It has frequently been observed that the worst performers on the first day will tend to improve their scores on the second day, and the best performers on the first day will tend to do worse on the second day. A class of students takes two editions of the same test on two successive days. The following is an example of this second kind of regression toward the mean.
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Hence, those who did well previously are unlikely to do quite as well in the second test even if the original cannot be replicated. On a retest of this subset, the unskilled will be unlikely to repeat their lucky break, while the skilled will have a second chance to have bad luck. In this case, the subset of students scoring above average would be composed of those who were skilled and had not especially bad luck, together with those who were unskilled, but were extremely lucky. Most realistic situations fall between these two extremes: for example, one might consider exam scores as a combination of skill and luck. if there were no luck (good or bad) or random guessing involved in the answers supplied by the students – then all students would be expected to score the same on the second test as they scored on the original test, and there would be no regression toward the mean. If choosing answers to the test questions was not random – i.e. No matter what a student scores on the original test, the best prediction of their score on the second test is 50. Thus the mean of these students would "regress" all the way back to the mean of all students who took the original test. If one selects only the top scoring 10% of the students and gives them a second test on which they again choose randomly on all items, the mean score would again be expected to be close to 50.
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Naturally, some students will score substantially above 50 and some substantially below 50 just by chance. Then, each student's score would be a realization of one of a set of independent and identically distributed random variables, with an expected mean of 50. Suppose that all students choose randomly on all questions.