Critical Value Calculator (Z, T, Chi-Square, F) + Tables

Creator: Kushagra VermaReviewer: Parker Rowland

A critical value is the cutoff that separates the rejection region from the rest of a hypothesis test. Pick z, t, chi-square, or F above, enter your significance level, and you get the exact value with a full worked solution. Need the number fast? For a 95% two-tailed z-test it is 1.96, for 99% it is 2.576, and the t and chi-square values sit in the reference tables further down.

This page walks through what each critical value means, how to find it by hand, and includes complete z, t, chi-square, and F tables you can read straight off.

What Is a Critical Value?

A critical value is the number that marks where the rejection region starts in a hypothesis test. If your test statistic lands past it, the result is statistically significant and you reject the null hypothesis.

Think of it as a fence line. On one side, your result is consistent with the null hypothesis. On the other side, in the rejection region, the result is too extreme to be chalked up to chance, so you reject the null.

Every critical value depends on three inputs:

• The test type (z, t, chi-square, or F), which sets the distribution.
• The significance level (\u03b1), usually 0.10, 0.05, or 0.01.
• The degrees of freedom, needed for t, chi-square, and F tests.

The four values you will meet most often are the t critical value (t-tests), the z critical value (z-tests), the chi-square critical value (goodness-of-fit and independence tests), and the F critical value (ANOVA and variance tests). The calculator above returns any of them, so you skip flipping through printed tables.

What Is the Critical Value for a 95% and 99% Confidence Level?

For a 95% confidence level the two-tailed z critical value is 1.96. For 99% it is 2.576, and for 90% it is 1.645. These are the most searched critical values in statistics.

Confidence intervals and two-tailed hypothesis tests share the same z critical values, because a 95% confidence interval leaves 5% split across both tails, exactly like a two-tailed test at \u03b1 = 0.05. Here are the numbers people look up most:

If you are building a confidence interval rather than running a test, use the two-tailed column. Our confidence interval calculator applies these values for you automatically.

How Do You Find a Critical Value?

To find a critical value, pick your test type, set the significance level (\u03b1), find the degrees of freedom if the test needs them, then read the value from a table or calculator based on whether the test is one- or two-tailed.

Here is the process by hand, step by step.

1. Choose the test. A t-test, z-test, chi-square test, and F-test each use a different distribution.
2. Set \u03b1. Most work uses 0.05. Stricter tests use 0.01, looser ones 0.10.
3. Find the degrees of freedom. For a t-test, df = n \u2212 1. Chi-square and F tests use their own df rules. The degrees of freedom calculator handles this if you are unsure.
4. Decide one- or two-tailed. A two-tailed test splits \u03b1 across both ends, so you look up \u03b1/2 in each tail. A one-tailed test keeps the full \u03b1 in one tail.
5. Read the value. Match your test, \u03b1, and df to a table entry, or let the calculator above return it.
6. Compare. If your test statistic falls beyond the critical value, you land in the rejection region and reject the null hypothesis.

Types of Critical Values

There is no single critical value that fits every test. The one you use depends on the distribution behind your data.

T Critical Value

Used in t-tests when the population standard deviation is unknown, which is most real-world cases. It depends on both the degrees of freedom and the significance level, and it is the go-to for small samples. As sample size grows, the t critical value slowly closes in on the z value.

Z Critical Value

Used in z-tests when the population standard deviation is known or the sample is large. It does not need degrees of freedom. The famous 1.96 for 95% and 2.576 for 99% are z critical values, which is why they show up in nearly every confidence interval.

Chi-Square Critical Value

Used in tests of independence and goodness of fit. It depends on degrees of freedom and is always positive, since chi-square deals with squared differences. Chi-square tests are right-tailed, so the full \u03b1 sits in the upper tail.

F Critical Value

Used in ANOVA and tests that compare two variances. It needs two degrees of freedom values, one for the numerator and one for the denominator, and like chi-square it is right-tailed. You can pull the exact number from our ANOVA calculator when running a full analysis.

Worked Examples: Finding Critical Values Step by Step

These are the exact kinds of problems students paste into search. Each answer below matches the calculator above. If you want the reasoning behind each lookup instead of just the number, our guide on how to find a critical value covers the Z, t, chi-square and F cases one at a time.

Example 1: z critical value for a 95% two-tailed test

Setup: \u03b1 = 0.05, two-tailed. Split \u03b1 into both tails, so each tail holds 0.025.

Read the value: The z value with 0.025 in the upper tail is 1.960.

Answer: The critical values are \u00b11.96. Reject the null if your z statistic is below \u22121.96 or above +1.96.

Example 2: t critical value, \u03b1 = 0.05, two-tailed, df = 15

Setup: A t-test with a sample of 16 gives df = 16 \u2212 1 = 15. Two-tailed, so each tail holds 0.025.

