What we discussed in the previous section was called a one-sample t-test. We took one sample of numbers, and compared the mean of that sample to some other number. That number is usually zero: e.g., in our t-test, we checked to see whether the average improvement for our sample of 20 students was greater than zero.
Another kind of t-test is the paired samples t-test (also sometimes called a dependent samples t-test). This happens when you have two sets of numbers (e.g., pre-test scores and post-test scores) and you compare their means (e.g., to see if the average post-test score is higher than the average pre-test score). For it to be a paired samples t-test, the scores have to come in natural pairs: for example, a pre-test score and a post-test score from the same person makes a pair. If your pre-test scores and post-test scores are from different people, you can't use a paired samples t-test. I won't talk much more about the paired samples t-test because it's actually literally the same thing as a one-sample t-test. Checking whether someone's post-test score is higher than their pre-test score is exactly the same thing as checking whether someone's {post-test minus pre-test} difference is higher than zero. (For example, imagine someone's pre-test score is 63 and their post-test score is 85. Mathematically, you could express this as "85 > 63", or you could express it as "85-63 > 0" ; these are the same.) Thus, every paired samples t-test could be restated as a one-sample t-test. In fact, the formula for a paired samples t-test is essentially the same as the formula for a one-sample t-test.
Last, there is an independent samples t-test. This is what you use when you want to compare two samples of data, and the data do not come in natural pairs. For example, if I want to see if PolyU students are taller than HKU students, I would need to use an independent samples t-test, because my measurements of PolyU students come from different people than my measurements of HKU students, and there's no meaningful way to pair the people up. The formula for an independent samples t-test follows the same logic as the one-sample t-test formula you learned earlier (its key parts are still the size of the group difference, the amount of variance, and the size of the sample), but some of the details are different. I usually just calculate it using statistical software.
When you use statistical software for doing an independent samples t-test, you will have to indicate whether your samples have [approximately] equal variance or [approximately] unequal variance. (For example, imagine you want to compare 20 cats and 20 dogs to see whether cats or dogs are heavier. Cats' weights don't vary a lot; all housecats are more or less the same size. Dogs' weights do vary a lot, though: there are some tiny dogs and some big dogs. Thus, these two groups have unequal variance.) In general, it's best to just always use the unequal variance test to be safe (in fact, some statistical software, like R, does this by default); in situations where you do have equal variance both tests are pretty much equally effective, and in situations where you have unequal variance the unequal-variance test is far more effective than the equal-variance test. So there's rarely a need to ever use the equal-variance test.
Additionally, when you use statistical software for any t-test (independent or one-sample), you will have to indicate whether you want a two-tailed test or a one-tailed test. A two-tailed test is what you use when you have no directional prediction: Group A might be higher than Group B or Group B might be higher than Group A, you have no idea, you just want to see if the groups are different in either direction. I think this situation is rare, but people still often use the two-tailed test anyway. A one-tailed test is what you use when you have a directional prediction: e.g., you expect that Group A is higher than Group B, and you have no interest in testing whether Group A is lower than Group B. If you have a directional prediction, an appropriate one-tailed test has a better chance of finding a significant p-value. However, you have to be careful that you're looking in the right tail (if you expect Group A to be higher than Group B but you accidentally put the two groups into the Excel formula in the wrong order, Excel might totally miss the effect). Understanding all the details of how this works is beyond the scope of the present class (but you can learn more about it in a statistics class). My recommendation is: if you are confident with what you're doing, use one-tailed tests when appropriate to have a better chance of finding the effects you're looking for; but if you're not confident or not sure what you're doing, just use a two-tailed test.
To learn how to do a t-test in Excel, see e.g. this site. (There are many other tutorials available for this sort of thing; this is just one I found. Note that the instructions here assume that you have activated the "Data Analysis Toolkit" in Excel. This is not activated by default, but can be activated with a few simple steps; it should be easy to Google for instructions on how. I usually don't use this toolkit, I usually just directly use the =T.TEST() function, but that function only reports the p-value, whereas the Data Analysis Toolkit also reports the t statistic and other relevant numbers that you might need.)
To learn how to do a t-test in R with as little use of R coding as possible, see t-test in R.pdf
To do a little practice doing a t-test with computer software, continue to the exercises below.
Recall the first activity in this module, where I listed the ages of some [imaginary] PolyU and HKUST graduate students:
Using whatever software you prefer, do a t-test to compare these two groups. You'll have to decide which kind of t-test to use. Once you've done the test, write down the p-value.
The answer is at the bottom of the page.
When you have finished these activities, continue to the next section of the module: "Choosing a test".
For this question I accept either p=0.096925231, or p=0.048462616. The first is what you get from a two-tailed test, and the second is what you get from a one-tailed test; whether you use a two-tailed or a one-tailed test depends on your original hypothesis. Note that both of these p-values come from the "unequal variance" t-test; the "equal variance" t-test never needs to be used.
by Stephen Politzer-Ahles. Last modified on 2021-05-15. CC-BY-4.0.