standart deviation

I'll let one of the GMAT tutors address this. They have said that the formula for standard deviation is not something that the GMAT tests for.

-Carrie

mariah wrote:94. 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

While we never need to calculate SD on test day, we may be tested on our general understanding of the concept.

In order to calculate SD, you need two pieces of information:

1) the number of terms in the set; and
2) the exact spacing of the set.

So, if we knew the full set, we could calculate SD. If we knew that we had a set of 5 consecutive even integers, we could calculate SD.

Let's look at the statements in the question posted:

Q: is the SD > 15000?

(1) range = 25000. This certainly isn't enough to calculate SD, but it does give us the maximum SD.

Standard deviation measures how spread out the terms are from the mean. If we have a range of 25000, what's our MAX standard deviation going to be?

Well, this is a very complicated question! In fact, after googling it for about 15 minutes (I'm not ashamed to admit that I google!), I still can't find a straight answer to the question. The fact that I can't find a simple answer through google makes me think that this question is WAY beyond the scope of the GMAT.


(2) mean = 150000. Mean by itself is irrelevant to SD: insufficient.

So, we're stuck with (a) and (e) (because statement (2) is completely useless, the answer can't possibly be (c)). My gut tells me that the answer is (e), but I've been unable to find any proof for a relationship between range and max SD, so I'm really not sure.

What's the source of this question?

May be you should go with the worst case scenario? I mean, say there are two elements in the set. If the mean of the set is 150.000 and the range is 25.000, so the higest possible standard deviation could be achieved with element a being 137500 and element b being 162.500.

This will give you two differences between the sets and the mean of 12.500 (12.500 and -12.500 which is the most disperse a set of items can be in this set). Calculating the SD for these two numbers can take some HUGE calculus (I did it on Excel). But if you look closely (as you always have to do in GMAT and is always I`m too lazy to do) calculating the SD of two or more numbers with the same absolute value regarding difference with me mean, is exactly that same absolute value. So, the SD for those two values is 12.500.

So, if the assumption is correct, and that is the escenario with the highest possible SD, then the highest possibe deviation is 12.500. And answer should be C.

IMO answer is A.

Standard deviation is a average of the dispersion of data about a mean value.

1). The range of the set is 25,000
sufficient. We just know the range. We have no idea about the mean.

Scenario 1[Max value of SD]:If we have just two numbers and if they are 25000 units apart from each other on the number line. Mean will be the mid-point. In this scenario SD will be max 12500 which is less than 15000.

Scenario 2[Min value of SD]: Lets say if we have more than 100,000 numbers in the set and two numbers from that list are 25000 units part from each other and rest are very closely distributed around the mean, in this scenario SD will be very minimal (less than 15000).

In both the scenarios we get SD less than 15000.

2). The mean of the set is 150,000.
Insufficient. We know the mean, but we have no idea about how many numbers are there in the set and how closely they are distributed. For Example, if we have just two numbers in the set and they are very closely distributed(149,999 & 150,001) then SD will be negligible. On the other hand if these two numbers are very far from each other on the number line(lets say the difference between the two numbers is 300,000) then SD will be more than 15000.

So Answer is A

Let me know if i am making a mistake.
Whats OA?

I think you are right, there is no need to know the mean for a worst-case-scenario once you get the range, cause with the range and only two elements you know the difference with the mean (the only thing, besides the number of elements, that you need to calculate SD) is just the difference between eacho of the two elements and the midpoint of the range. Yep, answer should be A.

OA E
I have collected it from the net.

OA E
I have collected it from the net.

As far as I know, SD can never be more than half the range if we use this equation to calculate SD.



However sometimes people use a different equation to calculate SD.



This second equation can give you SD more than half the range (Maximum 0.7071*Range).

If we use second equation for our problem, maximum value of SD can be 25000*0.7071, which is 17677.5 (>15000). But I think GMAT prefers the first equation. So by that equation maximum value of SD can be 12500.

hardik.jadeja wrote:As far as I know, SD can never be more than half the range if we use this equation to calculate SD.



However sometimes people use a different equation to calculate SD.



This second equation can give you SD more than half the range (Maximum 0.7071*Range).

If we use second equation for our problem, maximum value of SD can be 25000*0.7071, which is 17677.5 (>15000). But I think GMAT prefers the first equation. So by that equation maximum value of SD can be 12500.

The real point to take from this discussion is that this question seems WAY beyond the scope of what we're required to know about SD. Since the source wasn't posted, I'm guessing that it's not an official GMAT question, so this debate is really wasted if we're interested in the GMAT rather than advanced stats.