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Manager  Joined: 12 Mar 2009
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During an experiment, some water was removed from each of  [#permalink]

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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.
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Re: GMAT Prep DS Q  [#permalink]

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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.

For more on this check Standard Deviation chapter of Math Book: math-standard-deviation-87905.html

Collection of PS questions on SD: ps-questions-about-standard-deviation-85897.html
Collection of PS questions on SD: ds-questions-about-standard-deviation-85896.html

Hope it's clear.
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Re: GMAT Prep DS Q  [#permalink]

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vaivish1723 wrote:
Hi,
I have attached few gmat prep DS Qs. Please respond with explanations. Apology if there is duplications.

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.

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Re: GMAT Prep DS Q  [#permalink]

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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
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Re: GMAT Prep DS Q  [#permalink]

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vaivish1723 wrote:
Hi,
I have attached few gmat prep DS Qs. Please respond with explanations. Apology if there is duplications.

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.
At Veritas, we 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.

Attachment: Ques1.jpg [ 6.42 KiB | Viewed 51907 times ]

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

Attachment: Ques2.jpg [ 6.51 KiB | Viewed 51938 times ]

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.
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During an experiment, some water was removed from each of 6 water tanks. If the standard deviation of the volume 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.

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
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Re: During an experiment, some water was removed from each of  [#permalink]

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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.
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Re: During an experiment, some water was removed from each of  [#permalink]

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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.

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%.
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Re: During an experiment, some water was removed from each of  [#permalink]

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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
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Re: During an experiment, some water was removed from each of  [#permalink]

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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

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.
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Re: During an experiment, some water was removed from each of  [#permalink]

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I got u bunuel, the same percentage increase or decrease to each element in a list will not correspond to adding or subtracting the same number or constant to/from the different elements.
Thanks very much.
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Well I suppose this is actually quite easy. If you know only the basics of standard deviation, then it should be clear that if every tank loses 30% of its water, then the standard deviation also decreases by 30%. So A is sufficient, while B alone isn't as that information isn't comparable to information in the prompt.

Originally posted by geometric on 16 Jun 2012, 00:20.
Last edited by geometric on 16 Jun 2012, 00:44, edited 1 time in total.
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thanks for the explanation. Was confused on the percentage part, however can you elaborate a bit on how if the percentage is changed the SD remains the same. For instance if container 1 has say 11 litres, 2 has 17, 3 has 19 30 % of each will be different 30% of higher value will be higher and for lowest value will be lowest, so SD must change isn't it??
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Hi,

Standard deviation is defined as:
$$\sqrt{\frac {(x_{mean}-x_1)^2+(x_{mean}-x_2)^2+...+(x_{mean}-x_n)^2}n}$$

and as you know, increasing or decreasing a each term of the series, increases/decreases the mean by same value.
so,$$(x_{mean}-x_n)$$ will not change in case of addition/subtraction.

let say each term is multiplied by "a", mean as well as each term is multiplied by a and we get:
$$(ax_{mean}-ax_1)$$
Thus, standard deviation will change only in case of multiplication/division.

Now back to the question;
Using (1),
30% is removed, so what is left is 70% of water in tank, also, the average reduces to 70% of original.
Thus new standard deviation is 70% of 10 = 7. Sufficient.

Using (2),
Clearly, Insufficient.

Regards,
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1
riteshgupta wrote:
Can any one answer the below with a bit of detail, so that S.D concept is cleared????

During an experiment, some water was removed from each of 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 percent 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.

We have

Standard Deviation = $$\sigma =\sqrt{{\frac{1}{6}}*[(x_1 - M)^2 + (x_2 - M)^2 + ... + (x_6 - M)^2} = 10$$

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 = $$\sigma =0.7* \sqrt{{\frac{1}{6}}*[(x_1 - M)^2 + (x_2 - M)^2 + ... + (x_6 - M)^2} = 0.7*10 = 7$$

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.

Regards,

Shouvik.
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Re: During an experiment, some water was removed from each of  [#permalink]

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vaivish1723 wrote:
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.

IF WE MULTIPLY OR DIVIDE EACH ELEMENT WITH SAME FACTOR THEN RESULTANT SD is also same time multiplied or divided.

we have initial SD = 10

STATEMENT 1: RESULTANT QUANTITY IN EACH TANK WILL BE 0.7 TIMES THE INITIAL VOLUME...HENCE IN SHORT WE ARE MULTIPLYING EACH TERM WITH 0.7 HENCE SD = 10*0.7 =7
HENCE SUFFICIENT

STATEMENT 2:BY KNOWING AVERAGE WE CANNOT CALCULATE SD AS WE DONT KNOW HOW MUCH WATER IS TAKEN OUT FROM EACH TANK.
HENCE INSUFFIECIENT.

HENCE A
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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.
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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?
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Re: During an experiment, some water was removed from each of  [#permalink]

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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?

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
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Re: During an experiment, some water was removed from each of  [#permalink]

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Bunuel wrote:
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

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.

Option A states
(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.

How do we know volume of water at the beginning of experiment in each tank was same....
by referring as some water -its not discrete ...

Pleas help..Do you think it should be explicitly stated. Re: During an experiment, some water was removed from each of   [#permalink] 07 Sep 2019, 05:31
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