Complete guide
Confidence interval calculator for a mean or proportion
Choose mean or proportion, enter sample information and confidence level. Calculator24 returns lower and upper limits, estimate, standard error, margin and method.
What confidence means
Confidence describes the long-run performance of the interval procedure. A 95% method would contain the true parameter in about 95% of repeated samples under its assumptions.
Interval for a mean
When population standard deviation is unknown, use sample standard deviation and a t critical value with n−1 degrees of freedom.
x̄±t*·s/√nInterval for a proportion
The Wilson score interval stays within zero and one and behaves better than the simple p̂±z standard-error interval near boundaries.
Worked mean example
For x̄=50, s=10 and n=100, standard error is 1 and the 95% t multiplier is just under 2, giving limits near 48 and 52.
Width and sample size
Standard error shrinks with √n, so quadrupling sample size roughly halves the random-error component when other inputs remain fixed.
Interpretation limits
The interval does not repair selection bias, measurement error, dependence or an invalid sampling frame. It also does not assign a probability to a fixed parameter after the data are observed.
What changes interval width
| Change | Typical effect |
|---|---|
| Higher confidence | Wider |
| Larger sample | Narrower |
| More variability | Wider |
| Bias | Not repaired by width |
Frequently asked questions
Is there a 95% chance the parameter is inside?
In frequentist terms, 95% describes the long-run success rate of the method, not a probability assigned to the fixed parameter.
Why use t for a mean?
The population standard deviation is usually unknown and estimated from the sample.
Why use Wilson for a proportion?
It behaves better near zero or one than the basic symmetric normal interval.
Can confidence limits exceed the data range?
Mean intervals can; proportion limits are constrained to 0–100%.
Does a narrow interval prove accuracy?
No. Systematic bias can remain.
How does sample size affect width?
The random-error term usually scales approximately with one over the square root of n.
What to keep in mind
The mean interval uses a t critical approximation and assumes independent observations from an approximately normal population or an adequate sample. The proportion interval is Wilson; sampling design and bias are outside scope.
Results display up to 8 decimal places; exported numbers preserve calculation precision. This is a calculation summary, not an official certificate.