During an experiment, some water was removed from each of

Question

During an experiment, some water was removed from each of the 6 water tanks. If the standard deviation of the volumes of water in the tanks at the beginning of the experiment was 10 gallons, what was the standard deviation of the volumes of water in the tanks at the end of the experiment?

(1) For each tank, 30% of the volume of water that was in the tank at the beginning of the experiment was removed during the experiment.

(2) The average (arithmetic mean) volume of water in the tanks at the end of the experiment was 63 gallons.

Bunuel · Accepted Answer

During an experiment, some water was removed from each of the 6 water tanks. If the standard deviation of the volumes of water in the tanks at the beginning of the experiment was 10 gallons, what was the standard deviation of the volumes of water in the tanks at the end of the experiment?

(1) For each tank, 30% of the volume of water that was in the tank at the beginning of the experiment was removed during the experiment.
(2) The average (arithmetic mean) volume of water in the tanks at the end of the experiment was 63 gallons.

You should know that:
If we add or subtract a constant to each term in a set:
Mean will increase or decrease by the same constant.
SD will not change.

If we increase or decrease each term in a set by the same percent (multiply all terms by the constant):
Mean will increase or decrease by the same percent.
SD will increase or decrease by the same percent.

You can check it yourself:
SD of a set: {1,1,4} will be the same as that of {5,5,8} as second set is obtained by adding 4 to each term of the first set.

That's because Standard Deviation shows how much variation there is from the mean. And when adding or subtracting a constant to each term we are shifting the mean of the set by this constant (mean will increase or decrease by the same constant) but the variation from the mean remains the same as all terms are also shifted by the same constant.

So according to this rules statement (1) is sufficient to get new SD, it'll be 30% less than the old SD so 7. As for statement (2) it's clearly insufficient as knowing mean gives us no help in getting new SD.

Answer: A.

For more on this check Standard Deviation chapter of Math Book: https://gmatclub.com/forum/math-standar ... 87905.html

Collection of PS questions on SD: https://gmatclub.com/forum/ps-questions ... 85897.html
Collection of PS questions on SD: https://gmatclub.com/forum/ds-questions ... 85896.html

Hope it's clear.

KarishmaB · Answer

Standard deviation measures dispersion around the mean i.e. how far apart the values are from mean. The actual calculation of the Standard Deviation is not asked in GMAT but you need to theoretically understand the concept. e.g. If we are interested in SD of the following values: 2, 4, 5, 6, 8 Here, mean is 5. I encourage you to visualize the numbers on a number line. The diagram below shows the 5 numbers with their mean 5. SD measures how far the numbers are from their mean. Ques1.jpg If we add 10 to each of the numbers, the numbers become: 12, 14, 15, 16, 18 New mean is 15 but relative to the new mean, the numbers are still dispersed in the say way around 15. So SD for these numbers is the same as SD above. If we multiply/divide each number by some number, the SD changes. Look at the diagram below to understand why. If each number is multiplied by 3, the numbers are: 6, 12, 15, 18, 24 Ques2.jpg On the number line, now they are much farther from their mean 15. Hence their SD is greater than before. It is actually 3 times the initial SD. (Check out the formula of SD to see why.) In this question, initial SD was 10. When 30% of the water is removed from each tank, the leftover water is 70% i.e. 0.7*original volume of water. Since we are multiplying the original volume by 0.7, the SD will change. It will become 0.7*previous SD i.e. 0.7*10 = 7. mehdiov: As we see from above, if we remove the same quantity, the SD will not change. Here we removed a fraction of the original quantity of each. e.g. if one tank had 50 gallons, we removed 30% i.e. 15 gallons. If another had 100 gallons, we removed 30 gallons.

cipher · Answer

Ans A SD doesnt change when we add/subtract the same amount from all the values in the set. Please see this link by Bunuel, it will help in understanding concepts regarding SD. ps-questions-about-standard-deviation-85897.html

amannain1 · Answer

Hi Nitish,
Answer is indeed A but SD will change.
It should be now 30% less than before.

As bunuel has listed in point#7:
"If we increase or decrease each term in a set by the same percent:
Mean will increase or decrease by the same percent.
SD will increase or decrease by the same percent."

--
Aman

kraizada84 · Answer

This question is similar to :

set X=(A,B,C,D,E,F)
SD= 10

Whats the SD when 
set X=(A-a),(B-b),.....(F-f)

statement 1: 
a=0.3A
SIMILARILY FOR OTHERS

hence we can find the value for this list.
SUFFICIENT

statement 2: INSUFFICIENT

as only knowing AM at the end of operation couldnot give any information for the reductions in the value of individual element

hence A

imadkho · Answer

Dear Bunuel, your explanation is great, but if I got you right, then based on the first statement, the standard deviation of the volumes at the end of the experiment should be also 10 (and not 7), as it was at the beginning of the experiment.

