right relation b/w range and SD

[vipulgoyal]

Hi Experts,

Please shed some light on this , I hBeen trying to find right relation b/w range and SD, following q are related with this

Gave some numbers about money cost. Is the range of these numbers greater than $500?
1). The median is $1,000
2). Standard deviation is $500

Is the standard deviation of a certain set greater than 15,000?
1) The range of the set is 25,000
2) The mean of the set is 150,000

There was discussion At BtG on this but NO conclusion
https://www.beatthegmat.com/standart-dev ... 30305.html

[MartyMurray]

The standard deviation is basically the average distance between the elements and the mean.

We cannot determine the exact range from the standard deviation because there are many ways to have the same standard deviation and those ways can involve various ranges.

For instance the numbers 0, 0, 0, 100, 100, 100, have a mean of 50 and, an average distance from that mean of 50. 0 is 50 less than 50 and 100 is 50 more than 50. The range is 100.

These numbers also have a mean of 50 and an average distance of 50 from 50, but the range is now 200.

-50, 0, 50, 50, 100, 150

Standard deviation and average distance are not the same, but basically these examples illustrate why you can't get range from standard deviation.

At the same time, standard deviation does give some insight into range.

The range cannot be less than the standard deviation. For a set of numbers to have a certain standard deviation, some numbers in the set have to be at least that far from the mean, and other numbers will be on the other side of the mean. So the difference between those numbers will be >= the standard deviation.

Think about it, if the standard deviation is basically the average distance from the mean, some numbers will be at that average deviation and possibly some numbers will be an even greater distance from the mean, and to create that mean, you need numbers on the other side of the mean. The maximum two distances on each side of the mean add up to the range.

So the answer to your first question,

Gave some numbers about money cost. Is the range of these numbers greater than $500?
1). The median is $1,000
2). Standard deviation is $500

is B, because if the standard deviation is 500, then some numbers have to be at least 500 away from the mean and other numbers have to be on the other side of the mean. So the range is over 500.

Looking at your second question, I read somewhere that actually the range is always >= twice the standard deviation. This makes sense and I am pretty sure it's true, but while I doubt you would see the above question on the GMAT I am pretty much certain you would not see the below question on the GMAT. Still, let's assume that indeed the range is always >= twice the standard deviation and use that assumption to answer this question.

Is the standard deviation of a certain set greater than 15,000?
1) The range of the set is 25,000
2) The mean of the set is 150,000

Statement 2 is insufficient because the mean does not tell us about the distance from the mean.

Statement 1, however, tells us that the maximum possible standard deviation is 12,500, if our assumption is correct. So given that assumption statement 1 is sufficient.

Having said all this, I am skeptical about your needing to know much of it for the GMAT.

I think it's obvious that the range has to be at least equal to the standard deviation, and so I could see the first question showing up on the test. Still I doubt it will as, from what I have seen and heard, when the GMAT uses standard deviation it mostly asks about relationships and effects rather than exact numbers.

Getting the second question right involves using a relationship between range and standard deviation that is known by few and rarely discussed, if it even really exists. So while I doubt that the first question would appear on the test, I am close to certain that the second question, or one like it, will not appear on the GMAT any time soon, and probably will not ever.

[vipulgoyal]

thanks Marty,

Having said that it is less likely appearing (second one) in exam, I am going to end this quest, however my curiously arouse bcz i saw same q in three different sources( which appears to be authentic) while studying stats .. thanks one again

[Max@Math Revolution]

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 independent equations ensures a solution.

Gave some numbers about money cost. Is the range of these numbers greater than $500?
1). The median is $1,000
2). Standard deviation is $500

Is the standard deviation of a certain set greater than 15,000?
1) The range of the set is 25,000
2) The mean of the set is 150,000


Normally questions regarding the relationship between average, standard deviation, median etc or the relationship between average, range and median have (E) as the answer. There is, however, the rule that Standard deviation <= range/2. Hence, the question above, 500<=r/2, 1,000<=r, answering the question 'yes' and being sufficient, the answer becomes (B).
For the question below,
for condition 1, from SD<=25,000/2=12,500, we can figure out that the SD cannot be greater than 15,000, so 'no'. But the condition is sufficient, so the answer becomes (A).

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