Z Critical Value Calculator


Critical Value of Z (Right Tailed): 1.65
Critical Value of Z (Two Tailed): ± 1.96

Tool Overview: Z Critical Value Calculator

Stuck trying to interpret the results of a statistical test - specifically finding the critical values for a standard normal distribution? You've come to the right place. Our free statistics package is intended as an alternative to Minitab and other paid software. This critical value calculator generates the critical values for a standard normal distribution for a given confidence level. The critical value is the point on a statistical distribution that represents an associated probability level. It generates critical values for both a left tailed test and a two-tailed test (splitting the alpha between the left and right side of the distribution). Simply enter the requested parameters (alpha level) into the calculator and hit calculate.

What Is a Critical Value and How Do You Use It?

A critical value is a concept from statistical testing. If we are performing hypothesis testing, we will reduce our proposition down to a single pair of choices, referred to as the null hypothesis and the alternative hypothesis. The null hypothesis denotes what we will believe to be correct if our sample data fails the statistical test. As a matter of form, it should usually reflect the default state for your process (eg. expected from normal operations). The alternative hypothesis represents an atypical outcome for the process, in which case we infer that something occured. We will then identify when and from where we shall draw a sample to assess which of these two alternatives is most likely to be correct. We will calculate a test statistic for the sample which we will compare to the expected distribution of the statistic to assess the relative probability of the hypothesis being correct. Bear in mind that this entire process exists in a probabilistic universe; we cannot opine on truth but only likelihood. We will identify the most appropriate distribution for comparing this sample (see below on when to use a standard normal vs. a t distribution). The critical value is the point in that distribution at which we must accept the alternative hypothesis as being more likely.

Note: This particular calculator is designed to find the critical value for the mean of a standard normal distribution. If your sample size is small (and you're still looking at the mean), you should use the t statistic sample distribution (we have a separate t score calculator for the t critical value). Both of these assume you're comparing the mean of the sample distribution to a fixed value. If you're trying to make inferences comparing the means of two populations (eg. both are "moving targets" vs. specified at a certain level, you're going to want to use the f statistic. (We're working on a calculator for the f critical value; you can find a table in the back of any statistics text book or here.)

In the case of the Z critical value all we need to calculate the critical value is the significance level (the alpha value) for the test. The alpha value reflects the probability of incorrectly rejecting the null hypothesis. The Z critical value is consistent for a given significance level regardless of sample size and numerator degrees. Common confidence levels for academic use are .05 (95% confidence), .025 (97.5%), and .01 (99%). That being said, a wise analyst compares the benefits of the required confidence level against the costs of achieving it (eg. don't always default to alpha .05 or .01).

How To Find Critical Values of Z

This calculator is intended to replace the use of a Z value table while providing access to a wider range of possible values for you to work with. In the offline version, you use a z score table (aka a z table) to look up the critical value for the test based on your desired level of alpha. Remember to adjust the alpha value based on wether you are doing a single-tailed test or two tailed test. In this case, we can simply split the value of alpha in two since the standard normal distsribution is symmetric about its axis. From there, finding the critical values for your test is a matter of looking up the appropriate row and column in the table. Our critical values calculator automates this process, so all you need to do is enter your alpha value and the tool will find the critical values for you.

When to Use Standard Normal (Z) vs. Student's T distribution

This calculator requires you to have sufficiently large sample that you are comfortable the values of the mean will converge on the standard normal distribution via the central limit theorem. This genreally requires you to have 30+ observations. If you are working with a smaller sample, you should consider using the version we set up to find critical values of a t-distribution. In any event, to run the hypothesis test you compare the observed value of the statistic with the t value from the t distribution table.

About this Website

This calculator is part of a larger collection of tools we've assembled as a free replacement to paid statistical packages. The other tools on this site include a descriptive statistics tool, confidence interval generators (standard normal, proportions), linear regression tools, and other tools for probability and statistics. Many calculators allow you to save and recycle your data in similar calculations, saving you time and frustration. Bookmark us and come back when you need a good source of free statistics tools.

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