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Math Expert V
Joined: 02 Sep 2009
Posts: 62637
Which of the following distribution of numbers has the greatest standa  [#permalink]

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16 00:00

Difficulty:   45% (medium)

Question Stats: 65% (01:24) correct 35% (01:28) wrong based on 384 sessions

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Which of the following distribution of numbers has the greatest standard deviation?

(A) {-3, 1, 2}
(B) {-2, -1, 1, 2}
(C) {3, 5, 7}
(D) {-1, 2, 3, 4}
(E) {0, 2, 4}

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Re: Which of the following distribution of numbers has the greatest standa  [#permalink]

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Bunuel wrote:
Which of the following distribution of numbers has the greatest standard deviation?

(A) {-3, 1, 2}
(B) {-2, -1, 1, 2}
(C) {3, 5, 7}
(D) {-1, 2, 3, 4}
(E) {0, 2, 4}

For answer choice A, the mean = 0 and the deviations are 3, 1, 2
For answer choice B, the mean = 0 and the deviations are 2, 1, 1, 2
For answer choice C, the mean = 5 and the deviations are 2, 0, 2
For answer choice D, the mean = 2 and the deviations are 3, 0, 1, 2
For answer choice E, the mean = 2 and the deviations are 2, 0, 2

We don’t need to worry about the arithmetic means (they just help us calculate the deviation of each element from the mean); our focus should be on the deviations. The SD formula squares the individual deviations and then adds them, then the sum is divided by the number of elements and finally, we find the square root of the whole term. So if a deviation is greater, its square will be even greater and that will increase the SD.

If the deviation increases and the number of elements increases, too, then we cannot be sure what the final effect will be – an increased deviation increases the SD but an increase in the number of elements increases the denominator and hence, actually decreases the SD. The overall effect as to whether the SD increases or decreases will vary from case to case.

First, we should note that answers C and E have identical deviations and numbers of elements, hence, their SDs will be identical. This means the answer is certainly not C or E, since Problem Solving questions have a single correct answer.

Let’s move on to the other three options:

For answer choice A, the mean = 0 and the deviations are 3, 1, 2
For answer choice B, the mean = 0 and the deviations are 2, 1, 1, 2
For answer choice D, the mean = 2 and the deviations are 3, 0, 1, 2

Comparing answer choices A and D, we see that they both have the same deviations, but D has more elements. This means its denominator will be greater, and therefore, the SD of answer D is smaller than the SD of answer A. This leaves us with options A and B:

For answer choice A, the mean = 0 and the deviations are 3, 1, 2
For answer choice B, the mean = 0 and the deviations are 2, 1, 1, 2

Now notice that although two deviations of answers A and B are the same, answer choice A has a higher deviation of 3 but fewer elements than answer choice B. This means the SD of A will be higher than the SD of B, so the SD of A will be the highest. Hence, our answer must be A.
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Re: Which of the following distribution of numbers has the greatest standa  [#permalink]

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Bunuel wrote:
Which of the following distribution of numbers has the greatest standard deviation?

(A) {-3, 1, 2}
(B) {-2, -1, 1, 2}
(C) {3, 5, 7}
(D) {-1, 2, 3, 4}
(E) {0, 2, 4}

Hi Bunuel,

Both A and D can be the answer for the above question.
Math Expert V
Joined: 02 Sep 2009
Posts: 62637
Which of the following distribution of numbers has the greatest standa  [#permalink]

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Vyshak wrote:
Bunuel wrote:
Which of the following distribution of numbers has the greatest standard deviation?

(A) {-3, 1, 2}
(B) {-2, -1, 1, 2}
(C) {3, 5, 7}
(D) {-1, 2, 3, 4}
(E) {0, 2, 4}

Hi Bunuel,

Both A and D can be the answer for the above question.

