ExplanationUnderstanding p-Values and t-Values
When you learned about the t-test (independent, paired, or both) you saw that results are reported as a probability of the differences in the data occuring from the same population by chance. This probability is called the p-value. The p-value is not produced directly by the t-test, it is calculated in one further step, using the outcome of the t-test (which is called the t-value!).
These two values are related and they are explained in this section.
Most software packages report the t-value as part of the summary of the results of a t-test. The value of t is the result of putting the sample data through the formula for the t-test.
The t-value is related to the size of the difference between the means of the two samples you are comparing. The larger t is, the larger the difference. The t-value is not the most useful result to report, which is why we also report p-values.
A p-value answers this question: If my null hypothesis were true, what is the probability of getting a t-value at least as big as mine?. Obviously, the lower this value is, the less likely it is that you would find a difference like yours by chance.
The p-value helps you decide whether or not to accept the null hypothesis. You make this decision by deciding how low your p-value should be before you will reject the null hypothesis. This cut-off point is called the significance level and is usually set at 0.05 or 0.01.
Sometimes you will see the p value reported as an exact figure, for example, p=0.023. In other places, you will see a less than sign (<), for example, p<0.05. This is because computer programs can calculate the exact value of p, whereas looking up p-values in tables (which is covered at level two of this topic) only allows you to say that the p-value is below one of a few set values. Either method is acceptable, but we will use the less than sign because we use tables at level two. Note also, that software will sometimes report very low p-values, 0.00001, for example, or even p=0. Never report that p=0, this is a side effect of the limited accuracy of some software. If p is less than 0.001, then report p<0.001 rather than the exact p value.
Degrees of Freedom
The degrees of freedom (df) of a set of data relates to the number of values there are in that data.
The more data you have, the smaller your sampling error is likely to be. The degrees of freedom value takes this into account when calculating the p-value.
- For paired samples (within subjects design), for a paired t-test, df is the number of students minus 1;
- For independent samples (between subjects design), for an independent t-test, df is the number of students in sample 1 plus the number of students in sample 2 minus 2.
Reporting p-Values and t-Values
You report the results of a t-test in the following way:
Where df is the degrees of freedom of your data, t is the t-value you found and p is the p-value you found.
Here is an example: t(20)=2.8, p<0.01, which means that a t-test with 20 degrees of freedom produced a t-value of 2.8, which means that the probability of the difference being due to chance is less than 0.01. As we mentioned above, you can also report the exact p value, for example, t(20)=2.8, p=0.007