Many quake in their boots when they hear that there will be Statistics covered on the GRE. They run to their college stats textbooks, dust off the cover, roll up their sleeves, and start computing the standard deviations of a list of twenty, three-digit numbers. Stop, if this in anyway describes you.

The Statistics on the GRE is much simpler, and does not test your aptitude at crunching numbers as much as it does your ability to think about Statistics. That is you will rely more on intuition than computation on statistics questions on the GRE. You shouldn’t be so worried about how many statistics questions there are on the GRE, anyway.

To illustrate take a look at the following question.

1- The standard deviation on a test was 12 points, and the mean was 70. If the scores fell along a normal distribution and student X scored 95 points, then student X scored higher than approximately what percent of students?

- 2%
- 13%
- 48%
- 96%
- 98%

Answering this question correctly requires understanding standard distribution (that refers to the distribution of scores along the familiar bell-curve). To understand how standard deviation relates to the bell-curve take a look below:

Within 1 Standard Deviation Above the Mean= 34%

Within 1 Standard Deviation Below the Mean= 34%

Between 1 and 2 Standard Deviations Above the Mean = 13.5%

Between 1 and 2 Standard Deviations Below the Mean= 13.5%

Between 2 and 3 Standard Deviations Above the Mean = 2%

Between 2 and 3 Standard Deviations Below the Mean= 2%

In the problem above, 34% of students scored between 70 and 82. Likewise, 34% of students scored between 58 and 70. This symmetry is very important, and you will notice that the bell curve is symmetrical (or even) on both sides. So, given a large enough sample size, the number of students who scored three standard deviations below the average of 70 (34) is the same as the number who scored three standard deviations above the average (106).

Returning to the actual question, we want to find how many standard deviations above the average a score 95 of points is: 95 – 70 = 25, which is a tiny bit more than two standard deviations. The question is asking for an approximation, so we can round down 25 to 24.

Looking at table above, we can see that two standard deviations above the norm is better than 34% + 13.5%. The trick here is to not forget to account for the left side of the bell-curve, which is 50% (after all, half the score are on the left side and the other half on the right side—don’t forget the symmetry of the bell-curve).

That gives us a total of 50% + 47.5 = 97.5, which approximates to (E) 98%.

Let’s try another problem.

2. The reaction time of 1000 Rhesus monkeys was measured. The average time it took the monkeys to respond to a quickly moving object in their visual fields was .135 seconds, with a standard deviation of .021 seconds (assume a normal distribution). If one of the geriatric monkeys had a reaction time of .205 seconds, then that monkey’s reaction time is how many standard deviations from the mean?

- 0 – 1 standard deviations
- 1 – 2 standard deviations
- 2 – 3 standard deviations
- 3 – 4 standard deviations
- 4 – 5 standard deviations

This is exactly the sort of daunting problem that the GRE likes to throw at you. Believe it or not, there is very little math involved. Again, you want to rely on intuition more than math.

.205 – .135 = .07. If the standard deviation is .021, we can determine the number of standard deviations the monkey’s reaction time is from the mean: .07/.021, which equals approximately 3.4. Therefore (D) – the geriatric monkey’s reaction time is 3 – 4 standard deviations from the mean.

## Takeaway

To do well on statistics questions on the GRE, you have to rely more on intuition than on number crunching. Having a strong sense of standard distribution and how standard deviation relates to standard distribution will help you immeasurably.

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I’ve just looked at the standard deviation part in the math review of ETS official guide. They are using 14%, not 13.5%..

Hi Meghana,

You’re right! The truth is that there are varying degrees of specificity used across mathematics to talk about standard deviation. 14%, 13.5% or even 13.6% and others are used. They are all close enough that, for the purposes of the GRE, using any of these figures will allow you to answer correctly. 🙂

If in the question it is stated that a distribution is “Normal distribution” should i assume that the average of the distribution must be the median?

When a distribution is normal, the mean, median and mode in the population will all be equal

nailed em both.

Getting better at this!!

My gre is on saturday.

Wish me luck guys!!!

I’m curious what % ETS is using. Magoosh is using 34% + 34% for +- 1 SD. Manhattan GRE uses 33% + 36%, which then results in different answer choices depending on the questions that ask for percentiles at the edge of the 3rd SD.

