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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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A
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Which of the following distribution of numbers has the greatest standa 18 May 2016, 22:27
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Bunuel
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}
Show more


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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Which of the following distribution of numbers has the greatest standa 18 May 2016, 12:12
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Bunuel
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}
Show more

Hi Bunuel,

Both A and D can be the answer for the above question.
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Which of the following distribution of numbers has the greatest standa 18 May 2016, 12:25
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Vyshak
Bunuel
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.
Show more

Those two sets have different standard deviations.
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Which of the following distribution of numbers has the greatest standa 18 May 2016, 13:09
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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 18 May 2016, 21:20
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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.

Answer - A.

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

Posted from my mobile device
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Which of the following distribution of numbers has the greatest standa 19 May 2016, 20:38
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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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how did they calculate the sd for each option ?
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Which of the following distribution of numbers has the greatest standa 17 Sep 2018, 08:47
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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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Hi Moderators chetan2u, Gladiator59, generis,

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 20 Sep 2019, 11:42
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DarkHorse2019
Hi Moderators chetan2u, Gladiator59, generis,

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?
Show more


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

Your solution obviously fails here.

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.
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Which of the following distribution of numbers has the greatest standa 16 May 2020, 00:21
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VeritasKarishma
Bunuel
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.
Show more


Hello "VeritasKarishma"
Can you please explain why the mean=0 for option A, highlighted above. I have read that if the no. set is odd then the middle most no. is mean. correct me if i am wrong.

TIA
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VeritasKarishma
Bunuel
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.
Show more



Will it work if I just list differences between the number then divide by total number ?
e.g
(A) {-3, 1, 2} =4,1=2.5
(B) {-2, -1, 1, 2} =1,2,1=1.33
(C) {3, 5, 7} =2,2,2=2
(D) {-1, 2, 3, 4} =3,1,1=1.66
(E) {0, 2, 4} =2,2=2

we see option A's average difference is greater.
So can we solve the problem this way ?
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Which of the following distribution of numbers has the greatest standa 16 May 2020, 04:13
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jackfr2

Will it work if I just list differences between the number then divide by total number ?
e.g
(A) {-3, 1, 2} =4,1=2.5
(B) {-2, -1, 1, 2} =1,2,1=1.33
(C) {3, 5, 7} =2,2,2=2
(D) {-1, 2, 3, 4} =3,1,1=1.66
(E) {0, 2, 4} =2,2=2

we see option A's average difference is greater.
So can we solve the problem this way ?
Show more

Hi jackfr2

There is more error involved in your way so it's hard to say YES because it won't result in corrcet answer all the time. However, fundamentally this calculation which you have shown is also symbolic calculation of standard deviation

Standard Deviation is the Average Deviation of the terms from MEAN Value so it's recommended that you take the deviation of values from mean
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Greater the Range, greater the SD.
Range of : A= 5
B= 4
C=4
D= 5
E= 4

Now we have to chose between A & D. It is very clear that numbers in A are more sparsely distributed than in D.
Hence, Answer is A
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riagarg07
Greater the Range, greater the SD.
Range of : A= 5
B= 4
C=4
D= 5
E= 4

Now we have to chose between A & D. It is very clear that numbers in A are more sparsely distributed than in D.
Hence, Answer is A
Show more
Is this really true?
SD has to be less than range. So both can be less than 5 or 4. But that doesn’t mean one has to be greater than the other, right?
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riagarg07
Greater the Range, greater the SD.
Range of : A= 5
B= 4
C=4
D= 5
E= 4

Now we have to chose between A & D. It is very clear that numbers in A are more sparsely distributed than in D.
Hence, Answer is A
Is this really true?
SD has to be less than range. So both can be less than 5 or 4. But that doesn’t mean one has to be greater than the other, right?
Show more
Checked on this further.

Apparently, if it’s a normal distribution sd = range/4 approximately
They claim it’s because 95% of bell curve is covered in first 2 sd on both sides.

But, I see 99% covered in first 3 sd on either side of bell curve on most sites. So I do wonder why it’s not range/6?
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