Standard deviation

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Wed Sep 02, 2015 6:26 am

nikhilgmat31 wrote: Hi Mitch,

Can you please explain statement 1 by picking some numbers. I am not able to visualize any numbers to prove statement 1 is not sufficient.

All you need to know is this:
There is NO direct relationship between RANGE and SD.

A = 0,2,2,2,5
B = 0,0,2,4,4
In this case, the SD of A â‰ˆ 1.7, while the SD of B = 2, with the result that B has the greater SD.
Here, A has the smaller SD because three of the data points -- 2,2,2 -- are close to the mean of 2.2.

A = 0,0,2,5,5
B = 0,0,2,4,4
In this case, the SD of A â‰ˆ 2.5, while the SD of B = 2, with the result that A has the greater SD.
Here, A has the greater SD because four of the data points -- 0,0,5,5 -- are far from the mean of 2.4.

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Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Fri Sep 04, 2015 3:11 am

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and equations ensures a solution.

If sets A & B have the same number of terms, is the Standard deviation of set A greater than the Standard deviation of set B ?

1. the range of set A is greater than the range of B
2. sets A & B are both evenly spaced sets

If there is the relation between the range and standard deviation in the original condition, the answer is usually E. This is because we have too many variables to identify, and while we need to match the number of variables and equations, we only have 1 equation each in 1) and 2).

Using both 1) & 2) together, range=Max-min and the two sets A and B have evenly distanced elements. With bigger range, if the elements are evenly distanced the standard deviation is greater, thus the answer is yes. Therefore it is sufficient, and the answer is A.
For example, A={2,5,8}, B={1,2,3}. All elements in A are distanced by 3 while elements in b are distanced by 1, and the standard deviation is greater for B compared to A.

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nikhilgmat31: Legendary Member; Posts: 518; Joined: Tue May 12, 2015 8:25 pm; Thanked: 10 times

by nikhilgmat31 » Mon Sep 07, 2015 12:04 am

GMATGuruNY wrote:
nikhilgmat31 wrote: Hi Mitch,

Can you please explain statement 1 by picking some numbers. I am not able to visualize any numbers to prove statement 1 is not sufficient.
All you need to know is this:
There is NO direct relationship between RANGE and SD.

A = 0,2,2,2,5
B = 0,0,2,4,4
In this case, the SD of A â‰ˆ 1.7, while the SD of B = 2, with the result that B has the greater SD.
Here, A has the smaller SD because three of the data points -- 2,2,2 -- are close to the mean of 2.2.

A = 0,0,2,5,5
B = 0,0,2,4,4
In this case, the SD of A â‰ˆ 2.5, while the SD of B = 2, with the result that A has the greater SD.
Here, A has the greater SD because four of the data points -- 0,0,5,5 -- are far from the mean of 2.4.

Hi Mitch,

How you calculate SD so quickly and reach at SD value
A = 0,2,2,2,5
B = 0,0,2,4,4
In this case, the SD of A â‰ˆ 1.7, while the SD of B = 2

Please check if below one is correct.
1. calculate mean
2. Find squares of difference of each number from mean
3. Find sum of all the squares
4. Divide the sum by numbers of items in set. - It will give Variance
5. Square root of Variance is SD.

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gmatdriller: Master | Next Rank: 500 Posts; Posts: 418; Joined: Sun Jul 04, 2010 12:48 pmFollowed by: 3 members; Thanked: 6 times

by gmatdriller » Wed Sep 09, 2015 12:32 am

SD on the gmat would not require such an intensive calculation.

You can visualize the SD of a set of data points.
If A = 2,2,2 and B = 1,1,3
Mean of A = 2; Mean of B = 1.67

You would agree that the elements in A are closely packed towards the mean(2)
than are the elements in set B towards 1.67... So, SD for A < SD or B

Again if A = 1,2,3 and B = 2, 9, 10
On inspection, you can easily tell that SD for A < SD for B
Mean for A = 2; Mean for B = 7
But the elements in A are more closely parked towards 2 than are the elements in B
towards 7.

