# How to Find Percentiles for a *t*-Distribution

When you want to find percentiles for a *t*-distribution, you can use the *t*-table. A *percentile* is a number on a statistical distribution whose less-than probability is the given percentage; for example, the 95th percentile of the *t-*distribution with *n* – 1 degrees of freedom is that value of

whose left-tail (less-than) probability is 0.95 (and whose right-tail probability is 0.05).

The *t*-table shows right-tail probabilities for selected *t*-distributions. You can use it to solve the following problems.

Suppose you have a sample of size 10 and you want to find the 95th percentile of its corresponding *t-*distribution. You have *n* – 1= 9 degrees of freedom, so, using the *t*-table, you look at the row for *df* = 9. The 95th percentile is the number where 95% of the values lie below it and 5% lie above it, so you want the right-tail area to be 0.05. Move across the row, find the column for 0.05, and you get

This is the 95th percentile of the *t-*distribution with 9 degrees of freedom.

Now, if you increase the sample size to *n* = 20, the value of the 95th percentile decreases; look at the row for 20 – 1 = 19 degrees of freedom, and in the column for 0.05 (a right-tail probability of 0.05) you find

degrees of freedom indicate a smaller standard deviation and thus, the *t*-values are more concentrated about the mean, so you reach the 95th percentile with a value of *t* closer to 0.