Read the value: From the t table at df = 15 and \u03b1 = 0.05 two-tailed, the value is 2.131.

Answer: The critical values are \u00b12.131.

Example 3: chi-square critical value, \u03b1 = 0.05, df = 4

Setup: A goodness-of-fit test with 5 categories gives df = 5 \u2212 1 = 4. Chi-square is right-tailed, so all 0.05 sits in the upper tail.

Read the value: At df = 4 and \u03b1 = 0.05, the chi-square critical value is 9.488.

Answer: Reject the null if your chi-square statistic exceeds 9.488.

Example 4: F critical value for a one-way ANOVA

Setup: Three groups with 30 total observations. Numerator df = 3 \u2212 1 = 2. Denominator df = 30 \u2212 3 = 27. \u03b1 = 0.05.

Read the value: At df\u2081 = 2, df\u2082 = 27, \u03b1 = 0.05, the F critical value is 3.354.

Answer: Reject the null if your F statistic exceeds 3.354.

Example 5: t critical value, one-tailed, \u03b1 = 0.05, df = 24

Setup: A right-tailed t-test with a sample of 25 gives df = 24. All 0.05 stays in one tail.

Read the value: At df = 24, one-tailed \u03b1 = 0.05, the value is 1.711.

Answer: Reject the null if your t statistic exceeds 1.711.

Z Critical Value Table

The z critical value does not depend on degrees of freedom, so this short table covers every common case. Use the two-tailed column for confidence intervals and two-sided tests.

For the full breakdown of z scores and their tail probabilities, see the complete z table.

T Critical Value Table (df 1 to 120)

The t critical value shrinks as degrees of freedom rise. At high df it nearly equals the matching z value. Find your df in the first column, then read across to your \u03b1 and tail type.

A full layout with more \u03b1 levels lives on our t table page.

Chi-Square Critical Value Table

Chi-square tests are right-tailed, so the whole significance level sits in the upper tail. Match your degrees of freedom to your \u03b1.

The extended version sits on our chi-square table.

F Critical Value Table (\u03b1 = 0.05)

The F critical value needs two degrees of freedom: numerator (df\u2081, across the top) and denominator (df\u2082, down the side). This grid covers the most common ANOVA setups at \u03b1 = 0.05.

One-Tailed vs Two-Tailed: Which Critical Value Do You Use?

A two-tailed test splits \u03b1 across both ends of the distribution, so you look up \u03b1/2 in each tail. A one-tailed test puts the full \u03b1 in one tail, giving a smaller cutoff that is easier to cross.

The direction of your hypothesis decides this. If you only care whether a value is greater than (or only less than) something, use a one-tailed test. If you care about a difference in either direction, use two-tailed.

Here is why it matters: at \u03b1 = 0.05, the two-tailed z critical value is 1.96, but the one-tailed value is 1.645. The one-tailed test has more power to detect an effect in the direction you specified, but it cannot detect an effect the other way.

Quick check: Chi-square and F tests are always right-tailed by nature, so the one-tailed rule applies to them automatically. Only z and t tests give you the one- versus two-tailed choice.

When Should You Use Each Critical Value?

Picking the wrong distribution leads to the wrong conclusion. Use this guide.

• Use a t critical value when the sample is small and the population standard deviation is unknown. This covers most experiments and surveys.
• Use a z critical value when the population standard deviation is known or the sample is large. Common in quality control and confidence intervals.
• Use a chi-square critical value when testing categorical data for independence or goodness of fit.
• Use an F critical value when comparing variances or running ANOVA across multiple group means.

When you have your critical value, the next step is usually a full test. Our hypothesis testing calculator takes it from the cutoff to the final decision.

Frequently Asked Questions

How do you find a critical value?

Choose the test type (z, t, chi-square, or F), set your significance level, find the degrees of freedom if the test needs them, and decide one- or two-tailed. Then read the value from a table or the calculator above.

What is the critical value for a 95% confidence interval?

It is 1.96. A 95% confidence interval leaves 2.5% in each tail, and the z value for that is 1.96. For 99% it is 2.576, and for 90% it is 1.645.

What is a critical value in hypothesis testing?

It is the cutoff that separates the rejection region from the acceptance region. If your test statistic passes it, you reject the null hypothesis.

Is the critical value different for one-tailed and two-tailed tests?

Yes. A two-tailed test uses \u03b1/2 in each tail, so the cutoff is larger. A one-tailed test uses the full \u03b1 in one tail, giving a smaller cutoff.

How do you find the t critical value?

Calculate degrees of freedom (df = n \u2212 1 for a one-sample t-test), pick your \u03b1, choose one- or two-tailed, then read the value from the t table above or enter it in the calculator.

Why is the t critical value larger than the z value?

The t distribution has heavier tails to account for the extra uncertainty of estimating the standard deviation from a small sample. As df grows, that uncertainty shrinks and t approaches z.

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