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.

DanaJ 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.

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

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]

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.

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.

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

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

goelmohit2002 wrote:Is the range of numbers greater than 500 ?

1) The median of the numbers is 1000.
2) Standard deviation of numbers is 500.

OA after some discussion