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GRE Math: Percentiles and Quartiles

Fact: An 8 year old boy who is 4’5″ (53 inches) tall is in the 86th percentile for height for his age.

What on earth does that mean?  Well, the percentile of an individual tells you what percent of the population has a value of a variable is below that individual’s value of the variable.  For example, to say that a 4’5″ 8 year-old boy is in the 86th percentile for height for his age, we are saying: gather together all 8 year-old boys on Earth, and measure their heights; if you sort out all the 8 year-old boys who have a height less than 4’5″, they will comprise approximately 86% of the population.   That boy is taller than 86% of other boys his age – that means he’s in the 86% percentile.

Percentiles is a relatively unlikely topic to see on the GRE, but if it does show up, here are a few handy facts to have up your sleeve.

 

Biggest and Smallest

A few details to clarify.  The individual with the lowest value of the variable, with the minimum value, is not bigger than anyone, so the lowest percentile, the percentile of the rock-bottom minimum, is the 0th percentile.  If my score is in the 0th percentile, then I am not higher than anyone.

What’s trickier is the maximum score.  If my score is the highest score, I am higher than everybody else, but that’s not the 100th percentile, because in order to be higher than 100% of the population, higher than everyone, I would have to have a score higher than my own score: a paradox!  In fact, for this very reason, there’s no such thing as a 100th percentile.  The person with the highest score is higher than everybody else, but not higher than herself, so she’s in the 99th percentile.  If we are sticking with whole numbers, the 99th percentile is the highest possible percentile.  If we go to decimals, we can get higher with the 99.9th percentile (1 out of a 1000), the 99.99th percentile (1 out of 10000), etc.

 

Median and Quartiles

The median is the middle of a list: the median divides a list into an “upper half” and a “lower half.”    This means, the median is higher than the lower half of the population, higher than 50%, so the median is the 50th percentile.  Now, we have to be careful here.  On a list with only three members — e.g. {2, 4, 7} — the median is the middle number, here 4, but that number is higher than only one number out of three — so 4 is the 33rd percentile of that list.  In a technical sense, the median is not always the 50th percentile.

In some sense, though, that’s a specious objection.  When there are only 3 members on a list, nobody in their right mind talks about percentiles.  When the total number is less than a few hundred, there’s seldom talk of a percentile.  Percentiles, by their very nature, are a way to make sense of tens of thousands, even millions of individuals.  How many 8 year-old boys are there on Earth?  Who knows, but it’s certainly a very very large number.  That’s where percentiles are used in practice.

When the number of folks in the group is that large, then for all intents and purposes,the median is the 50th percentile.  If you are familiar with the idea of quartiles, then the first quartile is the 25th percentile and the third quartile is the 75th percentile, again, when the group sizes are truly huge.

 

Practice Questions

1) Sasha took a nationwide standardized test that is graded on a scale from 20 to 60.  Sasha got one of the best scores recorded on that this test.

 

Column A              Column B

Sasha’s score            the percentile of Sasha’s score

 

(A) The quantity in Column A is greater.

(B) The quantity in Column B is greater.

(C) The two quantities are equal.

(D) The relationship cannot be determined from the information given.

 

2) Alice took nationwide standardize test that is graded on a scale from 0 to 100.  Alice scored the highest score recorded on this test.

Column A               Column B

Alice’s score             the percentile of Alice’s score

 

(A) The quantity in Column A is greater.

(B) The quantity in Column B is greater.

(C) The two quantities are equal.

(D) The relationship cannot be determined from the information given.

 

3) A large distribution of score is normally distributed

Column A

score that’s one standard deviation above the mean

Column B

score that has the 80th percentile

(A) The quantity in Column A is greater.

(B) The quantity in Column B is greater.

(C) The two quantities are equal.

(D) The relationship cannot be determined from the information given.

 

Practice Questions Answers and Explanations

(1) B; (2) D; (3) A;

1) We know that Sasha is near the top of the scoring distribution, so that would mean a score with a percentile close to the 99th percentile.  Because of the scoring scale, the score is not going to be above 60, so the percentile is clearly bigger.  Answer = B.

2) Alice got the highest score, so by definition, that’s the 99th percentile.  What we don’t know is: how hard was this test?  What score was the highest score?  If it was a particularly challenging test, it could be that the highest score anyone achieved was only, say, a 73.  In that case, the percentile would be greater.  If, on the other hand, it was possible to get a perfect score, and Alice did in fact do that, then her score of a 100 would be greater than the percentile.  We don’t have enough information to decide.  Answer = D.

