What Is a Critical Value in Statistics?

A critical value is a threshold used in statistics to make decisions in hypothesis testing and to construct confidence intervals. It marks the boundary between where we reject or fail to reject a null hypothesis. For example, if your test statistic exceeds the critical value, you have enough evidence to reject the null hypothesis. Critical values depend on the chosen significance level (α), the type of test (one-tailed or two-tailed), and the probability distribution (Z, t, Chi-Square, or F).

Where Do Critical Values Come From?

Critical values are derived from probability distributions. Each distribution (Z, t, Chi-Square, F) has a set of critical values that correspond to different significance levels and degrees of freedom. The most common distributions used in practice are:

• Z (Standard Normal) distribution – used for large sample sizes (n ≥ 30) when the population standard deviation is known.
• t (Student's t) distribution – used for small sample sizes (n < 30) or when the population standard deviation is unknown.
• Chi-Square (χ²) distribution – used for tests involving variance, goodness of fit, or independence in categorical data.
• F distribution – used for comparing variances of two populations or in ANOVA.

For example, a Z critical value for a 95% confidence level two-tailed test is 1.96. This means that 95% of the data falls within ±1.96 standard deviations of the mean. If your test statistic is more extreme than 1.96 or -1.96, you reject the null hypothesis.

Why Critical Values Matter

Critical values are essential because they provide a clear, standardized rule for making statistical decisions. Without them, researchers would have to rely on gut feelings or arbitrary cutoffs. They allow you to control the risk of making a Type I error (rejecting a true null hypothesis). By choosing a significance level (e.g., α = 0.05), you set the critical value so that the probability of a Type I error is exactly 5%.

You can learn more about the process of finding these values in our How to Find Critical Values: Step-by-Step (2026) guide.

How Critical Values Are Used

In Hypothesis Testing

In hypothesis testing, you first calculate a test statistic (like a Z-score or t-score) from your sample data. Then you compare it to the critical value. The decision rule is:

• If |test statistic| > critical value (two-tailed) – Reject the null hypothesis.
• If test statistic > critical value (right-tailed) – Reject the null hypothesis.
• If test statistic < critical value (left-tailed) – Reject the null hypothesis.

The formula for a Z-test statistic is: \( Z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}} \)

For instance, suppose a school claims its students have an average IQ of 100. You take a random sample of 36 students and find an average IQ of 105, with a population standard deviation of 15. Your test statistic is: \( Z = (105 - 100) / (15 / √36) = 5 / 2.5 = 2.0 \). For a two-tailed test at α = 0.05, the critical value is 1.96. Since 2.0 > 1.96, you reject the null hypothesis and conclude the average IQ is different from 100.

In Confidence Intervals

Critical values also appear in confidence intervals. The margin of error is calculated as: \( \text{Margin of Error} = \text{critical value} \times \text{standard error} \). For a 95% confidence interval using the Z distribution, the critical value is again 1.96. The interval is then: \( \bar{X} \pm 1.96 \times \frac{\sigma}{\sqrt{n}} \).

To understand what these intervals mean, see our How to Interpret Critical Values (2026 Guide).

Common Misconceptions About Critical Values

• Critical value is the same as the p-value. No – the critical value is a fixed threshold set before conducting the test, while the p-value is the probability of observing the results (or more extreme) assuming the null is true. You compare the p-value to α, not to the critical value.
• One-tailed and two-tailed tests use the same critical value. False. For the same α, a two-tailed test splits the significance level equally on both tails, so the critical value is larger in magnitude than for a one-tailed test. For example, at α = 0.05, a two-tailed Z-test uses 1.96, while a one-tailed uses 1.645.
• Degrees of freedom don’t matter. They do – especially for t, Chi-Square, and F distributions. As degrees of freedom increase, critical values approach those of the Z distribution.

For a complete list of critical values for different distributions, check our Critical Values for Z, t, Chi-Square, F (2026) page.