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Percentiles

Now that we’ve talked about boxplots and the numbers associated with boxplots, we can talk about percentiles. Percentiles are one more way of discussing the position of an individual score in a large population. Now you may wonder, since we already have those numbers associated with boxplots, why do we need something else?

Why do we percentiles in addition to boxplots? Boxplots. Well, pretend those are more precise than the boxplot numbers. For example, think about GRE math scores. On the GRE math section, a score of 160 is between the third quartile and the max. A score of 167 is between the third quartile and the max.

Well saying it like that makes it sounds like those two scores are equivalent, as if they’re more or less interchangeable, and of course that’s not the case at all. When we talk about percentiles, the difference becomes clear. A score of 160 is the 78th percentile. A score 167 is the 95th percentile. Well holy smokes, that’s a big difference.

Let’s talk about what that means. So, let’s back up. First of all in the abstract, if a score is in the 40th percentile in a large distribution, this means the score is larger than 40% of the distribution. It is larger than 40% of the scores.

And in general, that pattern holds. I’ll state it in terms of a variable. If a score is the pth percentile of a distribution, that score is larger than p percent of the scores in the distribution. And of course, p could be anything between one and 100. Let’s go back to those GRE scores and look at them again.

A score of 160 on the GRE math section is the 78th percentile. So what that means is, if you score 160 on your GRE math, you have scored higher than 78% of GRE takers on the math section. So, 70, 78%, that is slightly bigger than 75, so you’ve done better than slightly more than three quarters of the people who take the GRE. Not bad.

But, consider 167. 167 is the 95th percentile. You get 167 as your GRE Math score, that means you’ve done better than 95% of the people who take the GRE. Or another way to say that is, you have a math score that is in the top 5%. Now that’s very special, and that is much more impressive than 160.

And percentiles is what allows us to talk about this subtle, but important, difference. Now a few things to keep in mind about this. First of all, percentiles only make sense for large distributions. Everyone taking the GRE, everyone with a driver’s license in the state of California, every household in the nation of Japan.

Something like that, a gigantic population. It doesn’t make sense to talk about percentiles when you have a list of five or ten, something like that. It just doesn’t make sense to use it for short lists. It is only used for populations. Notice that the percentile of the minimum score, the lowest score, is a 0th percentile.

So, let’s think about this. Suppose there’s a test, and suppose I get the absolute lowest score on that test. Well, if I get that score, it means that my score is higher than zero percent of the population, and therefore I would be the 0th percentile. So that’s more or less straightforward, the lowest score is a 0th percentile. Here’s the one that’s tricky though.

The percentile of the highest score is the 99th percentile. In other words, there’s no such thing as 100th percentile. Now why would this be? Suppose I get the highest score on the test. Well, to be the 100th percentile, I would have to have a score that is a, that is higher than 100% of the scores.

And the problem with that is, my own score is one of the scores, so to be the 100th percentile, I’d have to get a score that was higher than my own score, and of course that’s impossible. So, the highest you can get in whole numbers is the 99th percentile. Now sometimes people get into very fine distinctions. We can talk about the 99.9th percentile, the 99.99th percentile, something along those lines, but the 100th percentile is impossible.

Keep in mind there’s a very close correlation between the numbers that we just talked about in boxplots and percentiles. Of course the minimum we just said is the 0th percentile, the maximum is the 99th percentile. And approximately, the first quartile is the 25th percentile, the median is the 50th percentile, the third quartile is the 75th percentile.

And I say approximately. If we’re only talking about a thousand or a couple thousand people, these numbers are gonna be approximate, they’ll be much more accurate the larger the population is. By the time you’re talking about a population that’s the size of an entire nation, these equations here will be exact.

Keep in mind also, we can talk about how this fits in with the normal distribution, which you talked about a few videos ago. So, these are the actual numbers that you saw in the normal distribution video. Well let’s think about this, the center number, of course, is the mean, the median, and the mode. So that number right in the middle, the mean, that is the 50th percentile.

Well, if we go one standard deviation below the mean, so drop down 34%, that means that this number here must be the 16th percentile. You’re there, you’re higher than 16% of the population. If we go up a standard deviation above the mean, then we add 34%, and so that means here, we’re at the 84th percentile. And if we go another standard deviation above the mean, we add 13.5, that puts us at the 97.5 percentile.

And so if you are two standard deviations above the mean, you are scoring higher than 97.5% of the population. Incidentally, if you are three standard deviations above the mean, you are well above the 99th percentile. And so that is how percentiles work on the normal distribution. Finally, this is a very subtle and difficult idea to understand.

Halfway between percentiles is not the same as halfway between scores. What do I mean by that? Well consider this, this theoretical practice question. If six, 660 is the 70th percentile of a normal distribution, so I’ve got a normal distribution. And 760 is the 80th percentile, then what can we say about the 75th percentile?

So this is kind of a vaguely worded question. This might be worded much more precisely, for example, as a quantitative comparison. But the basic idea is, we have a normal distribution. And we’re up on the upper arm of the normal distribution, so we’re at the part of the normal distribution where it’s descending like this.

And somewhere here is 660, that’s a 70th percentile. And somewhere here is 760, and this is the 80th percentile. So, we know that between here, this is all 10% of the population. And we want to know, if we go right in the middle, how does that number right in the middle compare to the 75th percentile? Now the mistake would be to think that the, the percentile that’s in the middle is exactly the same place as the score that’s in the middle.

And problems like this, people fall into that trap all the time. But let’s think about this very carefully. Where are there more people? Toward this side or toward this side? Well, we can see the normal distribution is sloping down here. And so we can see, there’s many more people on this side and many fewer people on this side.

And in fact, if we just think of the area of this whole thing, where would we draw a vertical line to cut this area in half? And I’m going to guess it would be around here, approximately. In other words, I’m gonna say that this area here is approximately equal to this area here. We have this sort of short, chunkier area and then this long, thin area.

And they are about the same. And so notice that the, the place where we put the line that cuts the area in half, that would actually be the 75th percentile. So, 5% would be on one side, and 5% would be on the other side. And notice that this is lower than the score right between them. So the score right between them would be 710, exactly between 660 and 760.

And so the 75th percentile has to be lower than 710. So again, this is a very subtle problem, but this could come up on some of the more difficult math questions on the GRE.