Does it matter at all, or can you confirm what value ETS is using to test?

Alberto,

ETS will definitely use 34%. The actual number is 34.13%. ETS would never take such liberties, rounding down to 33%. Though, for the most part, using 33% would probably not affect your answer in most cases.

Hope that helps!

Hello Chris,

Just a quick question on question 1. To solve the question dont we need to first be sure it is indeed a “normal distribution” as isnt only the normal distribution has this nice bell shape with one standard deviation within 34% about the mean? As in a quant comparison question I would assume the answer would have been D as we would require normal or approximately normal? Do let me know where I have gone wrong 🙂 Thanks!

Oh my yes–you are totally right :). I’m used to dealing with statistics in a GRE context, so every spread of scores seems to fall on a normal distribution. But the test will say “normal distribution”. I’ll make the change to the post now. Thanks!

Hi Chris,

For all our sakes, I’m divulging this : when I first passed my GRE last July, I got a question that required computing a standard deviation. Well, almost. It gave you the standard deviation of three numbers, and asked for its new value when the numbers were changed in a certain manner.

Now the question may have been part of the experimental section or could’ve been solved without otherwise knowing the formula. I still wonder how though.

So Chris could you please weigh in on the matter and explain how I should’ve come up with the answer without using the formula?

Thanks!

Thanks for divulging 🙂

I really can’t say for sure, but I’d be surprised if the GRE had a question in which the only way one could reliably solve it is by using the Standard Deviation formula. Inferential thinking is typically also rewarded. That is, SD on the GRE is about having this sense of what it means for any given set of numbers to have a greater standard deviation than another set.

For instance, if I have (3, 4, 5, 7) and (2,3,4,6.1), the standard deviation is greater in the second set (notice the difference between 6.1 – 4 is greater than that of 5 and 3). I am sure the problem you saw was more complex, but again the inferential approach would have been successful as well.

All this said, I think it safe to memorize the SD formula – just in case.

Let me know if you have any more questions…and to think of it, all of this has inspired a blog post: How to solve Standard Deviation without using the big, nasty formula. Look for it soon :).

Looking forward to it then 🙂

Hey Chris,

I’m stuck on the first problem. I understand every step until we account for the left side of the bell curve. If the bell curve is symmetrical, then shouldn’t it be 47.5+47.5? Where did the 50% come from?

Hi SB,

It is indeed symmetrical, 47.5% on either side of the mean gives us a 95% range.

In this problem, however, the entire left half of the curve represents test scores below the average. Therefore, if we get a test score that is 2 standard deviations above the mean, it will be better than all the test scores below it. This includes the scores between our score and the average AND all of the scores below the average (not just 47.5% of the scores below the average).

If we wanted to know what percent of score were within 2 standard deviations of the mean in either direction, then we would take 47.5% on each side. In this case we’re just concerned with everything below a certain value.

I hope that helps!

Thanks Jonah :).

Very clearly explained. I should have probably been clearer where the 50% came from.

Thanks again!

Sorry for the confusion :). Since we are accounting for all the scores below 70 (the mean), that number represents half of all scores. Hence the 50%. Also check out Jonah’s comment below. I think he does a good job of explaining the concept :).

Hi Chris,

I tried solving the first problem in a different way, and wanted to quickly check if conceptually it made sense . Based on this I get 96% as the answer,

I took the standard % distribution of a bell curve,

So

68% 1’st sd,

95% 1’st and 2’nd SD

99.7 % 1’st , 2’nd, 3’rd SD.

Of course 4’sd will cover 100% of the population.

Now I plotted a graph and if the SD is 12′ then 2’nd sd should include marks 94 and below and for the 3’rd SD the limit is 106 (94+12). Given mean 70.

So if the student x gets 95 points the. He sits little ahead of the 95% limit of the population.95% corresponds to 94 points / marks.

99.7 % correspon to 106 marks .

Now I kept on dividing this space ( between the 2’nd and the 3’rd SD , I.e between 95% and 99.7% which also corresponds to between 96 and 106 mArks ). The spot exactly in the MIDDLE of this is 100 marks that is 96.4%

So the lower limit is 94 marks corresponding to 95% and upper limit of this tiny space is 100 marks corresponding to 96.4% .. I

And 95 sits between these boundaries so I conclude its closest to 96% .

