Confidence Interval CalculatorFree online math calculator — instant results, no sign-up
Interval for a mean.
About the Confidence Interval Calculator
This calculator determines the confidence interval and margin of error for a population mean based on sample statistics. Researchers, students, and analysts use it to estimate the plausible range of an unknown population average given an observed sample.
- Sample mean
- The calculated average of the collected sample data points, expressed in your measurement units.
- Std. deviation
- The standard deviation of the sample or population, representing data dispersion. Must be greater than 0.
- Sample size
- The total number of observations in the sample (n). Must be a positive integer.
- Confidence
- The chosen confidence level: 90%, 95%, or 99%, corresponding to standard normal critical z-values.
Formula and method
e = z × (s / √n) Lower Bound = m − e Upper Bound = m + e
- mSample mean
- sStandard deviation
- nSample size
- zCritical value from the standard normal distribution (1.645 for 90%, 1.96 for 95%, 2.576 for 99%)
- eMargin of error
The tool selects a predetermined z-score based on the selected confidence level (1.645, 1.96, or 2.576) and computes the standard error by dividing standard deviation by the square root of sample size. It then multiplies the standard error by the z-score to find the margin of error, which is added to and subtracted from the sample mean. Results are rounded to 4 decimal places.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Sample mean50
- Std. deviation8
- Sample size40
- Confidence95%
Results
- Interval47.5208 – 52.4792
- Margin of error± 2.4792
How to use the Confidence Interval Calculator
- 1Select the critical z-score corresponding to the chosen confidence level (1.645 for 90%, 1.96 for 95%, or 2.576 for 99%).
- 2Calculate the standard error of the mean: SE = s / √n.
- 3Multiply the critical value by the standard error to obtain the margin of error: e = z × SE.
- 4Subtract and add the margin of error to the sample mean to define the interval [m − e, m + e].
Reading your result
The margin of error indicates the expected half-width variation around the sample mean at your selected confidence tier. The resulting range represents the interval that would contain the true population mean across repeated sampling under standard normal assumptions.
Limitations
This calculator uses standard normal z-critical values rather than Student's t-distribution critical values, which makes it less accurate for small sample sizes (typically n < 30) where the population standard deviation is unknown. It also assumes random sampling and an approximately normal sampling distribution.
Common questions
Why does this calculator use z-scores instead of t-scores?+
It applies a standard normal z-approximation using fixed critical values (1.645, 1.96, and 2.576). For small samples where the population standard deviation is unknown, a Student's t-distribution is theoretically preferred.
How does increasing the sample size affect the confidence interval?+
Because sample size is in the denominator under a square root (√n), increasing n decreases the standard error, resulting in a smaller margin of error and a narrower interval.
Why is a 99% confidence interval wider than a 95% confidence interval?+
Higher confidence requires capturing a larger proportion of the distribution, which demands a higher critical z-value (2.576 versus 1.96) and therefore produces a wider margin of error.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
Further reading
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Results are estimates for informational purposes and are not financial, medical or legal advice.