3) Here, it might be helpful to brush up on Normal Distribution.  On a normal distribution, it’s always true that 68% of the populations lies within one standard deviation of the mean.  That means, half of that, 34%, lie between the mean and one standard deviation above the mean.  The score that is one standard deviation is higher than the 34% between the mean and one standard deviation above the mean, as well as than the 50% below the mean.  That means, a score that lies one standard deviation above the mean is the 50 + 34 = 84th percentile.  Thus, it’s higher than a score in the 80th percentile.  Answer = A.

About the Author

Mike McGarry is a Content Developer for Magoosh with over 20 years of teaching experience and a BS in Physics and an MA in Religion, both from Harvard. He enjoys hitting foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Follow him on Google+!

22 Responses to GRE Math: Percentiles and Quartiles

  1. mori May 28, 2014 at 12:06 pm #

    There are two explanation for almost the same question in ETS and Manhattan.
    Is there anyone who can help me?

    • Mike
      Mike May 28, 2014 at 1:35 pm #

      Dear Mori,
      This blog is not a place to ask for questions about outside material. Here’s what I’ll recommend. You can post your question as a new thread on GMAT Club, in the Magoosh forum:
      http://gmatclub.com/forum/magoosh-324/
      Even though that’s a GMAT forum, we will answer your GRE math questions.
      Does this make sense?
      Mike :-)

  2. akanksha October 8, 2013 at 10:21 pm #

    Hi Mike

    Can you please explain this question:
    A large distribution of score is normally distributed

    Column A: score that’s one standard deviation above the mean
    Column B: score that has the 80th percentile

  3. Abhishek September 4, 2013 at 5:56 am #

    Hi Mike,

    Just a question, if 10 people appear for a exam the highest percentile can be 90 ((100)-100/10) and if 1000 people give an exam highest percentile would be 99.9 ((100)-100/1000). Please let me know if my understanding is right or wrong

    • Mike
      Mike September 4, 2013 at 9:53 am #

      Dear Abhishek,
      Yes, that’s quite right.
      Mike :-)

      • BHargav September 8, 2013 at 6:36 am #

        hi Mike,In dat case…if the list consists as u explained in the above (2,4,7).,the 50th percentile will be the meadian 4 na,but how it could be 33rd percentile…explain me.??

        • Mike
          Mike September 8, 2013 at 12:06 pm #

          BHargav,
          The percentile is a measure that is used to discuss the position of an individual within a population — a population consisting of thousands or hundreds of thousand or more. In that context, the only context in which percentile is truly meaningful, the median is always the 50th percentile.
          It’s absurd to talk about the percentile of a list with only three members. The GRE will not ask about percentile in a context in which it is absurd.
          Does this make sense?
          Mike :-)

  4. VAMSEE September 2, 2013 at 3:10 am #

    Hi Mike,
    The definition for percentile given by you is “the percentile of an individual tells you what percent of the population has a value of a variable is below that individual’s value of the variable. ” But what about the values that are equal to the value being considered?
    For example consider the below values:
    1,2,3,4,5,6,6,6,7,8
    What is the percentile of value 6 among the values given above?
    Is it 50 or 70

    • Mike
      Mike September 2, 2013 at 11:20 am #

      Dear Vamsee,
      That’s a great question. The percentile of 6 is the 50th percentile, because each 6 is above just 50% of the population.
      Having said that, keep in mind that there’s something a bit absurd about talking about percentiles when there are only 10 on the list. The use of primary percentiles is to help keep track of where a particular number falls within a population of hundreds or thousands or more. If you have 10 numbers, you can just look at the whole list — you don’t need fancy metrics to tell you what’s where.
      Does this make sense?
      Mike :-)

  5. Arun August 19, 2013 at 9:48 am #

    Hey Mike,
    Fist of all, excellent article!!
    I have a query regarding the first question. What if all of the test takers got the “same” highest score? Then Sasha would be in the zero percentile zone as nobody is technically under her, right? I know I am being stupid here and the case is only theoretical. But should we have to take such cases into account? Please forgive my ignorance if I am asking a really stupid question!

