Critical Value Calculator: z, t, Chi-Square, F

Q: How do I find a critical value in Excel?

Use =NORM.S.INV(1−α/2) for a two-tailed z, =T.INV.2T(α, df) for a two-tailed t, =CHISQ.INV.RT(α, df) for a right-tailed chi-square and =F.INV.RT(α, df1, df2) for a right-tailed F. This calculator gives the same values and shows the rejection region.

How to use the critical value calculator

Choose the distribution your test uses: z, t, chi-square or F.
Enter the significance level α, such as 0.05 (for a 95% confidence level) or 0.01.
For t and chi-square enter the degrees of freedom; for F enter both the numerator and denominator df.
Pick two-tailed, left-tailed or right-tailed to match your alternative hypothesis.
Click Find critical value to get the cutoff, the shaded rejection region and the decision rule.

Critical value formulas

A critical value is a quantile of the test's distribution. With cumulative distribution function F:

Right-tailed: c^* = F⁻¹(1 − α)

Left-tailed: c^* = F⁻¹(α)

Two-tailed: F⁻¹(α/2) and F⁻¹(1 − α/2)

c^*: The critical value: the boundary of the rejection region
α: The significance level: the total probability in the rejection region when the null hypothesis is true
F⁻¹: The inverse cumulative distribution (quantile) function of z, t, χ² or F
df: Degrees of freedom, needed for t (one), χ² (one) and F (numerator and denominator)

For z and t, which are symmetric, the two-tailed values are ± the same number. Chi-square and F aren't symmetric, so their two cutoffs differ.

Worked example

A researcher runs a two-tailed t-test with 10 degrees of freedom at α = 0.05. What is the critical value?

Two-tailed: split α between the tails
α/2 = 0.05 / 2 = 0.025
Find the t value with 0.975 of the area to its left
t_{0.975, 10} = 2.228
Decision rule
Reject H₀ if |t| > 2.228

What it means: if the researcher's t statistic is above 2.228 or below −2.228, the result is significant at the 5% level. A t of 2.1 would fall just short, so it wouldn't be significant.

How to interpret a critical value

Beyond the critical value: the test statistic is in the rejection region, so reject H₀. The result is statistically significant at level α.
Not beyond it: fail to reject H₀. The data are consistent with the null hypothesis at that level.
Smaller α, larger cutoff. Moving from α = 0.05 to 0.01 pushes the critical value further out, making significance harder to reach.
Common z values to remember: 1.645 (90%), 1.960 (95%) and 2.576 (99%) for two-sided confidence intervals.

Prefer printed tables? See the z-table, t-table and chi-square table.

Frequently asked questions

What is a critical value?

A critical value is the cutoff on a test statistic’s distribution that separates the results you’d call ordinary from those you’d call significant. If your test statistic lands beyond it, in the rejection region, you reject the null hypothesis at your chosen significance level α. Critical values also set the width of confidence intervals.

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

For z (the normal distribution), it is 1.960, because 95% of the area lies between −1.96 and 1.96. For t it depends on the degrees of freedom: 2.228 with df = 10, 2.042 with df = 30 and closer to 1.96 as df grows. Choose α = 0.05 and two-tailed in the calculator.

Should I use z or t?

Use z for tests of proportions, or of means when the population standard deviation is known. Use t for means when the standard deviation is estimated from the sample, which is the usual case. Chi-square is for goodness-of-fit, independence and variance tests, and F is for ANOVA and comparing two variances.

What is the difference between one-tailed and two-tailed critical values?

A two-tailed test splits α between both tails, so each tail gets α/2 and the cutoff is further out (1.960 for α = 0.05 with z). A one-tailed test puts all of α in one tail, so its cutoff is closer to the center (1.645). Use one-tailed only when your hypothesis states a direction in advance.

Critical value or p-value: which should I use?

They always lead to the same decision. The critical-value approach compares your statistic with a cutoff; the p-value approach compares a probability with α. Most software and journals report p-values because they show how strong the evidence is. Get one from your statistic with the p-value calculator.

How do I find a critical value in Excel?

Use =NORM.S.INV(1−α/2) for a two-tailed z, =T.INV.2T(α, df) for a two-tailed t, =CHISQ.INV.RT(α, df) for a right-tailed chi-square and =F.INV.RT(α, df1, df2) for a right-tailed F. This calculator gives the same values and shows the rejection region.