mistakes
- Reading "95% confidence" as a 95% probability that the true value lies in this specific computed interval — the FAQ explains the probability describes the method across repeated sampling, not any one interval.
Frequently asked questions
What does 95% confidence actually mean?
That the procedure captures the true parameter 95% of the time across repeated sampling. Any single interval either contains it or does not — the probability describes the method, not the specific interval.
Should I use t or z?
t when the population standard deviation is unknown, which is nearly always. The two converge above about 30 observations.