What Is Statistics Confidence Interval?
A confidence interval gives a range of plausible values for a population mean or proportion. It is built from the sample estimate, its standard error and a critical value.
Choose a mean or a proportion, enter your summary statistics and a confidence level. You get the interval, margin of error, standard error and critical value, a comparison of the 90, 95 and 99 percent intervals and the sample size that would halve the width.
Below are the formulas, a worked example and answers on how to read a confidence interval.
How to Use the Statistics Confidence Interval Calculator
- Choose mean or proportion.
- Enter the summary statistics.
- Set the confidence level.
- Compare the 90, 95 and 99 % intervals.
Statistics Confidence Interval Formula
Interval for a mean
Wilson interval
Statistics Confidence Interval Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| x̄ | Sample mean | |
| s | Sample standard deviation | |
| n | Sample size | |
| t* | Critical value |
Statistics Confidence Interval Worked Example
A sample of 30 has mean 52.4 and SD 8.1. The standard error is 8.1 ÷ √30 = 1.479 and t* (29 df, 95 %) = 2.045, so the margin of error is 3.024 and the interval is 49.38 to 55.42.
With the example values loaded in the calculator above, the steps are:
- Standard error = s ÷ √n = 8.1 ÷ √30 = 1.47885.
- t* for 95 % with 29 df = 2.04523.
- Margin of error = t* × SE = 3.02459.
- Interval = 52.4 ± 3.02459 = 49.3754 to 55.4246.
Result: 95 % confidence interval = 49.3754 to 55.4246; Sample mean = 52.4; Margin of error = 3.02459.
How to Read a Confidence Interval
A sample of 30 with mean 52.4 and SD 8.1 gives a 95% interval of 49.38 to 55.42. The interval is the estimate plus or minus a margin of error, here 3.02.
The 95% describes the method, not this single interval. If you repeated the study many times, about 95% of intervals built this way would contain the true mean. The table on this page shows how the same data give narrower intervals at lower confidence and wider ones at higher confidence.
Where Statistics Confidence Interval Is Used
Estimating a mean
Build an interval for a population mean from a sample mean, SD and n.
Estimating a proportion
Use the number of successes and n for a poll or conversion rate.
Comparing levels
See how 90%, 95% and 99% change the width.
Planning
Find the sample size that would halve the width.
Common Statistics Confidence Interval Mistakes to Avoid
- For a mean, the t critical value depends on the sample size. It is above 1.96 for small samples.
- A wider interval is not a mistake. It means more uncertainty, usually from a smaller sample.
- Do not read the interval as "95% of the data". It describes the mean, not individual values.
- Halving the width needs about four times the sample size.
Statistics Confidence Interval Terms and Definitions
- Margin of error
- Half the width of the interval.
Statistics Confidence Interval FAQs
How do I calculate a confidence interval?
Compute the standard error, multiply it by the critical value for your confidence level, and add and subtract that margin from the sample estimate. For a mean the critical value is from the t distribution with n − 1 degrees of freedom.
What does 95 % confidence mean?
It means the procedure captures the true value in about 95% of repeated samples. It does not mean there is a 95% probability that this one interval contains it.
Why does a higher level give a wider interval?
To be more sure of capturing the true value, you need a larger critical value, which widens the margin. At 99% the interval is wider than at 95% for the same data.
How can I narrow the interval?
Increase the sample size, reduce the variability of the measurements, or accept a lower confidence level. Quadrupling n halves the width.