Statistics How To

Z Interval: Simple Definition, Formula & Worked Example

Confidence intervals >

“Z-interval” could refer to either:

  1. A specific type of confidence interval (see below), or
  2. A TI-83 calculator option to find a confidence interval for the mean using the Z Interval. If you’re using the TI-83, see this article on confidence intervals for step by step TI-83 Z Interval instructions.

You may want to read this article first: What is a confidence interval?

What is a Z Interval?

A z interval is a specific type of confidence interval which tells you a range where you can expect a particular mean or proportion to fall. It can be calculated from a known standard deviation.

Z-Interval vs. T-Interval

In general, you want to choose to calculate z-intervals when:

  • Your sample size is 30 or greater,
  • You know the standard deviation of your sample.

When you don’t know the standard deviation and still want to find a confidence interval, calculate a t-interval instead. The concept is the same, except that you use a different table to calculate the interval: the z-table for a z-interval and a t-table for a t-interval. For more on the specific differences between the “T” and “Z”, see: T-scores vs. Z-scores.

How to Calculate the Z Interval

A z interval for a mean is given by the formula:

The formula may look a little daunting, but the individual parts are fairly easy to find:

Note: If zα/2 is new to you, read all about zα/2 here. Most of the time, you won’t have to do any calculations because the most common values are already known:

z interval

For most statistical tests, you’ll probably be using one of four confidence intervals (90%, 95%, 98% and 99%). The z alpha/2 for each confidence level is always the same:

For example, if you’re trying to find a z-interval with a 90% confidence level and a 5% alpha (significance) level, then use 1.645 in the above equation.


Suppose we want to calculate a 95 percent confidence interval for a situation where we know:

  • Mean (μ) = 50,
  • Standard deviation = 5,
  • Number of data points in the sample is 81.

For a 95 % confidence interval, the above table tells us that z0.025 = 1.96.

Now we have all the data to plug into our formula, and our confidence interval will be:

[50- (1.96) (5/9), 50 + (1.96)(5/9)]

We can simplify this to:

[48.91, 51.09]

which is our confidence interval or z interval.


Pennsylvania State University Department of Statistics Online Programs. STAT 414 Intro Probability Theory. Lesson 30 Confidence Intervals for One Mean. Retrieved from and on May 30, 2018.

Xie, Yao. Lecture Notes: Confidence Interval. Retrieved from on May 30, 2018.

Walker, Jerimi Ann. Confidence Intervals for the Mean- By Hand. Retrieved from on May 30, 2018.


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Statistical concepts explained visually - Includes many concepts such as sample size, hypothesis tests, or logistic regression, explained by Stephanie Glen, founder of StatisticsHowTo.

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