Am not an expert, but this is the way I approach this sort of question.
I stand to be corrected, pls.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Wed Sep 09, 2015 6:29 am

nikhilgmat31 wrote:
GMATGuruNY wrote:
nikhilgmat31 wrote: Hi Mitch,

Can you please explain statement 1 by picking some numbers. I am not able to visualize any numbers to prove statement 1 is not sufficient.
All you need to know is this:
There is NO direct relationship between RANGE and SD.

A = 0,2,2,2,5
B = 0,0,2,4,4
In this case, the SD of A â‰ˆ 1.7, while the SD of B = 2, with the result that B has the greater SD.
Here, A has the smaller SD because three of the data points -- 2,2,2 -- are close to the mean of 2.2.

A = 0,0,2,5,5
B = 0,0,2,4,4
In this case, the SD of A â‰ˆ 2.5, while the SD of B = 2, with the result that A has the greater SD.
Here, A has the greater SD because four of the data points -- 0,0,5,5 -- are far from the mean of 2.4.

Hi Mitch,

How you calculate SD so quickly and reach at SD value
A = 0,2,2,2,5
B = 0,0,2,4,4
In this case, the SD of A â‰ˆ 1.7, while the SD of B = 2

Please check if below one is correct.
1. calculate mean
2. Find squares of difference of each number from mean
3. Find sum of all the squares
4. Divide the sum by numbers of items in set. - It will give Variance
5. Square root of Variance is SD.

You do NOT need to know how to calculate SD.
Generally, problems about SD can be solved conceptually.

Some key concepts:
1. A small SD means that the data points are clustered close to the mean.
2. A large SD means that the data points are spread far from the mean.
3. If SD=0, then all of the data points are equal.
4. Generally, there is no direct relationship between the MEAN of a set and its SD; a set with a greater mean can have a smaller SD, and vice versa.
5. Generally, there is no direct relationship between the RANGE of a set and its SD; a set with a greater range can have a smaller SD, and vice versa.
6. If every data point increases or decreases by a constant k, then the SD does not change.
7. If every data point increases or decreases by the same PERCENTAGE, then the SD also increases or decreases by that percentage.

A more extreme case to illustrate Concept 5 and prove that Statement 1 is insufficient:
A = {0, 50, 50, 50, 50, 50, 100}
B = {1, 1, 1, 50, 99, 99, 99}
Here:
Both sets have a mean of 50.
Range of A > Range of B.
SD of A < SD of B.
Set A clearly has the SMALLER SD because most of its data points are EQUAL to the mean of 50, while most of the data points in B are FAR from the mean of 50.

A = {0, 0, 0, 50, 100, 100, 100}
B = {1, 50, 50, 50, 50, 50, 99}
Here:
Both sets have a mean of 50.
Range of A > Range of B.
SD of A > SD of B.
Set A clearly has the GREATER SD because most of its data points are FAR from the mean of 50, while most of the data points in B are EQUAL to the mean of 50.

As the case above illustrates, we can prove that statement 1 is insufficient WITHOUT calculating the SD for each set of values.

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nikhilgmat31: Legendary Member; Posts: 518; Joined: Tue May 12, 2015 8:25 pm; Thanked: 10 times

by nikhilgmat31 » Wed Sep 09, 2015 11:17 pm

Thanks Mitch, You are real Hero of GMAT.

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skgmat: Junior | Next Rank: 30 Posts; Posts: 10; Joined: Mon Aug 10, 2015 11:26 am

by skgmat » Sat Sep 12, 2015 9:13 pm

GMATGuruNY wrote:
Ramit88 wrote:If sets A & B have the same number of terms, is the Standard deviation of set A greater than the Standard deviation of set B ?

