About Z Confidence Interval Calculator
This z confidence interval calculator gives an interval for a population mean when the population standard deviation (sigma) is known, using the normal distribution rather than t. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
A known sigma is rare in practice. It happens with a well-characterised measuring instrument, an exam with a published standard deviation, or textbook problems. When it applies, the interval uses z instead of t and the sample size does not change the critical value.
The page opens on a mean of 100, a sigma of 15 and 25 observations. Change the level to 99% and the margin of error grows from 5.88 to 7.73.
How to use Z Confidence Interval Calculator
- Choose what you want an interval for: a mean (standard deviation unknown or known), a proportion or a difference of means.
- Enter raw data or summary statistics. For a proportion enter the number of successes and the sample size.
- For a known-sigma interval, type the population standard deviation.
- Pick the confidence level: 90%, 95% or 99%.
- Read the interval and the margin of error, then open the working to see each formula.
Features
- Mean with unknown sigma (t interval) or known sigma (z interval).
- Proportion with the Wald and Wilson score methods, compared side by side, with a small-sample warning.
- Difference of two means, with the Welch method by default.
- Raw data or summary statistics.
- Margin of error, critical value, standard error and degrees of freedom shown.
- A bell curve with the central confidence region shaded.
- Step-by-step KaTeX working and a plain-English reading of the interval.
Z interval for a mean
The critical value comes from the standard normal curve.
x̄ ± z(α/2) · σ / √nWorked examples
mean 100, σ 15, n 25, 95% gives 94.12 to 105.88, z = 1.96
mean 100, σ 15, n 25, 99% gives 92.273 to 107.73, z = 2.5758
Privacy: how it works
- The interval is calculated by code in your own browser tab. Your numbers are not sent to Ulkaa or any other server.
- There is no account and nothing is stored once you close the tab.
Tips
- Use the t interval unless you truly know the population standard deviation. Knowing it is rare outside textbooks.
- For a proportion, prefer Wilson. Use Wald only when you are matching a textbook answer.
- If a difference interval contains 0, the data are consistent with no difference at that confidence level.
- Quadrupling the sample size roughly halves the margin of error.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your data are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your data are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
When is it right to use z instead of t?
When the population standard deviation is known, not estimated from the sample. If you only have the sample standard deviation, use the t interval.
Do I need to enter the sample standard deviation?
No. Only the mean, the sample size and sigma are used in this mode.
More Confidence Interval Calculator pages
- Confidence Interval CalculatorGet a 90%, 95% or 99% confidence interval for a mean, a proportion or a difference of means, with every step shown.
- 95% Confidence Interval CalculatorType a mean, a standard deviation and a sample size and get the 95% interval and its margin of error.
- Confidence Interval for the Mean CalculatorPaste a list of values and get the confidence interval for the population mean.
- Confidence Interval for a Proportion CalculatorEnter the number of successes and the sample size to get a Wilson or Wald interval.
- Confidence Interval for a Difference of MeansEnter two groups and get the interval for the difference between their population means.