t Critical Value Table

The most common mistake is the rule "n ≥ 30 means use Z." This is false. The choice depends on whether the population standard deviation (σ) is known or estimated from the sample (s). Sample size alone does not determine the distribution. A t critical value is the threshold in the t-distribution that defines the rejection region for a null hypothesis when σ is unknown. You need t* for a given df and alpha, and you need to know which df to use for your test. Here is the t critical value table, the df formulas for common tests, and the real rule for choosing between Z and t.

The t critical value table below gives two-tailed critical values for α = 0.10, 0.05, and 0.01. Each row is a degrees of freedom (df) value. As df increases, the t critical value approaches the Z critical value. At df = ∞, the t critical value equals the Z critical value exactly.

For a one-tailed test at the same α, use the one-tailed critical value from the same df and α (e.g., for α = 0.05 one-tailed, use the value from the α = 0.10 two-tailed column).

Values are from the NIST/SEMATECH e-Handbook of Statistical Methods, Section 1.3.6.7, which publishes tables of critical values for normal, t, chi-square, and F distributions. The convention in textbooks such as OpenIntro Statistics (4th ed.) and Moore, McCabe & Craig's Introduction to the Practice of Statistics is to use the same values.

The degrees of freedom (df) for a t-test depend on the test design. Use the correct formula or your critical value will be wrong.

One-Sample and Paired t-Tests

df = n - 1. For a sample of 25 observations, df = 24. The two-tailed t critical value at α = 0.05 for df = 24 is 2.0641 from the table.

For a paired design, n is the number of pairs. A study with 15 paired observations has df = 14. The two-tailed t critical value at α = 0.05 for df = 14 is 2.1448.

Pooled and Welch's Two-Sample t-Tests

For a pooled two-sample t-test (equal variances), df = n₁ + n₂ - 2. For group sizes n₁ = 12 and n₂ = 10, df = 20. The two-tailed t critical value at α = 0.05 for df = 20 is 2.0860.

For Welch's t-test (unequal variances), use the Welch-Satterthwaite formula to calculate approximate df. The formula is:
df = ( (s₁²/n₁ + s₂²/n₂)² ) / ( ( (s₁²/n₁)² / (n₁ - 1) ) + ( (s₂²/n₂)² / (n₂ - 1) ) ).
Round the result down to the nearest integer. This df is then used to look up the critical value. For example, if the formula gives df = 27.4, use df = 27. The two-tailed t critical value at α = 0.05 for df = 27 is 2.0527.

The real rule is simple: use Z when the population standard deviation (σ) is known. Use t when σ is unknown and estimated from the sample standard deviation (s). This rule applies regardless of sample size.

The "n ≥ 30" shortcut is a misinterpretation of the Central Limit Theorem. At n = 30, the t critical value (2.0423) is still 4% larger than the Z critical value (1.9600). Using Z at n = 30 when σ is unknown inflates the Type I error rate. The difference only disappears at df = ∞, where t and Z are identical.

For proportions, the Z test uses a known null proportion, so Z is appropriate. For a single mean or a difference in means with σ unknown, use t. For goodness-of-fit or independence tests on categorical data, use chi-square critical values. For ANOVA or regression F-tests, use F critical values.

The choice of distribution determines the critical value. Compare your test statistic to the correct critical value: reject the null hypothesis if the test statistic exceeds the critical value (right-tailed), if it is less than the critical value (left-tailed), or if its absolute value exceeds the critical value (two-tailed). P-values are an alternative to critical values: reject H₀ if p-value < α.

If your exact df is not in the table, use one of these three methods.

Interpolation

For a df between two table values, linear interpolation gives a close estimate. For example, to find the two-tailed t critical value at α = 0.05 for df = 35: the table gives 2.0423 for df = 30 and 2.0211 for df = 40. The value for df = 35 is approximately 2.0423 + ( (35 - 30) / (40 - 30) ) * (2.0211 - 2.0423) = 2.0317. The exact value from software is 2.0301, a difference of 0.0016.

Use the Next Lower df

Using the next lower df gives a slightly larger critical value, making the test more conservative (less likely to reject the null). This is a safe choice for hypothesis testing. For df = 35, using df = 30 gives 2.0423 instead of 2.0301.

Use Software

Excel and Google Sheets have the T.INV function. For a two-tailed test at α = 0.05 with df = 35, use T.INV(0.975, 35) to get the exact value 2.0301. For a one-tailed test, use T.INV(0.95, 35) to get 1.6896. The TI-84 Plus CE has the invT function, which works the same way. If you need chi-square or F critical values, use CHISQ.INV.RT or F.INV.RT in Excel/Sheets. The TI-84 does not have built-in chi-square or F inverse functions as of OS 5.8 (2023).

Before you look up the t critical value, confirm that you are using the correct tail. A two-tailed test at α = 0.05 uses the same critical value as a one-tailed test at α = 0.025. The table above gives two-tailed critical values. For a one-tailed test, use the value from the column for α (two-tailed) that is double your one-tailed α. For example, for a one-tailed test at α = 0.05, use the two-tailed α = 0.10 column. The t critical value for df = 30, one-tailed α = 0.05, is 1.6973.

If you use the wrong tail, you will either reject the null too often (one-tailed value for a two-tailed test) or fail to reject when you should (two-tailed value for a one-tailed test). The difference is 0.345 for df = 30 at α = 0.05, which is enough to change a decision.