Bunuel · Answer

If we add or subtract a constant to each term in a set:
SD will not change.

If we increase or decrease each term in a set by the same percent (multiply all terms by the constant):
SD will increase or decrease by the same percent.

Since (1) says that  for each tank 30%  of the water was removed then the SD will decrease by the same 30%.

imadkho · Answer

Bunuel, it seems I am not getting you correctly. You are saying that SD will not change if a constant is added to/subtrated from the members of any list (I agree), so how come you are also telling me that SD will go down by 30% by the end of the experiment. I think it should also stay the same.
thanks

Bunuel · Answer

In this case we are not subtracting a constant from each term, we are decreasing each term by some percent (multiplying by 0.7) and  if we increase or decrease each term in a set by the same percent (multiply all terms by the constant): SD will increase or decrease by the same percent .

Edvento · Answer

We have Standard Deviation = \sigma =\sqrt{{\frac{1}{6}}* Here M is the mean. The subscript 6 is for the 6 water tanks

Now, Evaluating statement 1 only For each tank 30% of the water was removed. The mean becomes (1- \frac{30}{100})M = 0.7M Also each of x_1, x_2, ..., x_6 becomes 0.7x_1, 0.7x_2, ..., 0.7x_6 respectively. Hence, Standard Deviation = \sigma =0.7* \sqrt{{\frac{1}{6}}* Hence this statement alone is sufficient. Choices B, C, E are eliminated. Evaluating statement 2 only The mean after the reduction, M = 63 gallons. We have no idea of the initial value of M or x_1, x_2, ..., x_6 or how these values have changed. Hence, this statement alone is insufficient. Choice D is ruled out. Answer is A. Regards, Shouvik.

NoHalfMeasures · Answer

Here is a neat rule I keep handy when dealing with statistics problems on the GMAT:
” If X is added/subtracted to/from every element of a set, all 3 measures of Central Tendency- mean, median, mode- will be added/subtracted by X, whereas measures of Dispersion- range, interquartile range and standard deviation, variance will be unaffected. On the other hand, if every element is multiplied by X, both measures of central tendency and dispersion will be multiplied by X” -

Hope it helps others.

nidhiprasad · Answer

Question : If the mean is 10 and number of numbers is n . Now the new mean is 15 and number of numbers remain same . How does this information affect the SD. is this information sufficient to comment on SD?

KarishmaB · Answer

No. SD does not depend on the actual mean. It depends on the distance of the numbers from mean and the number of numbers. Check this post: https://gmatclub.com/forum/during-an-ex ... ml#p831904

Gijmaja · Answer

I seem to misunderstand an important information. Did you assume that the water tanks were of the same size? Nowhere does the problem state that.

Now imagine if tanks were of the following volumes: 4, 6, 8, 10. St. dev is 2.58.
reduce the water in the tank by 30%: 2.8, 4.2, 5.6, 7. St. dev is 1.81.

Can you expand on this matter?

KarishmaB · Answer

There was no assumption made about the size of the tank nor about the amount of the water in each tank.

The volume of water in each tank could have been any random number. What we do know is that their SD is 10.

What happens when each value in the list is reduced by 30% i.e. it becomes 70% of its initial value? Each value will be multiplied by 0.7. Then the mean will be multiplied by 0.7 too and the SD will get multiplied by 0.7 too. 
The same is illustrated by your example too since 1.81 is 70% of 2.58.

So when we multiply each value of a list by the same positive number, the SD gets multiplied by the same number too. 
So if the old SD of a set is 10 and if each value is reduced by 30% i.e. multiplied by 0.7, the new Sd will also get multiplied by 0.7 i.e. it will become 7.

nikitathegreat · Answer

In the first statement, do we mean 30% water of all the six tanks were removed from each of the tank or 30% water from each tank. If we mean latter, then in that scenario, each tank can have different volume of water and 30% of each volume of water would mean different. Please help me explain the language

Mrinal98 · Answer

Bunuel  statement 1 says "30% of the volume of water that was in the tank at the beginning of the experiment was removed". Are we to assume that each of the 6 tankers had equal gallons of water as it is nowhere mentioned in the question. 
if we take a different value for each of the 6 tankers then the "30%" value will not be a constant and hence statement 1 would not be sufficient.

Please correct me where I'm thinking wrong..

Bunuel · Answer

Yes, we are not told that each tank contained an equal amount of water, but it does not matter. Regardless of the initial amounts in the tanks, after the removal, they would contain 0.7 times their original volume. This means that the standard deviation would become 0.7 times its original value. When each term in a set is multiplied by a constant, like 0.7 in this case, the standard deviation also gets multiplied by the same constant.

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