Can I do Paired T-test for averaged data I will make my question more obvious here. My data consists of each students making 5 attempts to say, throw a ball in a hole back to back. My interest is to see, how many times they succeed, whether the average speed of throwing differs between passed and failed attempts. So I will also have the measurement on the speed of throwing. I know I can do a repeated measure ANOVA to test the mean difference of speed between throws, but that's not my intention. I was wondering if I could do a pair T-test to see the mean difference between successful and unsuccessful attempts. Can I take the average of the successful and unsuccessful attempts for each student and run a paired T-test (if normality is satisfied) on these data?
Any suggestion is appreciated! Let me know as well if you have any questions. Thanks!
 A: The paired t-test should be used where a single condition changes within a setting that otherwise remains the same. For example, blood pressure measurements within the same individual before and after starting a medicine.
Here you seem to be interested in the number of successful attempts at throwing a ball into a hole, and whether the speed of the throw affects the probability of success. The measurements are not independent because each individual throws five times (and I am ignoring any practice effect where the accuracy would improve from the first to the fifth throw).
A simple representation would be a two level model with a logistic link so the outcome is 1 for success and 0 for failure, and the explanatory variables include the measure of speed, and a dummy variable for the individual (to handle the within individual correlation).
A: You can compare the speeds of the successful and unsuccessful throws and do so with a design that recognizes pairing, but there are challenges you need to address.  First, you may have missing data, as when one person misses all the throws.  Second, the averages will have a smaller variance than individual throws, and the variance will be smaller when you average more throws.  Different variances for different observations violates one of the assumptions for a t-test (paired or not).  So, you might consider a non-parametric statistical test.