Those two sets have different standard deviations.
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Re: Which of the following distribution of numbers has the greatest standa  [#permalink]

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1
A => sd =\sqrt{14/3}
B => sd =\sqrt{10/4}
C => sd =\sqrt{8/3}
D => sd =\sqrt{14/4}
E => sd =\sqrt{11/3}

So the biggest is A.
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Which of the following distribution of numbers has the greatest standa  [#permalink]

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Let me try this one:

First, we find the mean for every set. After you find the mean, look for distance between mean and outliers (numbers on boundary of distribution - MAX |m-n|, where m is mean and n is any number in set). You can ignore choices B, C, and E based on that understanding. Remaining two choices have same distance = 3.

Next, we look at density of distribution in remaining two options. More numbers between mean and outlier, more denser the set become. More denser the set become, less deviation it provides. Based on that understanding, D should have lesser standard deviation between remaining two.

Comments welcome. Its been ages since I last tried SD.

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Re: Which of the following distribution of numbers has the greatest standa  [#permalink]

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IMHO 'A' should be the answer.

Lets look at average and Std deviation respectively in every choice

A) 0 / 5
B) 0/ 4
C) 3/4
D) 2/5
E) 2/4

Clearly 'A' has the greatest SD _________________
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Re: Which of the following distribution of numbers has the greatest standa  [#permalink]

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how did they calculate the sd for each option ?
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GMAT 2: 680 Q49 V33 Re: Which of the following distribution of numbers has the greatest standa  [#permalink]

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SD is a measure to represent how much the datapoints of a dataset are dispersed from the mean position of that set. Mathematically SD is the sq root of the avg of the squares of the differences of each datapoint from the mean of that particular dataset..........hmmm.... that's complicated wording !!

Essentially , Here were are looking for a data set with larger range and lesser datapoints.
largest Range : A & D
now D has 4 observations & A has 3 . so n/4<n/3
Hence , Set A will have largest SD.
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Which of the following distribution of numbers has the greatest standa  [#permalink]

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Any thumb rule method/ shortcut to do these standard deviation questions within 90 seconds?

I found the method by VeritasKarishma useful but again I don't know if it can be done within 90 seconds on a test day, & ofcourse I agree calculating mean and standard deviation for each of the options is going to be more time consuming...

I approached the problem in the following manner:

SD shows how much variation there is from the mean in a given set, in the problem here we need to find the set with the highest SD - meaning the set were the data points are the most widespread is the answer.

I have just calculated gaps or differences between consecutive elements of each set.

A) 4,1
B) 1,2,1
C) 2,2
D) 3,1,1
E) 2,2

As we see, choice A has the highest deviation between 2 consecutive elements, this is the answer. Moderators, Is this method okay?
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Which of the following distribution of numbers has the greatest standa  [#permalink]

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DarkHorse2019 wrote:

Any thumb rule method/ shortcut to do these standard deviation questions within 90 seconds?

I found the method by VeritasKarishma useful but again I don't know if it can be done within 90 seconds on a test day, & ofcourse I agree calculating mean and standard deviation for each of the options is going to be more time consuming...

I approached the problem in the following manner:

SD shows how much variation there is from the mean in a given set, in the problem here we need to find the set with the highest SD - meaning the set were the data points are the most widespread is the answer.

I have just calculated gaps or differences between consecutive elements of each set.

A) 4,1
B) 1,2,1
C) 2,2
D) 3,1,1
E) 2,2

As we see, choice A has the highest deviation between 2 consecutive elements, this is the answer. Moderators, Is this method okay?

Let's say set A = { 2, 5, 5, 5 }
Set B = { 0, 2, 4, 6 }

When you look as distance between two points, you forget to realize that that distance could be constant and could keep increasing indefinitely.
{ 0, 2, 4, 6, 8, 10 ... inf }

What you need to and definitely need to do is find the span of the set, individual distance don't mean anything when looking for the span.

Now, if you think you could just add the distances, you'd be wrong there too, just try the question with that strategy. Which of the following distribution of numbers has the greatest standa   [#permalink] 20 Sep 2019, 10:42
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