Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized for You

we will pick new questions that match your level based on your Timer History

Track Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 350,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

Set X has 5 integers: a, b, c, d, and e. If m is the mean [#permalink]
14 Jan 2012, 21:40

2

This post was BOOKMARKED

00:00

A

B

C

D

E

Difficulty:

85% (hard)

Question Stats:

30% (01:31) correct
70% (00:56) wrong based on 27 sessions

Set X has 5 integers: a ,b ,c ,d , and e. If m is the mean and D, where D = sqrt{ [(a-m)^2+(b-m)^2+(c-m)^2+(d-m)^2+(e-m)^2]/5}, is the standard deviation of the set X, is D>2?

(1) a ,b ,c ,d , and e are different integers (2) m is an integer not equal to any elements of the set

Re: Standard Deviation and Mean [#permalink]
15 Jan 2012, 13:31

2

This post received KUDOS

Expert's post

Hi, there! I'm happy to help with this!

First, I'll say this is a very difficult question. If a question of this difficulty appears on the GMAT, it will be one of the harder questions, a question for which you probably will spend a tad more than the allotted 2 min/ question (if you are efficient on the easy questions, you will have a little more time for the more difficult questions like this).

Question: Set X has 5 integers: a ,b ,c ,d , and e. If m is the mean and D, where D = sqrt{ [(a-m)^2+(b-m)^2+(c-m)^2+(d-m)^2+(e-m)^2]/5}, is the standard deviation of the set X, is D>2?

Notice, this is the ordinary definition of standard deviation. If standard deviation is not a familiar topic, I would suggest reviewing it a bit in whatever review material you have.

Notice, also, the real crux of the question is: can we make the standard deviation small enough? There's no problem making a standard deviation way more than 2 -- for example, if the set is {100, 200, 300, 400, 1000}, then it satisfies both statements and has D >2. It's always easy to pick big numbers and numbers widely spaced out to increase the standard deviation. The trick is: can we make the numbers close enough together so that it's still possible that D is less than or equal to 2?

Statement #1: a ,b ,c ,d , and e are different integers This eliminates the most obvious choice for a small standard deviation, viz, make all five numbers equal. If all the numbers of a set are equal, the standard deviation is zero, but that's not a possibility here. Well, if they have to be all different, the closest they can still be is consecutive, for example: X = {1, 2, 3, 4, 5}. Then, m = 3, and D = sqrt(((1-3)^2 + (2-3)^2 + (3-3)^2 + (4-3)^2 + (5-3)^2)/5) = sqrt((4 + 1 + 0 + 1 + 4)/5) = sqrt(10/5) = sqrt(2), which is less than 2. So, given statement #1, it's possible make a choice that has a D less than 2, and it's always possible to make other choices and make D huge, so there's no way to decide whether or not D > 2. Statement #1, by itself, is insufficient.

Statement #2: m is an integer not equal to any elements of the set This is a tricky statement. It tells us (a) that the mean is an integer, not something true for a general set of five integers, and (b) this integer, the mean, is *not* equal to any of the five elements of the set. So, we can't have all five integers the same (since the mean would also be the same as them), and we can't have five consecutive integers. If we want a small standard deviation, we need the numbers still close together, so make some of them the same, with a kind of "hole" in the middle for the mean. For example, the set {1, 1, 1, 3, 3} does not have a integer mean, but the set {1, 1, 1, 3, 4} does have an integer mean: m = 2. (We were leaving the entry 2 vacated on the list, hoping the mean would land there.) So, this latter set, X = {1, 1, 1, 3, 4} satisfies statement #2, and has a standard deviation of D = sqrt(((1-2)^2 + (1-2)^2 + (1-2)^2 + (3-2)^2 + (4-2)^2)/5) =sqrt((1 + 1 + 1 + 1 + 4)/5) =sqrt(8/5), which is less than 2. Again, given statement #2, it's possible make a choice that has a D less than 2, and it's always possible to make other choices and make D huge, so there's no way to decide whether or not D > 2. Statement #2, by itself, is insufficient.

