Calculate a confidence interval for a mean using sample mean, standard deviation, sample size, and confidence level, with margin of error and step-by-step work.
For mean 72, standard deviation 8, sample size 64, and 95% confidence, the confidence interval is about 70.04 to 73.96.
Enter summary statistics to calculate an interval.
How to use the confidence interval calculator
Enter the sample mean, standard deviation, sample size, and confidence level. The calculator finds the standard error, multiplies it by the critical z value, and builds the lower and upper confidence interval bounds.
The mean, standard deviation, and sample size should come from the same sample.
Some classes require t intervals instead of z intervals when the population standard deviation is unknown, especially for smaller samples.
Confidence interval formula
CI = x̄ ± z × s ÷ √n. The quantity s ÷ √n is the standard error. The quantity z × standard error is the margin of error.
Worked example
For a sample mean of 72, standard deviation of 8, sample size of 64, and 95% confidence, z = 1.96. The standard error is 8 ÷ √64 = 1. The margin of error is 1.96 × 1 = 1.96. The interval is 72 ± 1.96, or 70.04 to 73.96.
What a confidence interval means
A 95% confidence interval does not mean there is a 95% chance the fixed true mean is inside one finished interval. In repeated sampling with the same method, about 95% of such intervals would capture the true mean. For practical reading, a narrower interval gives a more precise estimate, assuming the data and method are reliable.
| Change | Effect on interval |
|---|---|
| Higher confidence level | Wider interval. |
| Larger standard deviation | Wider interval. |
| Larger sample size | Narrower interval. |
| Smaller margin of error | More precision, usually requiring more data. |
Common mistakes
- Using standard deviation from one sample with the mean from another sample.
- Confusing standard deviation with standard error.
- Reporting too many decimals and implying more precision than the data supports.
- Using a z interval when the assignment expects a t interval.
- Interpreting confidence as certainty about a single finished interval.
Confidence interval calculator FAQs
What is the margin of error?
The margin of error is the distance from the sample mean to either end of the interval. It equals the critical value multiplied by the standard error.
What is standard error?
Standard error estimates how much sample means vary from sample to sample. For this calculator, it is standard deviation divided by the square root of sample size.
When should I use 95% confidence?
95% is a common default for classwork and general reporting. Use the level required by your teacher, study plan, or research protocol.
Why does a higher confidence level make the interval wider?
Higher confidence uses a larger critical value, so the margin of error becomes larger.
Is this calculator for proportions?
No. This version is for a mean using mean, standard deviation, and sample size. Proportion intervals use a different formula.
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Reviewer
This educational math content is reviewed under the SolverCalc reviewer rule by Prof. Dr. Khalil Mudassar. WordPress adds the author bio automatically, so this page does not include a duplicate author box.
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