The t-distribution is similar to the standard normal distribution. However, unlike the standard normal distribution, it is actually a family of probability distributions. That is, it's not a single probability distribution but rather a collection of many probability distribution. Each individual probability distribution in the t-distribution depends on something called a degrees of freedom. So a t-distribution with 1 degree of freedom is different than a t-distribution with 2 degrees of freedom and a t-distribution with 2 degrees of freedom is different from a t-distribution with 3 degrees of freedom and so on. As the degrees of freedom increases, the t-distribution gets closer to the standard normal distribution. In fact, a t-distribution with infinite degrees of freedom is identical to the standard normal distribution.
The key difference between the standard normal table and the t-distribution table is that the standard normal table gives the area to the left of the given z-value while the t-distribution table gives the area to the right of the given t-value. That is, the standard normal table gives the lower tail area while the t-distribution table gives the upper tail area. In the t-table, the degrees of freedom are given in the first column while the areas in the upper tail are given in the first row. Note that there area only a few different upper tail areas given: .20, .10, .05, .025 and .01. This is due to the limitation of listing the probabilities in only a couple of pages of paper. There is a similar limitation in the standard normal table, as the table only gives z-values up to two decimal places.
To use the t-table, simply match the degrees of freedom with the area in the upper tail. For example, matching up 6 degrees of freedom with an area in the upper tail area of .05, you get a t-value of 1.9443. This means that under a t-distribution with 6 degrees of freedom, the area to the right of 1.9443 is .05. Recall that for a continuous probability distribution, like the t-distribution, area is synonymous with probability. So for a t-distribution with 6 degrees of freedom, the probability of getting a t-value greater than 1.9443 is .05. This can be written as $P(t \geq 1.9443) = .05$.
| Area in the Upper Tail | ||||||
| df | .20 | .10 | .05 | .025 | .01 | .005 |
| 5 | 0.920 | 1.476 | 2.015 | 2.571 | 3.365 | 4.032 |
| 6 | 0.906 | 1.440 | 1.943 | 2.447 | 3.143 | 3.707 |
| 7 | 0.896 | 1.415 | 1.895 | 2.365 | 2.998 | 3.499 |
Calculating the area to the left for the t-distribution requires additional stepssince the t-table gives you the area to the right. In order to get the area to the left, you have to subtract the area to the right from 1. So, for example, if you want to find the area to the left of 2.110 under a t-distribution with 18 degrees of freedom, you have to start by matching up 18 degrees of freedom and 2.110, which gives an area of .025. However, this is the area in the upper tail, or right, and not the area to the left. So you have to subtract this area from 1, which gives you .975.
| Area in the Upper Tail | ||||||
| df | .20 | .10 | .05 | .025 | .01 | .005 |
| 17 | 0.863 | 1.333 | 1.740 | 2.110 | 2.567 | 2.898 |
| 18 | 0.862 | 1.330 | 1.734 | 2.101 | 2.552 | 2.878 |
| 19 | 0.861 | 1.328 | 1.729 | 2.093 | 2.539 | 2.861 |
Finding the area between two t-values is a bit trickier than the area to the left or the right. Suppose you want to find the t-values such that 80 percent of the values falls between them with 23 degrees of freedom. This would mean that there is 20% of the values fall in the two tails, leaving an area of 10% in each. A percentage of 10 corresponds to an area, or probability, of .10. So matching up 23 degrees of freedom with .10, we get a t-value of 1.321. This means that the other t-value is -1.321, since the t-distribution, like the standard normal distribution, is symmetric. So our two t-values with an area of 80% between them are -1.321 and 1.321.
| Area in the Upper Tail | ||||||
| df | .20 | .10 | .05 | .025 | .01 | .005 |
| 22 | 0.858 | 1.321 | 1.717 | 2.074 | 2.508 | 2.819 |
| 23 | 0.858 | 1.319 | 1.714 | 2.069 | 2.500 | 2.807 |
| 24 | 0.857 | 1.318 | 1.711 | 2.306465 | 2.492 | 2.797 |
The t-distribution has many important applications. Confidence intervals and hypothesis tests about the population mean require the use of the t-distribution when the population standard deviation is unknown. In both these cases, the degrees of freedom is equal to the sample size minus one (df = n - 1). In regression analysis, the t-distribution is used when testing for a significant relationship between the dependent and independent variables. This test is known as a t-test and the degrees of freedom here is equal to sample minus the number of independent variables (p) minus one (n - p - 1).
| Hypothesis Testing |
| $ t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}} $ |
Note that the t-distribution is sometimes referred to as the student's t-distribution. This naming has to do with the history of the discovery of the t-distribution. The inventor of the t-distribution, William Gossett, was an employ of the Guinness Brewing factory when he came up with the t-distribution. However, under his contract, he was not allowed to publish any scientific articles. So he published the t-distribution under the anonymous name "student" and so the the t-distribution is sometimes referred to as the student's t-distribution.