Combined Statements #! & #2: This combination is particularly tricky. Neither of the sets we concocted under the separate statements will work here. We need five different integer, and the mean is going to be an integer not equal to the members of the set. This means, if we want a small standard deviation, we need to construct a set of integers close to each other, but with a "hole" in the middle where the mean will fall. If I try {1, 2, 3, 5, 6}, that has a mean of 3.4, a sum of 17. To have a mean that's an integer, the sum must be a multiple of 5. We are shooing for a sum of 20, which would be a mean of 4, which is why we are intentionally excluding 4 from the list. We could just add 3 the highest number to get {1, 2, 3, 5, 9}, but having one particularly huge number increases the standard deviation more than having a few sorta big numbers. Outliers enormously inflate the standard deviation. Starting with {1, 2, 3, 5, 6}, I can't add anything to the lower numbers without either duplicating a number in the set, or getting 4, which I trying to exclude. I can add +1 to 5 and +2 to 6, to get {1, 2, 3, 6, 8}. I believe that is as close as you can make a set of five integers with the constraints that (a) all are different, (b) the mean is an integer, and (c) the integer-mean is not a member of the set. With this set, X = {1, 2, 3, 6, 8}, the mean is m = 4, and the standard deviation is D = sqrt(((1-4)^2 + (2-4)^2 + (3-4)^2 + (6-4)^2 + (8-4)^2)/5) =sqrt((9 + 4 + 1 + 4 + 16)/5 = sqrt(34/5) > sqrt(6) > 2. In other words, we have constructed the tightest, most compact group of five integers permitted under the combined conditions, and even with our best choice, we are unable to make the standard deviation less than 2. The two conditions, combined, absolutely require that the standard deviation is more than 2. The combined statements are sufficient.

Answer = C

Does that make sense? Please let me know if you have any questions on what I've said here.

Set X has 5 integers: a, b, c, d, and e. If m is the mean [#permalink]
22 Feb 2012, 21:22

Set \(X\) has 5 integers: \(a\) , \(b\) , \(c\) , \(d\) , and \(e\) . If \(m\) is the mean and \(D\) , where \(D = \sqrt{\frac{(a-m)^2+(b-m)^2+(c-m)^2+(d-m)^2+(e-m)^2}{5}}\) , is the standard deviation of the set \(X\) , is \(D \gt 2\) ?

\(a\) , \(b\) , \(c\) , \(d\) , and \(e\) are different integers \(m\) is an integer not equal to any elements of the set \(X\)

Strategy for reaching answer C. Can eliminate A B & D. Stuck between C & E. If testing numbers how to select appropriate numbers???????? how to be sure of the minimum range???

Set \(X\) has 5 integers: \(a\) , \(b\) , \(c\) , \(d\) , and \(e\) . If \(m\) is the mean and \(D\) , where \(D = \sqrt{\frac{(a-m)^2+(b-m)^2+(c-m)^2+(d-m)^2+(e-m)^2}{5}}\) , is the standard deviation of the set \(X\) , is \(D \gt 2\) ?

\(a\) , \(b\) , \(c\) , \(d\) , and \(e\) are different integers \(m\) is an integer not equal to any elements of the set \(X\)

Strategy for reaching answer C. Can eliminate A B & D. Stuck between C & E. If testing numbers how to select appropriate numbers???????? how to be sure of the minimum range???

Let me note though that I wouldn't worry about this question at all, since you won't see anything like this on the real test.

Generally for the GMAT you only need to understand the concept of SD: you won't be asked to actually calculate the standard deviation of a set on the GMAT. So, what is the main thing you should know about it? Standard deviation shows how much variation there is from the mean, how widespread a given set is. So, a low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values.

Re: Set X has 5 integers: a, b, c, d, and e. If m is the mean [#permalink]
30 Jan 2015, 16:41

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________

Re: Set X has 5 integers: a, b, c, d, and e. If m is the mean [#permalink]
30 Jan 2015, 18:29

Expert's post

Hi All,

Standard Deviation is a relatively rare subject on Test Day (you'll likely see it just once). While you will be tested on the 'concept' of Standard Deviation (how "spread out" a group of numbers is, relative to the average of the group), the GMAT will NEVER ask you to calculate it. As such, the Standard Deviation question that you might see on Test Day will not look like this one.

Re: Set X has 5 integers: a, b, c, d, and e. If m is the mean [#permalink]
31 Jan 2015, 19:54

Quote:

Hi, there! I'm happy to help with this!

First, I'll say this is a very difficult question. If a question of this difficulty appears on the GMAT, it will be one of the harder questions, a question for which you probably will spend a tad more than the allotted 2 min/ question (if you are efficient on the easy questions, you will have a little more time for the more difficult questions like this).

Question: Set X has 5 integers: a ,b ,c ,d , and e. If m is the mean and D, where D = sqrt{ [(a-m)^2+(b-m)^2+(c-m)^2+(d-m)^2+(e-m)^2]/5}, is the standard deviation of the set X, is D>2?

Notice, this is the ordinary definition of standard deviation. If standard deviation is not a familiar topic, I would suggest reviewing it a bit in whatever review material you have.

Notice, also, the real crux of the question is: can we make the standard deviation small enough? There's no problem making a standard deviation way more than 2 -- for example, if the set is {100, 200, 300, 400, 1000}, then it satisfies both statements and has D >2. It's always easy to pick big numbers and numbers widely spaced out to increase the standard deviation. The trick is: can we make the numbers close enough together so that it's still possible that D is less than or equal to 2?

