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Question:

300 test results are integers ranging from 15 to 75, inclusive. Dominick’s result is clearly in the 80th percentile of those results, not the 79th or the 81st.

Quantity A: Number of other test results in the same percentile as Dominick’s

Quantity B: Maximum number of other test-takers with the same result as Dominick

Compare Quantity A and B.

Solution

Since the number of test results is divisible by 100, the percentiles cleanly divide the total into 100 percentile groups of 300 ÷ 100 = 3 results each. That is, there are 3 results in each percentile. So 2 other results are in the same percentile as Dominick’s. Quantity A is 2. Dominick’s result is clearly in the 80th percentile, not the 79th or the 81st. So it must be possible to distinguish the 80th percentile (that group of 3) from the “next-door” percentiles. Say Dominick got a 58. How many other people could have gotten a 58? Maybe no one, maybe one, maybe two — but if three other people got a 58, then you’d have a total of four people with the same result. In that case, it would be impossible to assign Dominick’s result definitively to the 80th percentile and not the neighboring percentiles. So the maximum number of other test-takers with the same result as Dominick is 2. Quantity B is also 2.

My Query

  1. Is this a normal distribution? If this was a Normal Distribution would this be a valid explanation? I mean, in a normal distribution, wouldn't most of the 300 test scores be concentrated around the mean, so each of those percentile points around the mean would have more of those 300 test scores?

  2. How do I picture this in a graph? The frequency of the test scores in Y-Axis and the integers ranging from 15-75 along the X-Axis? If that's the case, then wouldn't the graph be a flat line, because every score has only 3 test-taker (frequency).

  3. In general I want to know, how can we know, that as there is 300 test results, they must divide into equal percentile blocks.

Thanks.

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  1. No, it is not a normal distribution. We have no information about the distribution but its range and Dominick's result. It may or may not be bell-curved, but the outcomes are non-negative integers, which is too restrictive for a normal distribution.

  2. If you had more information about the distribution, you could indeed represent it with such a graph. It would not be a flat line in the general case: there are $3$ test-takers per percentile, not per possible score ! Maybe no-one add a $75$, maybe $20$ people had a $62$.

  3. This is basically the definition of percentiles: each of them contains $1$ percent of the observations.

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There are only $61$ possible different scores, so it is clear that the $300$ scores are not divided into $100$ blocks of $3$ scores each.

But it is possible (in fact, in reality quite likely) that people who got scores closer to the $50$th percentile are in blocks of scores much larger than those who are around the $80$th percentile.

The distributions of a finite set of test scores also do not generally form a perfect bell curve. It can be expected that there will be a few places where more than the expected number of people got the same score, or where fewer than expected got the same score. So for there to be one particular score that only three people received is not too unusual.

In any case, the question here is not asking you for a model of the test scores. It is not even asking how many people had the same score as Dominick. I believe it is asking whether one of the following numbers can be greater than, equal to, or less than the other:

The number of other scores (numbers in the range $15$ to $75$, excluding the score that Dominick got) that were received by someone and are in the $80$th percentile.

The number of people (other than Dominick) in the $80$th percentile.

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