This t-distribution table provides the critical t-values for both one-tailed and two-tailed t-tests, and confidence intervals. Learn how to use this t-table with the information, examples, and illustrations below the table.
How to Use the T-Distribution Table
Use the t-distribution table by finding the intersection of your significance level and degrees of freedom. The t-distribution is the sampling distribution of t-values when the null hypothesis is true. Learn more about the T Distribution: Definition and Uses.
Significance Level (Alpha α): Choose the column in the t-distribution table that contains the significance level for your test. Be sure to choose the alpha for a one- or two-tailed t-test based on your t-test’s methodology. Learn more about the Significance Level and One- and Two-Tailed Tests.
Degrees of freedom (df): Choose the row of the t-table that corresponds to the degrees of freedom in your t-test. The final row in the table lists the z-distribution’s critical values for comparison. Learn more about Degrees of Freedom.
Critical Values: In the t-distribution table, find the cell at the column and row intersection. When you are performing a:
- Two-tailed t-test: Use the positive critical value AND the negative form to cover both tails of the distribution.
- One-tailed t-test: Use the positive critical value OR the negative value depending on whether you’re using an upper (+) or lower (-) sided test.
Learn more about: How T-tests Work, test statistics, critical values, and How to do T-Tests in Excel
Tables for other statistics include the z-table, chi-square table, and F-table.
Examples of Using the T-Distribution Table of Critical Values
Suppose you perform a two-tailed t-test with a significance level of 0.05 and 20 degrees of freedom, and you need to find the critical values.
In the t-distribution table, find the column which contains alpha = 0.05 for the two-tailed test. Then, find the row corresponding to 20 degrees of freedom. The truncated t-table below shows the critical t-value.
The t-table indicates that the critical values for our test are -2.086 and +2.086. Use both the positive and negative values for a two-sided test. Your results are statistically significant if your t-value is less than the negative value or greater than the positive value. The graph below illustrates these results.
Now, suppose you perform a one-sided t-test with a significance level of 0.05 and 20 df.
In the t-distribution table, find the column which contains alpha = 0.05 for the one-tailed test. Then, find the row corresponding to 20 degrees of freedom. The truncated t-table below shows the critical t-value.
The row and column intersection in the t-distribution table indicates that the critical t-value is 1.725. Use either the positive or negative critical value depending on the direction of your t-test. The graphs below illustrate both one-sided tests. Your results are statistically significant if your t-value falls in the red critical region.
Using Critical T-values to Calculate Confidence Intervals
To calculate a two-sided confidence interval for a t-test, take the positive critical value from the t-distribution table and multiply it by your sample’s standard error of the mean. Then take the sample mean and add and subtract the product from it to calculate the upper and lower interval limits, respectively.
For a one-sided confidence interval, either add or subtract the product from the mean to calculate the upper or lower bound, respectively.
The confidence level is 1 – α.
Kennedy. Rex says
thanks very much..Its helpful to me
In my problem the sample size is 36 so the DF is 35. I am using a two-tailed test with an alpha of .05. DF 35 is not on the table – what do I do?
Jim Frost says
There are two standard approaches.
One is to use interpolation, which figures out the in-between value. In this case, it’s simple to interpolate because it’s exactly halfway between the critical values of 2.042 and 2.021 for 30 and 40 DF, respectively. The difference is 2.042 – 2.021 = 0.021. Divide that by two for half the distance (0.0105) and then add that to the lower value of 2.021 + 0.0065 = 2.0315. Of course, we’d use the positive and negative values for a two-sided test. That’s an approximation by interpolating the table values.
Another alternative is to go with the more conservative value, which is the smaller DF. In that case, you’d use the critical value for 30 DF, which +/- 2.042. If it is significant with the lower DF, then you know it’s significant for the actual, somewhat higher DF.
And, of course, in this day and age, you could use a t-value calculator, which produces +/- 2.030108. That’s the most exact value.
Hi Jim, I think the DF has to be used in this case. This is a t distribution and DF = n-1. So, DF for the critical values would be t (0.05 or 0.025), 19. Not df:20. Thanks. Merve
Jim Frost says
My examples use 20 DF, and you can see that DF is the first column in the table. In a 1-sample t-test, 20 DF corresponds to a sample size of 21 because for this test DF = n – 1. However, for a 2-sample t-test, the 20 degrees of freedom corresponds to a sample size of 22 because for that test DF = N₁ + N₂ – 2. Regardless of the test, the critical values that I show are accurate for 20 DF.