The Question: A basketball player has a chance of 85% (p) on his free throw shoots. He participates in three free throw competitions. In the first competition he makes 8 ($k_1$) free throw shots out of 10 ($n_1$) , in the second he makes 12 out of 15 and in the third he makes 16 out of 20. In which competition was he relatively the best or the worst respectively?
My answer: I defined the performance of the ith competition by $$ P_i = k_i - E_i(X) \\ $$ where $k_i$ is the number of successful shots made and $$E_i(X) = n_ip = 0.85*n_i$$
The actual question:
Does it make sense to normalize $P_i$, i.e. divide by the standard deviation $S_i = \sqrt[]{n_ipq}$ of the ith competition/Benoulli process, or is the above definition of Performance sufficient?
Or am I on a completely wrong track here and I don't even have to consider the different standard deviations, because they have nothing to do with this?