## Relation between Range and Standard Deviation

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goelmohit2002
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### Relation between Range and Standard Deviation

Hi All,

Can someone please tell is there any relation that exists between Range and standard deviation ? If yes, then please tell what is that ? If possible kindly give one example too.

Thanks
Mohit
Ben Ku
ManhattanGMAT Staff

Posts: 817
Joined: Sat Nov 03, 2007 7:49 pm

### Re: Relation between Range and Standard Deviation

Mohit,

Range is the the difference between the largest and smallest values in a set of data. The Standard Deviation is a measure of how far the data points are spread out. One SD above and below the average represents about 68% of the data points (in a normal distribution).

There is not a direct relationship between range and standard deviation. But because both are measures of spread, the range can help (depending on the data) to draw conclusions about the SD.

An example is:
45, 45, 45, 45, 45
In this case, the Range is 0. Since they're all the same values, there is no deviation, so the SD is also 0.

If the data set were: 43, 44, 45, 46, 47, the range here is 4. You can see the data is more spread out than the first example, so the SD will be larger than 0 (in this case, it's about 1.6)

If the data set were 41, 43, 45, 47, 49, here the range is 8. This set of data is even more spread out; the SD is 3.2.

However, range is not tied to the SD. If the data set were: 41, 45, 45, 45, 49, the range is still 8, but the distribution is less spread out, so the SD would be less than 3.2; it's in fact 2.8.

Hope that helps
Ben Ku
Instructor
ManhattanGMAT
goelmohit2002
Students

Posts: 226
Joined: Sat Jul 04, 2009 8:40 am

### Re: Relation between Range and Standard Deviation

Ben Ku wrote:Mohit,

Range is the the difference between the largest and smallest values in a set of data. The Standard Deviation is a measure of how far the data points are spread out. One SD above and below the average represents about 68% of the data points (in a normal distribution).

There is not a direct relationship between range and standard deviation. But because both are measures of spread, the range can help (depending on the data) to draw conclusions about the SD.

An example is:
45, 45, 45, 45, 45
In this case, the Range is 0. Since they're all the same values, there is no deviation, so the SD is also 0.

If the data set were: 43, 44, 45, 46, 47, the range here is 4. You can see the data is more spread out than the first example, so the SD will be larger than 0 (in this case, it's about 1.6)

If the data set were 41, 43, 45, 47, 49, here the range is 8. This set of data is even more spread out; the SD is 3.2.

However, range is not tied to the SD. If the data set were: 41, 45, 45, 45, 49, the range is still 8, but the distribution is less spread out, so the SD would be less than 3.2; it's in fact 2.8.

Hope that helps

Hi Ken,

somewhere I read that SD <= Range/2

Please tell is this indeed the case or not ? If yes, then is it relevant for GMAT ?
RonPurewal
Students

Posts: 19747
Joined: Tue Aug 14, 2007 8:23 am

### Re: Relation between Range and Standard Deviation

goelmohit2002 wrote:Hi Ken,

somewhere I read that SD <= Range/2

Please tell is this indeed the case or not ? If yes, then is it relevant for GMAT ?

hi, mohit.

actually, unless the test authors do a complete about-face on this issue, you'll NEVER need to worry about the NUMERICAL VALUE of the standard deviation, AT ALL.

there have been several problems (4-6, i can't remember the exact number) that have given a numerical value to the standard deviation.
in ALL of these problems, the concept of standard deviation was completely irrelevant. you could take "standard deviation" and replace it with "pink flamingo", and the problem wouldn't change.
see here for an example:
post17297.html

--

the only thing you'll have to know about the numerical value of the standard deviation is how to REDUCE or INCREASE it.
specifically, you can reduce it, guaranteed, by adding numbers that are closer to the mean than are any of the existing numbers.
you can increase it, guaranteed, by adding numbers that are farther from the mean than are any of the existing numbers.
goelmohit2002
Students

Posts: 226
Joined: Sat Jul 04, 2009 8:40 am

### Re: Relation between Range and Standard Deviation

RonPurewal wrote:
goelmohit2002 wrote:Hi Ken,

somewhere I read that SD <= Range/2

Please tell is this indeed the case or not ? If yes, then is it relevant for GMAT ?

hi, mohit.

actually, unless the test authors do a complete about-face on this issue, you'll NEVER need to worry about the NUMERICAL VALUE of the standard deviation, AT ALL.

