t table of critical values (and how to read it)

What t* critical values are

A t critical value, written $t^*$ (say "t-star"), is a multiplier that comes from the t distribution. You use it in two places: building a confidence interval and running a t test. In the confidence interval formula statistic $\pm$ (critical value)(standard error), $t^*$ sets how wide the interval is. In a test you compare your t statistic against $t^*$ to see whether the result sits far enough into the tail to be significant.

Every $t^*$ depends on two things: how confident you want to be (or how small a tail area you are testing) and the degrees of freedom. A t table, also called a t distribution table, is a grid of these values with degrees of freedom down the side and tail probabilities or confidence levels across the top.

How degrees of freedom work

Degrees of freedom, written $df$, tell you which row of the t table to use. The t distribution is a family of curves, one for each value of $df$. Smaller $df$ means heavier tails and a larger $t^*$; larger $df$ makes the curve look more like the normal curve.

For a single sample of size $n$ (a one-sample mean, or the mean of paired differences), the rule is:

So a sample of $n = 25$ has $df = 24$. Here $n$ is the sample size. You lose one degree of freedom because you estimate the population mean $\mu$ (Greek letter "mu") with the sample mean $\bar{x}$ ("x-bar") before computing the sample standard deviation $s$.

How to read the t distribution table for a 95% CI with df = 24

Suppose you are building a 95% confidence interval from a sample of $n = 25$. Two steps get you to $t^*$.

1. Find your row. With $df = n - 1 = 24$, go to the row labeled 24.
2. Find your column. A 95% interval is two-tailed, so the leftover 5% splits into two tails of 2.5% each. Use the column headed 95% confidence, which is the same as upper-tail probability 0.025.

Where that row and column meet you read $t^* = 2.064$. That is the number you multiply by the standard error to get the margin of error.

One-tail vs two-tail columns

Most t tables label each column two ways: an upper-tail probability and a confidence level. The upper-tail probability is the area in one tail beyond $t^*$. Let $\alpha$ ("alpha") be the significance level of a test.

• For a two-tailed test at level $\alpha$, or a confidence interval at level $C = 1 - \alpha$, use the column with upper-tail probability $\alpha/2$. A 95% interval uses 0.025.
• For a one-tailed test at level $\alpha$, use the column with upper-tail probability $\alpha$. A one-tailed test at $\alpha = 0.05$ uses 0.05.

With $df = 24$, the 0.05 column gives $t^* = 1.711$ and the 0.025 column gives $t^* = 2.064$. Same row, different tail area, different critical value. Reading the wrong column is the most common t-table mistake.

What happens to the t critical value as df grows

As degrees of freedom increase, the heavy tails of the t distribution thin out and $t^*$ shrinks toward the matching z critical value $z^*$ ("z-star") from the standard normal curve. For 95% confidence, $z^* = 1.960$.

By $df = 100$ the t value 1.984 is within about 0.024 of $z^*$. That is why large-sample t procedures give almost the same interval as a z procedure. The t value stays a little bigger, so the interval is slightly wider to account for estimating the population standard deviation $\sigma$ ("sigma") with the sample standard deviation $s$.

Look up t* without the table

You do not have to read the table by hand every time. The confidence interval calculator and the one-sample t test calculator look up $t^*$ for the degrees of freedom and confidence level you enter. To decide whether a problem calls for t instead of z in the first place, read t test vs z test.