DanaJ wrote:The formula for range is R = xn - x1 (greatest value of set - smallest value of set).
The formula for standard deviation is... pretty long and hard to explain. All you should remember here is that the standard deviation is the average distance from the mean of the set to the set's components.
So you know the following:
- R = distance from greatest to smallest (basically longest distance in set)
- SD = average distance from mean (which is somewhere in the middle) to an element of the set
As you can probably guess from the above, R will always be greater than SD. I'm not sure if indeed R/2 > SD and I'm not even going to go as far as to pretend I want to try to demonstrate it, seen as how the formula for SD is impossibly long and convoluted (general case of n elements in set).
However, if you remember that R > SD, it will be more than enough to solve the problem at hand.
Edit: After some philosophical 3 a.m. thinking, the R/2 > SD formula may not be that hard to demonstrate. Will try maybe tomorrow.
Range of numbers
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mehravikas
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Thanks that helps...
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mehravikas
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With that we should remember that SD is directly proportional to the range. If the range of a set changes, SD will change accordingly.
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goelmohit2002
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Thanks DanajDanaJ wrote: Edit: After some philosophical 3 a.m. thinking, the R/2 > SD formula may not be that hard to demonstrate. Will try maybe tomorrow.
Will be really helpful if you can help in the same....
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goelmohit2002
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DanaJ
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It's R > SD.
I've tried to demonstrate that R/2 >= SD, but with no luck, I'm afraid, since the SD formula just gives me headaches. I have noticed that, for a set of two elements (i.e. the most basic set), you have R/2 = SD. The next step is to assume that R/2 >= SD for a set of n - 1 elements and then use this assumption to prove that a set of n elements also has this property. I couldn't do that.
However, since you do have R/2 = SD for a set of two, it's not impossible for the formula of R/2 >= SD to be true. I've asked my econometrics teacher about it on his blog and am now awaiting results
I've tried to demonstrate that R/2 >= SD, but with no luck, I'm afraid, since the SD formula just gives me headaches. I have noticed that, for a set of two elements (i.e. the most basic set), you have R/2 = SD. The next step is to assume that R/2 >= SD for a set of n - 1 elements and then use this assumption to prove that a set of n elements also has this property. I couldn't do that.
However, since you do have R/2 = SD for a set of two, it's not impossible for the formula of R/2 >= SD to be true. I've asked my econometrics teacher about it on his blog and am now awaiting results
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goelmohit2002
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Thanks. Please let us know too what your teacher replies back.DanaJ wrote:It's R > SD.
I've tried to demonstrate that R/2 >= SD, but with no luck, I'm afraid, since the SD formula just gives me headaches. I have noticed that, for a set of two elements (i.e. the most basic set), you have R/2 = SD. The next step is to assume that R/2 >= SD for a set of n - 1 elements and then use this assumption to prove that a set of n elements also has this property. I couldn't do that.
However, since you do have R/2 = SD for a set of two, it's not impossible for the formula of R/2 >= SD to be true. I've asked my econometrics teacher about it on his blog and am now awaiting results
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goelmohit2002
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Hi Danaj,
From your post.....looks like the following one is correct relation....Please confirm if I am correct or not...
R/2 >= SD.
Thanks
Mohit
From your post.....looks like the following one is correct relation....Please confirm if I am correct or not...
R/2 >= SD.
Thanks
Mohit
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goelmohit2002
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Please do tell what your teacher tells.....as I guess this is a important stuff which majority of the people are not clear about.DanaJ wrote:I know for sure that R > SD, but I'm still waiting to see if my teacher says that R/2 >= SD. [/b]
no need to get into SD formula - 1SD on either side of mean gives a range of 68% of the items in the set.
Assume we have 2 numbers
0 and 100 then the mean is 50 and the SD is <50 call is x
if we add another 100 to the set (0, 100, 100)
mean is 66 and SD is <x call it y
if we add another 100 to the set (0, 100, 100, 100)
mean is 75 and SD is <y call it z
if we keep adding 100 indeffinitely
the mean will approach 100 and the SD will approach 0
though we have not changed the range in this example
so I think dana is right with her assumption, SD < Range, don't think it can ever be <=
I would even go so far as to say SD appraoches 0.5*range - I can't think of an example where this case would not hold
Assume we have 2 numbers
0 and 100 then the mean is 50 and the SD is <50 call is x
if we add another 100 to the set (0, 100, 100)
mean is 66 and SD is <x call it y
if we add another 100 to the set (0, 100, 100, 100)
mean is 75 and SD is <y call it z
if we keep adding 100 indeffinitely
the mean will approach 100 and the SD will approach 0
though we have not changed the range in this example
so I think dana is right with her assumption, SD < Range, don't think it can ever be <=
I would even go so far as to say SD appraoches 0.5*range - I can't think of an example where this case would not hold
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DanaJ
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I tend to agree with your point. However, the "68%" only holds true for a normal distribution (the bell curve). Not all sets follow that pattern; actually, most small sets do not. A normal distribution is used particularly for large populations (i.e. the height of all the people in the world).m&m wrote:no need to get into SD formula - 1SD on either side of mean gives a range of 68% of the items in the set.
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goelmohit2002
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Yes, this is a very good example....yes it is sure from above example that SD < R/2....Now the thing boils down to finally which one is correct relation to rememberm&m wrote:no need to get into SD formula - 1SD on either side of mean gives a range of 68% of the items in the set.
Assume we have 2 numbers
0 and 100 then the mean is 50 and the SD is <50 call is x
if we add another 100 to the set (0, 100, 100)
mean is 66 and SD is <x call it y
if we add another 100 to the set (0, 100, 100, 100)
mean is 75 and SD is <y call it z
if we keep adding 100 indeffinitely
the mean will approach 100 and the SD will approach 0
though we have not changed the range in this example
so I think dana is right with her assumption, SD < Range, don't think it can ever be <=
I would even go so far as to say SD appraoches 0.5*range - I can't think of an example where this case would not hold
is
1. SD < R
or 2. SD <= R
or 2. SD < R/2
or 4. SD <= R/2
Which one is the correct one to remember ?
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Mystery Machine
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DanaJ,DanaJ wrote: However, since you do have R/2 = SD for a set of two, it's not impossible for the formula of R/2 >= SD to be true. I've asked my econometrics teacher about it on his blog and am now awaiting results
Thanks for clearing much of the air......
I wanted to share a rule that I have read somewhere(but cant remember the place now). It states that "The maximum possible value of S.D. can be HALF of Range".
i.e. MAX(S.D)=R/2
Would be happy if that helps...
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goelmohit2002
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One small thing if SD <= R/2....then why the same is not applicable at the below thread....
https://www.beatthegmat.com/in-which-ran ... tml#184015
or do u guys think that:
a) deviation is something different than standard deviation...then how to calculate the same and reach to the answer.
b) or u guys think that the OA at the above thread is incorrect ?
Kindly share your opinion please.
https://www.beatthegmat.com/in-which-ran ... tml#184015
or do u guys think that:
a) deviation is something different than standard deviation...then how to calculate the same and reach to the answer.
b) or u guys think that the OA at the above thread is incorrect ?
Kindly share your opinion please.
