DIY Cost Tools
Confidence Interval Calculator
MATH & STATISTICS CALCULATOR
Estimate a two-sided confidence interval for a mean using a normal critical value, sample mean, standard deviation, and sample size.
How it works
The calculator divides the standard deviation by the square root of the sample size to find the standard error. It multiplies that value by the selected normal critical value and adds and subtracts the resulting margin of error from the sample mean.
Formula
Standard error = s ÷ √n
Margin of error = z × standard error
Confidence interval = sample mean ± margin of error
Important limits
- This tool uses a normal z critical value. When the population standard deviation is unknown and the sample is small, a t interval is usually more appropriate.
- The observations should be independent and the sampling method should be appropriate for statistical inference.
- Strong skew, outliers, clustered samples, complex survey designs, repeated measures, or biased sampling may make this interval misleading.
- A 95% confidence level does not mean there is a 95% probability that a fixed parameter lies inside one already-computed interval.
Frequently asked questions
What does the confidence level mean?
Under the model assumptions, a method that produces 95% intervals would capture the true parameter in about 95% of repeated samples.
Why does a larger sample make the interval narrower?
The standard error divides by the square root of the sample size, so increasing the sample size reduces sampling uncertainty when the standard deviation stays the same.
Should I enter sample or population standard deviation?
Use the population standard deviation when it is genuinely known. A sample standard deviation can be used as an approximation for large samples; smaller samples typically call for a t critical value.
Review notes
The default 95% example was checked by hand: standard error 1.50, margin 2.94, and interval 69.06 to 74.94. Zero, negative, nonnumeric, decimal, and small-sample inputs were also tested. Last reviewed September 30, 2026.