there have been several problems (4-6, i can't remember the exact number) that have given a numerical value to the standard deviation.
in ALL of these problems, the concept of standard deviation was completely irrelevant. you could take "standard deviation" and replace it with "pink flamingo", and the problem wouldn't change.
see here for an example:
post17297.html

--

the only thing you'll have to know about the numerical value of the standard deviation is how to REDUCE or INCREASE it.
specifically, you can reduce it, guaranteed, by adding numbers that are closer to the mean than are any of the existing numbers.
you can increase it, guaranteed, by adding numbers that are farther from the mean than are any of the existing numbers.

Thanks Ron !!...

Actually there was one DS problem that I came across that was testing this concept of SD <= Range/2

Actually that question was not asking about the exact value....but was testing the knowledge of above formula.

Not sure whether the same is indeed tested concept in GMAT or not.....

Can you please tell what indeed is the case with GMAT ?
goelmohit2002
Students

Posts: 226
Joined: Sat Jul 04, 2009 8:40 am

### Re: Relation between Range and Standard Deviation

Hi Ron,

The question that I came across is posted here:

http://www.beatthegmat.com/range-of-numbers-t43516.html

was really confused....this relation between SD and Range I never read and is not even there in Manhattan Strategy guide.
RonPurewal
Students

Posts: 19747
Joined: Tue Aug 14, 2007 8:23 am

### Re: Relation between Range and Standard Deviation

goelmohit2002 wrote:
RonPurewal wrote:
goelmohit2002 wrote:Hi Ken,

somewhere I read that SD <= Range/2

Please tell is this indeed the case or not ? If yes, then is it relevant for GMAT ?

hi, mohit.

actually, unless the test authors do a complete about-face on this issue, you'll NEVER need to worry about the NUMERICAL VALUE of the standard deviation, AT ALL.

there have been several problems (4-6, i can't remember the exact number) that have given a numerical value to the standard deviation.
in ALL of these problems, the concept of standard deviation was completely irrelevant. you could take "standard deviation" and replace it with "pink flamingo", and the problem wouldn't change.
see here for an example:
post17297.html

--

the only thing you'll have to know about the numerical value of the standard deviation is how to REDUCE or INCREASE it.
specifically, you can reduce it, guaranteed, by adding numbers that are closer to the mean than are any of the existing numbers.
you can increase it, guaranteed, by adding numbers that are farther from the mean than are any of the existing numbers.

Thanks Ron !!...

Actually there was one DS problem that I came across that was testing this concept of SD <= Range/2

Actually that question was not asking about the exact value....but was testing the knowledge of above formula.

Not sure whether the same is indeed tested concept in GMAT or not.....

Can you please tell what indeed is the case with GMAT ?

i guarantee you that the gmat will not test anything remotely resembling this sd vs. range/2 thing.

that's the problem that arises when forums don't require you to post a source (and don't segregate into different folders by source): you can't tell genuine/official problems from random problems that are worthless, or even counterproductive, in preparation for the real test.
goelmohit2002
Students

Posts: 226
Joined: Sat Jul 04, 2009 8:40 am

### Re: Relation between Range and Standard Deviation

RonPurewal wrote:i guarantee you that the gmat will not test anything remotely resembling this sd vs. range/2 thing.

that's the problem that arises when forums don't require you to post a source (and don't segregate into different folders by source): you can't tell genuine/official problems from random problems that are worthless, or even counterproductive, in preparation for the real test.

Thanks a lot Ron !!!!

Really helps !!!
Ben Ku
ManhattanGMAT Staff

Posts: 817
Joined: Sat Nov 03, 2007 7:49 pm

### Re: Relation between Range and Standard Deviation

I agree with Ron that you will never be asked to find the numerical value of the SD (I did so in my original response only to illustrate the meaning of the SD). You should know the following:

(1) You will never be asked to find the numerical value of the standard deviation
(2) You should know the meaning of SD: the more spread out a data is, the larger the SD. If all data have the same value, the SD is 0.
(3) You should know that that SD is related to the variance (the SD is the square root of the variance)
(4) Many questions that involve the number of SD above/below the mean, you can simply substitute "pink flamingo."

The formula of the SD=range/2 is completely false. Don't ever use it.
Ben Ku
Instructor
ManhattanGMAT