Re: The outcome of a standardized test is an integer between 151
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05 Jul 2021, 19:11
There are 400 test scores distributed among 50 possible outcomes
(integers between 151 and 200, inclusive, which number 200 – 151 + 1 = 50 integers).
There is an average of 400 ÷ 50 = 8 scores per integer outcome, and there are 400 ÷ 100 = 4 scores in each percentile.
So, if all the scores were
completely evenly distributed with exactly 8 scores per integer, there would
be two percentile groups per integer outcome (0th and 1st percentiles at 151,
2nd and 3rd percentiles at 152, etc.). In that case, all 50 integers from 151 to
200 would correspond to more than one percentile group.
Reduce the number of integers corresponding to more than one percentile
group by bunching up the scores. Imagine that everyone gets a 157. Then that
integer is the only one that corresponds to more than one percentile group (it
corresponds to all 100 groups, in fact). However, don’t reduce further this
way. This gives exactly 1 integer, so the minimum number of integers
corresponding to more than one percentile group is 1, which is Quantity A.
As for Quantity B, though, a particular integer may have no percentile groups
corresponding to it. In the previous example, if everyone gets a 157, then no
one gets a 158, or a 200 for that matter. So the minimum number of percentile
groups corresponding to a score of 200 (or to any other particular score) is 0,
which is Quantity B.