Statement #1: a ,b ,c ,d , and e are different integers This eliminates the most obvious choice for a small standard deviation, viz, make all five numbers equal. If all the numbers of a set are equal, the standard deviation is zero, but that's not a possibility here. Well, if they have to be all different, the closest they can still be is consecutive, for example: X = {1, 2, 3, 4, 5}. Then, m = 3, and D = sqrt(((1-3)^2 + (2-3)^2 + (3-3)^2 + (4-3)^2 + (5-3)^2)/5) = sqrt((4 + 1 + 0 + 1 + 4)/5) = sqrt(10/5) = sqrt(2), which is less than 2. So, given statement #1, it's possible make a choice that has a D less than 2, and it's always possible to make other choices and make D huge, so there's no way to decide whether or not D > 2. Statement #1, by itself, is insufficient.

Statement #2: m is an integer not equal to any elements of the set This is a tricky statement. It tells us (a) that the mean is an integer, not something true for a general set of five integers, and (b) this integer, the mean, is *not* equal to any of the five elements of the set. So, we can't have all five integers the same (since the mean would also be the same as them), and we can't have five consecutive integers. If we want a small standard deviation, we need the numbers still close together, so make some of them the same, with a kind of "hole" in the middle for the mean. For example, the set {1, 1, 1, 3, 3} does not have a integer mean, but the set {1, 1, 1, 3, 4} does have an integer mean: m = 2. (We were leaving the entry 2 vacated on the list, hoping the mean would land there.) So, this latter set, X = {1, 1, 1, 3, 4} satisfies statement #2, and has a standard deviation of D = sqrt(((1-2)^2 + (1-2)^2 + (1-2)^2 + (3-2)^2 + (4-2)^2)/5) =sqrt((1 + 1 + 1 + 1 + 4)/5) =sqrt(8/5), which is less than 2. Again, given statement #2, it's possible make a choice that has a D less than 2, and it's always possible to make other choices and make D huge, so there's no way to decide whether or not D > 2. Statement #2, by itself, is insufficient.

Combined Statements #! & #2: This combination is particularly tricky. Neither of the sets we concocted under the separate statements will work here. We need five different integer, and the mean is going to be an integer not equal to the members of the set. This means, if we want a small standard deviation, we need to construct a set of integers close to each other, but with a "hole" in the middle where the mean will fall. If I try {1, 2, 3, 5, 6}, that has a mean of 3.4, a sum of 17. To have a mean that's an integer, the sum must be a multiple of 5. We are shooing for a sum of 20, which would be a mean of 4, which is why we are intentionally excluding 4 from the list. We could just add 3 the highest number to get {1, 2, 3, 5, 9}, but having one particularly huge number increases the standard deviation more than having a few sorta big numbers. Outliers enormously inflate the standard deviation. Starting with {1, 2, 3, 5, 6}, I can't add anything to the lower numbers without either duplicating a number in the set, or getting 4, which I trying to exclude. I can add +1 to 5 and +2 to 6, to get {1, 2, 3, 6, 8}. I believe that is as close as you can make a set of five integers with the constraints that (a) all are different, (b) the mean is an integer, and (c) the integer-mean is not a member of the set. With this set, X = {1, 2, 3, 6, 8}, the mean is m = 4, and the standard deviation is D = sqrt(((1-4)^2 + (2-4)^2 + (3-4)^2 + (6-4)^2 + (8-4)^2)/5) =sqrt((9 + 4 + 1 + 4 + 16)/5 = sqrt(34/5) > sqrt(6) > 2. In other words, we have constructed the tightest, most compact group of five integers permitted under the combined conditions, and even with our best choice, we are unable to make the standard deviation less than 2. The two conditions, combined, absolutely require that the standard deviation is more than 2. The combined statements are sufficient.

Answer = C

Does that make sense? Please let me know if you have any questions on what I've said here.

I do agree that this is very hard question, and thank you very much for your answer:D However, in my opinion, your method of choosing some sets of numbers to support your logic: this method should only be used carefully. For some questions, this method can lead to correct answer quickly. But for some questions, especially with gmat learner who do not have strong background in math, this method can be a nightmare for them: they choose the wrong excample numbers, and lead them to wrong answer.

To sum it up, I strongly appreciate your work, and I still hope we can try to find other ways to solve this question :D

gmatclubot

Re: Set X has 5 integers: a, b, c, d, and e. If m is the mean
[#permalink]
31 Jan 2015, 19:54

This week went in reviewing all the topics that I have covered in my previous study session. I reviewed all the notes that I have made and started reviewing the Quant...

I was checking my phone all day. I wasn’t sure when I would receive the admission decision from Tepper. I received an acceptance from Goizueta in the early morning...

I started running as a cross country team member since highshcool and what’s really awesome about running is that...you never get bored of it! I participated in...