Please let me know if my approach is correct.

Many thanks.

Ps. I love vocab Wednesdays 🙂

I noticed the exact same thing and wanted to comment but you beat me to it. I think it’s a mistake. This example definitely hasn’t followed the 68,95,99 rule properly.

Hi Ammar and Meera,

I can see your logic and you can use a similar one, but there is one thing that you have to be careful with. First, when we say that 95% is within the first and second SD, the remaining 5% is divided between the two ‘tails’ of the normal graph. That is, 2.5% of the remaining percent is at the lower end of the bell curve, and only 2.5% is at the upper end. So a person who sits at the second positive SD away from the curve (94 points) is actually in the 97.5th percentile (2.5% from the bottom tail of the graph, plus 97.5% from the first and second SD). If a person scored more than 94 points on the test, then they must be just above the 97.5%. The only possible answer is 98%.

We also have to be careful with ‘dividing’ the spaces between the standard deviations. The values between two SDs are not uniform, and if you are between the second and third SD, more of the values will be closer to the second SD than the third one. If you look at the normal graph divided into standard deviations you will see this! So if you need to pinpoint a value between two SDs, you must be careful that you remember that the values will tend towards one end or another.

Hey Chris!

Thanks for your previous reply. Please answer the following:-

1) I am done vit std deviation too, its relation with bell curve(d 3 no’s viz 68,95,99), the concept of probability distribution,freq distribution,etc. Which part I havnt touchd yet?

2) Also, what is the difficulty level of Manhattan GMAT quants, compared to actual GRE?

3) Which has an equal level? Manhattan GRE verbal or which verbal material?

Thank-you 🙂

1) As far as the bell curve goes that is sufficient.

2) MGMAT quants are much more difficult than actual GRE questions.

3) MGRE tends to be pretty difficult for verbal. It’s comparable to test.

Hope that helps!

Hey Chris!

How you doing?

Please tell me what concepts pertaining to Data analysis should be studied for revised GRE. I am done with the following:-

1) Mean,mode,median, concept of random variable.

2) Normal distribution(Bell curve)

3) quartiles,interquartile range,percentile.

What more should be learnt?

Thanks!

You are doing a great job 🙂

Hi Deepak,

Good question! I think you pretty much covered everything except Standard Deviation. Otherwise, your list is comprehensive :).

Hi Chris,

As the theme of the week goes (ETS PAPER BASED TEST) I thought to include a question from new version of paper based.test by ETS:

Section 5 question 8:-

Thefrequencydistributionsshownaboverepresenttwogroupsofdata.Eachofthedata valuesisamultipleof10.

QuantityA 8.Thestandarddeviationof distributionA

QuantityB Thestandarddeviationof distributionB

It may not be clear from the description and moreover there weee two bar graph involved which I am unable to paste here

Thanks

Hi Aman,

I’ve actually already recorded this video and it should be up soon. Doing such a graph-based problem without the graph is very difficult, so look for the questions on youtube soon!

Hi Chris,

What abt calculating standard deviations? Would we be asked to do that?

TnR

Prem

Prem,

Only in rare instances – it seems the GRE is more focused on testing one’s sense of standard deviation vs. one’s ability to actually compute it.

well explained

Thank you!

Hey Chris,

Would the GRE really phrase a question like this? We learned in my college Statistics class that just because one value is “better” (in this case faster) doesn’t mean that the answer has to be “below the mean”. It’s perfectly plausible in statistics to have higher numbers mean “worse” results but still refer to them as being “above the mean”. I think somebody who calculated the answer as 3-4 standard deviations above the mean would have a legitimate gripe if the answer was marked incorrect.

Ooh…I was thinking about that when I was writing the question and wanted to make sure it wasn’t ambiguous… but you’re right. ‘Better’ is arbitrary. I will make the changes. Thanks for your sharp eye!

Typo or careless error? In the monkey problem, the reaction time is .215, but when you solved the problem, you used .205.

I feel better about myself now, seeing that even Chris makes mistakes sometimes!

Yes, I am human :).

I’ll make the changes. Thanks for catching the oversight!