# The Difference Between T-Values and P-Values in Statistics

Two terms that students often get confused in statistics are t-values and p-values.

To understand the difference between these terms, it helps to understand t-tests.

Broadly speaking, there are three different types of t-tests:

• One-sample t-test: Used to test whether a population mean is equal to some value.
• Two-sample t-test: Used to test whether two population means are equal.
• Paired samples t-test: Used to test whether two population means are equal when each observation in one sample can be paired with an observation in the other sample.

We use the following steps to perform each test:

• Step 1: State the null and alternative hypothesis.
• Step 2: Calculate the t-value.
• Step 3: Calculate the p-value that corresponds to the t-value. For each test, the t-value is a way to quantify the difference between the population means and the p-value is the probability of obtaining a t-value with an absolute value at least as large as the one we actually observed in the sample data if the null hypothesis is actually true.

If the p-value is less than a certain value (e.g. 0.05) then we reject the null hypothesis of the test.

For each type of t-test, we’re interested in the p-value and we simply use the t-value as an intermediate step to calculating the p-value.

The following example shows how to calculate and interpret a t-value and corresponding p-value for a two-sample t-test.

### Example: Calculate & Interpret T-Values and P-Values

Suppose we want to know whether or not the mean weight between two different species of turtles is equal. We go out and collect a simple random sample of 12 turtles from each population with the following weights:

Species #1: 301, 298, 295, 297, 304, 305, 309, 298, 291, 299, 293, 304

Species #2: 302, 309, 324, 313, 312, 310, 305, 298, 299, 300, 289, 294

Here’s how to perform a two-sample t-test using this data:

Step 1: State the null and alternative hypothesis.

First, we’ll state the null and alternative hypotheses:

• H0μ1 = μ2 (the two population means are equal)
• H1μ1 ≠ μ2 (the two population means are not equal)

Step 2: Calculate the t-value.

Next, we’ll enter the weights of each sample of turtles into the Two Sample t-test Calculator and find that the t-value is -1.608761.

Step 3: Calculate the p-value.

We can also use the Two Sample t-test Calculator to find that the p-value that corresponds to a t-value of -1.608761 is 0.121926.

Since this p-value is not less than .05, we fail to reject the null hypothesis.

This means we do not have sufficient evidence to say that the mean weight of turtles between the two populations are different.

Notice that we simply used the t-value as an intermediate step to calculating the p-value. The p-value is the true value that we were interested in, but we had to first calculate the t-value.