    • Mike
      Mike August 19, 2013 at 10:22 am #

      Arun,
      In that question, the test is “nationwide”, which implies that thousands (if not hundreds of thousands) of people take it. This is a subtle thing about populations and variation among population. If I give a test to class of ten students, yes, it’s possible that everyone will get the same score: that would be a very poorly written test. In the real world, it cost millions of dollars to sponsor a nationwide test. In practice, no one is going to spend millions of dollar to give a test, and then not even check the quality, so that the test is so poor that everyone gets the same score. If ETS did that even once, they would be out of business! The “everyone gets the same score” scenario is a mathematical abstraction that ignores the hard tangible economic realities of the situation. Does this make sense?
      Mike :-)

      • P R Siddharthan September 18, 2014 at 5:39 am #

        I am sorry to say this, but it does not.
        Even I hosted a word wide test online (on edX opencourseware) only 2 took the test. So the highest score got a percentile of 50 only .
        How can general knowledge on how a nation wide test is conducted influence my answer.
        Should I expect such ambiguous questions in GRE?

        • Mike
          Mike September 18, 2014 at 2:21 pm #

          Dear PR,
          My friend, we are getting into some subtleties of English here. Yes, anything online is available to anyone in the world with internet, so, say, the reply I am writing to you now — technically, anyone in the world could see it, but it would grossly misleading to say I have written a “worldwide” response. The words “worldwide” or “nationwide” don’t merely connote the possibility that a large number of people could have access to it: by contrast, these words connote an actual use or recognition or fame or popularity that is, in fact, wide spread among a large number of people. They connote that a substantial fraction of the relevant population actually does use or see something.
          It is factually correct to say that, for example, Coca Cola has worldwide popularity or recognition, or that Coke is a worldwide company. People really do recognize that logo everywhere. Magoosh has customers all over the world, but it would be grossly misleading to say that we have “worldwide” popularity — the latter would suggest that almost wherever you went, vast majority of people would recognize the brand.
          When Gandhi marched to the sea or when Nelson Mandala was released from prison, those legitimately were events of worldwide significant: they actually meant a great deal, and were deeply meaningful to, a large number of people on every part of the planet. If I say or do something and post it on social media, theoretically, anyone in the world could look at it, but even if a couple dozen friends of mine in different corners of the world say that it is meaningful, it would egregiously deceptive for me to say that that my activity was an event of worldwide significance. To call something “worldwide,” it must be true for a very large number of people in many different parts of the world.
          Much in the same way, the word “nationwide” does not merely mean that anyone in the nation could take the test, that maybe 12 or 15 people dispersed across the country have taken it. That would be a deceptively misleading use of the what the word “nationwide” means. The word “nationwide” always connotes popularity and widespread usage throughout the nation. There’s no specific number associated with the word, but if only a few dozen are using the item, then “nationwide” would a patently incorrect word, and using it could very well be construed as a deliberate way to deceive. Hilary Clinton legitimately has nationwide recognition in America: almost everyone in America, and many people in other parts of the world, know exactly who she is. People may like her or not, but everyone recognized that name. My blogs are on the web, and theoretically, everyone in America could read my blogs and find out about me, but it would be patently false for me to claim “nationwide” recognition for myself.
          Again, if I were to make the unfounded claim that Magoosh had “worldwide recognition,” that easily could be construed as a deliberate attempt to mislead or misrepresent on my part. People would be completely justified in saying that I was lying if I were to say that. Even there there are people all over the world that recognize and use Magoosh, the sheer numbers and sheer percentage of the relevant populations are not nearly enough to justify the wide level of inclusion that the word “worldwide” implies.
          The GRE never intentionally misleads students, so if they used the word “nationwide” or “worldwide,” they would only use them in their legitimate senses, the sense of sizable chunk of the population making use of something. The way they are used here models the use they could have on the GRE.
          Does all this make sense?
          Mike :-)

  6. VAMSEEDHAR REDDY August 7, 2013 at 4:05 am #

    2) Alice took nationwide standardize test that is graded on a scale from 0 to 100. Alice the highest score recorded on that this test.

    In the above question, the mentioned looks similar to gre, isn’t it. Gre is graded on a scale 260 to 340. In the gre the topper gets a score of perfect 340 which is 99th percentile. Doesnt’ it mean that grading is done on the basis of percentiles. But according to your answer, if the exam is tough,the score may be low. please clarify the process of gradation you mentioned.

