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Range and Standard Deviation

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by prachi18oct » Fri Oct 10, 2014 1:30 am
Hello Experts,

Can you please tell me the relation between S.D and range.
I read somewhere that S.D = range/2
but its not always true as S.D <= Range/2

So in D.S questions, if one of the statements have range given, is it sufficient to find the S.D ?

Sample Question:-

Set X and Y have 5 numbers, respectively.Is the S.D of X > S.D of Y?
1) Range of X > Range of Y
2) Average of X is greater than that of Y

What should be the answer here?

Also, If the values are same then the range = 0 and so is the S.D.
But if the values are different in a list 43,44,45,46,47 then range = 4 but S.D = 1.6(< Range/2)

Please explain.
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by Matt@VeritasPrep » Sat Oct 11, 2014 10:44 am
There isn't a set relationship between SD and range, but it IS always true that Range/2 ≥ S.D., for ANY set of numbers. Here's a nice proof if you're interested.

You CANNOT find the standard deviation given simply the range. Consider the sets {2,6} and {2, 2.1, 2.2, 2.3, ..., 5.8, 5.9, 6}. They have the same range, but they obviously don't have the same standard deviation! The average of a set is mostly irrelevant to the S.D., since the S.D. depends on RELATIVE size (the distance between each term and the mean), not ABSOLUTE size.
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by Mathsbuddy » Fri Oct 17, 2014 7:54 am
In response to below, RANGE = LARGEST VALUE - LOWEST VALUE and does not consider the individual values in between. STANDARD DEVIATION looks at the difference between each value and the mean (average) value. Therefore, they cannot be directly related.
prachi18oct wrote:Hello Experts,

Can you please tell me the relation between S.D and range.
I read somewhere that S.D = range/2
but its not always true as S.D <= Range/2

So in D.S questions, if one of the statements have range given, is it sufficient to find the S.D ?

Sample Question:-

Set X and Y have 5 numbers, respectively.Is the S.D of X > S.D of Y?
1) Range of X > Range of Y
2) Average of X is greater than that of Y

What should be the answer here?

Also, If the values are same then the range = 0 and so is the S.D.
But if the values are different in a list 43,44,45,46,47 then range = 4 but S.D = 1.6(< Range/2)

Please explain.
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