Confidence Interval One Sample Z Test

Sample Mean

The average value observed in your sample.

Population Standard Deviation

The known standard deviation of the full population, not just your sample.

Sample Size

Total number of people or items in your sample.

Confidence Level (%)

Common choices are 90, 95, or 99.

About This Test

What Is a One-Sample Z Confidence Interval?

This builds a confidence interval for a population mean when the population's standard deviation is already known — for example from extensive historical records or a well-established manufacturing process. If you only know your sample's standard deviation, you want the T Confidence Interval calculator instead, which accounts for that extra uncertainty.

When to Use It

Quality control processes with a long-established, known standard deviation
Standardized tests where the population standard deviation is published
Any situation where the population standard deviation is genuinely known, not estimated

The Formula

\[ \bar{X} \pm z_{\alpha/2}\,\frac{\sigma}{\sqrt{n}} \]

Worked Example

A bottling machine has a known population standard deviation of 2 ml from years of records. A sample of 40 bottles has a mean fill of 500 ml. What's the 95% confidence interval for the true average fill?

\( \bar{X} = 500, \ \sigma = 2, \ n = 40 \)
Critical value for 95%: \( z_{\alpha/2} = 1.96 \)
Margin of error: \( 1.96 \times 2/\sqrt{40} \approx 0.62 \)
95% confidence interval: (499.38, 500.62)
A large sample and a small known standard deviation give a very tight, precise interval.

Your Results

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