    • Mike
      Mike August 7, 2013 at 9:34 am #

      Dear Vamseedhar,
      You are correct, highest percentile is 99%, so for most tests the highest grade is 99th percentile. This question is asking us to compare the numerical grade, to the numerical percentile. For example, for the GRE, the number for the highest score (340) would always be higher than the number for the highest percentile, because 340 > 99. If a test when from, say, 1 to 12, then in all likelihood, the number for the highest score (12) would be less than the number for the highest percentile, 12 < 99. For a test that goes from 0 to 100, Maybe Alice got a score of 100, which was 99th percentile, so (score) > (percentile), or maybe Alice's score was only 78 but it was still 99th percentile, so then (score) < (percentile). We are comparing one number to another number. Does this make sense?
      Mike :-)

      • VAMSEE August 8, 2013 at 11:45 pm #

        Dear Mike,
        Thank you for quick reply. But your answer does not clarify my doubt. What I said was that in an exam like gre a 99 percentiler gets a perfect score. In the question given since the grade is scaled from 0 to 100
        a 99 percentiler should get 100 if it is similar to gre. May be the question does not clearly mention if it was absolute grading or relative grading(like gre). In case of relative grading, in the question given, the score of the student will be greater than percentile,I think. I hope you clarify this.

        • Mike
          Mike August 9, 2013 at 10:34 am #

          Dear VAMSEE,
          First of all, I’m not sure what grading scheme you have in mind when you say “absolute grading”, but anything that involves percentiles, by definition, is comparing you to the rest of the population and hence is relative.
          Percentiles are always always always about comparing you to the rest of the population. They never have any meaning apart from that. To know the percentile of any single score, on any test, you have to have information about how the entire rest of the population did.
          Question #2 says: “Alice scored the highest score recorded on this test.” We don’t know the numerical value of that — maybe 100, or maybe something much lower. We do know, because this score is the highest score, it’s the 99th percentile.
          Does this make sense?
          Mike :-)

  7. Soumya May 22, 2012 at 8:14 am #

    Hi Mike. My original thoughts for the first question was similar to the answer given here, that Sasha’s score is probably in the the 90th percentile or above. Then I realized that the number of test takers was not included (and that thinking she was in the 90th percentile was an arbitrary guess). So, how do we really know what percentile Sasha is in? What if there were two test takers? That would mean that Sasha could either be in the 0th or 50th percentile. Right? This would mean that the answer should be D. Perhaps I read the question wrongly and made a mistake in my analysis. What do you think? This is a similar thought process used in your explanation for Question 2 (but the opposite situation).

    • Mike
      Mike May 23, 2012 at 9:52 am #

      Dear Soumya —
      In both questions, the word “nationwide” was designed to connote that a large number of people took the test — maybe 1000, maybe 1000000, but some large number. It simply wouldn’t make sense to call the test “nationwide” if only two people took it —do you see what I mean? Furthermore, the fact that Sasha got “one of the highest score” means there was more than one highest score. That makes it impossible for there to have been only two test takers, and again suggests a large test-taking pool. Therefore, we know some reasonably large number of people took the test, and if Sasha got one of the highest scores, it’s well above the 90th percentile. Does this make sense?
      Mike :)

      • Soumya May 23, 2012 at 11:10 am #

        Hi Mike,

        Yes, that definitely makes sense. This is the kind of little mistake I keep making! How frustrating. I got a 790 on the SAT Math back in the day and now I cannot break 155-160 on practice GREs (which has been verified by my Magoosh math score estimate), all because of this silly lack of attention.

        On that note — how would you compare SAT and GRE math? Is this dissonance in my scores expected or abnormal? I should note that in that time, I had completed both my Bachelor’s and Master’s in Engineering in a very rigorous field at a highly ranked university. Perhaps I need to go back to practicing mental math and just focus on attention to detail. Any enlightenment in this matter would be appreciated. Anyways, thanks for the reply!

        Soumya

        • Mike
          Mike May 23, 2012 at 2:25 pm #

          Dear Souyma,
          First of all, about attention and lack thereof, take a look at these two earlier blogs:
          1) http://magoosh.com/gmat/2012/overcome-gmat-exam-anxiety-breathe/
          2) http://magoosh.com/gmat/2012/beating-gmat-stress/
          GRE math is, on average, slightly harder than SAT; the content is almost identical, but the GRE has a lower concentration of easy questions and a higher concentration of hard questions. For someone with degrees in Engineering, the difficulty level of the GRE math simply is not an issue. Instead, it’s all about attention to detail, catching distinctions, not overlooking key phrases, etc. In short,it’s not math you need to learn as much as mindfulness, which is discussed in those two posts.
          Does that make sense?
          Mike :)


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