1. the range of set A is greater than the range of B
2. sets A & B are both evenly spaced sets

[spoiler]ANS:C[/spoiler]

but i read somewhere if sets have same no. of digits then the set with higher range have greater Standard deviation .. can someone please explain this to me
Standard deviation describes how much the values in a set deviate from the mean. A larger standard deviation indicates that the values are deviating more -- getting farther away from -- the mean. So the question can be rephrased:

Do the values in set A deviate more from the mean than the do values in set B?

Let SD = standard deviation.

Statement 1:
We know that the distance between the biggest and smallest values in A is greater than the distance between the biggest and smallest values in B.
But we don't know the mean, and to determine which set has a greater SD, we need to know how all the numbers in each set -- not just the biggest and smallest -- are deviating from the mean.
Insufficient.

Statement 2:
When values are evenly spaced, the mean = the median, and all the values are symmetrical about the median.
For example, if m = median, and all the values are consecutive even or odd integers, the set will look like this:

...m-6, m-4, m-2, m, m+2, m+4, m+6...

But to determine which set has a greater SD, we need to know in each set the distance between each successive pair of values. For example:

If A = consecutive even integers = {2,4,6} and B = consecutive multiples of 3 = {3,6,9}, then the values in B deviate more from the mean and B has the larger SD.
If A = consecutive multiples of 3 = {3,6,9} and B = consecutive even integers = {2,4,6}, then the values in A deviate more from the mean and A has the larger SD.
Insufficient.

Statements 1 and 2:
A and B have the same number of values.
A and B are both evenly spaced sets, so the values in each set are symmetrical about the mean.
The range in A is greater.
For A to have a greater range, the distance between each successive pair in A must be greater than the distance between each successive pair in B. In other words, the values in A are more spread out.
Thus, the values in A are deviating more from the mean, and A has a larger SD.
Sufficient.

The correct answer is C.

Great explanation! Easy to follow and understand.
Just happened across this really old post while searching for problems testing std. deviation

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tickelmepinc: Newbie | Next Rank: 10 Posts; Posts: 1; Joined: Tue Oct 18, 2016 9:58 am

by tickelmepinc » Tue Oct 18, 2016 10:01 am

@GMATGuruNY

I understand your explanation for why the SD of A must be bigger.

However, have you considered the case in which both sets have the same spacing between each integer?

Example:
A {8,12,16,20,24} R = 16
B {1,4,8,12,16} R = 15

R of A is > R of B
Both are evenly spaced sets, satisfying both conditions.

Is SD of A still bigger than SD of B?

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Tue Oct 18, 2016 11:36 am

tickelmepinc wrote:@GMATGuruNY

I understand your explanation for why the SD of A must be bigger.

However, have you considered the case in which both sets have the same spacing between each integer?

Example:
A {8,12,16,20,24} R = 16
B {1,4,8,12,16} R = 15

R of A is > R of B
Both are evenly spaced sets, satisfying both conditions.

Is SD of A still bigger than SD of B?

B is not evenly spaced:
The difference between 1 and 4 = 4-1 = 3, while the difference between 4 and 8 = 8-4 = 4.

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Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members; Location: Grand Central / New York; Thanked: 470 times

by Jay@ManhattanReview » Thu Feb 02, 2017 10:41 pm

Can you please explain statement 1 by picking some numbers. I am not able to visualize any numbers to prove statement 1 is not sufficient.

Hi Nikhil,

We have statement 1: The range of set A is greater than the range of set B.

Say, A: {10, 15, 15, 15, 15, 20}; Range = 10; Mean = 15

and B: {10, 10, 10, 18, 18, 18}; Range = 8; Mean = 14

We see that in set A, there 4 terms that have no deviations w.r.t their mean (15), while in set B all the 6 terms have deviations w.r.t their mean (14). Thus, it is clear that SD for set B would be more than that for set A.

SD is a measure of the spread of terms w.r.t their mean. Closer the terms, less is the deviation.

Hope this is clear.

See another question on SD: https://www.beatthegmat.com/standard-dev ... tml#788974

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