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# The Department of Environmental Protection measured the volu

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Manager
Joined: 11 Aug 2011
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The Department of Environmental Protection measured the volu  [#permalink]

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Updated on: 18 May 2014, 04:11
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00:00

Difficulty:

55% (hard)

Question Stats:

57% (01:38) correct 43% (01:33) wrong based on 212 sessions

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The Department of Environmental Protection measured the volume of water in 10 similarly sized reservoirs in State X and found that the standard deviation of their volumes at the start of the year was 4 million cubic gallons. Was the standard deviation of those 10 volume measurements lower at the end of the year?

(1) At the end of the year, the average volume of the water in the 10 reservoirs had decreased by 20%.

(2) The percent decrease in the volume of the water in each reservoir during the year was the same.

I have issues solving such problems.
Can someone please let me know how do we solve such problems.

Originally posted by akhil911 on 18 May 2014, 04:09.
Last edited by akhil911 on 18 May 2014, 04:11, edited 1 time in total.
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Re: The Department of Environmental Protection measured the volu  [#permalink]

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18 May 2014, 04:10
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Official solution provided by Veritas prep

This difficult data sufficiency question tests an important number property relating to standard deviation: if the terms in a set are each decreased by a fixed percentage, then the standard deviation decreases by that same percentage (unless the terms are all the same to begin with – then the standard deviation would be zero before the change and zero after). In the question stem you learn that the standard deviation of 10 measurements is 4 million at the beginning of the year (note – this guarantees that they are not all the same), and you must decide if that standard deviation had decreased at the end of the year. In statement (1) you learn that the average volume of the 10 reservoirs decreased by 20% over the year. Importantly this does NOT mean that each term decreased by 20%. A decrease in the average of all 10 volumes might yield a smaller standard deviation, but it could easily result in a larger standard deviation. Consider a sample set of 5, 10, and 15, which has an approximate standard deviation of 4. If the average of those 3 numbers is decreased by 20%, then the new set could be 5, 4, and 15 (average went from 10 to 8) and this set clearly has a higher standard deviation just by inspection (approximately 5 if you do the calculation). Remember that you do not need to calculate the standard deviation – just realize that you do not know how the set was changed to decrease the average, so you do not know the effect on the standard deviation. Statement (1) is not sufficient. With statement (2) you must be very careful as it appears to give you the information that was lacking in statement (1): it proves that each one had the same percentage decrease. At this point, many students will pick (C) as they fit together so nicely. However statement (2) is actually sufficient by itself: if the volume of each reservoir decreased by the same percentage (and the volume for each was not the same to begin with – something you know from the question stem) then the standard deviation MUST be lower after that change. It guarantees that the set “shrunk” during the year, reducing the dispersion and thus the standard deviation. Answer is B – statement 2 alone is sufficient.
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Re: The Department of Environmental Protection measured the volu  [#permalink]

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18 May 2014, 06:39
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akhil911 wrote:
The Department of Environmental Protection measured the volume of water in 10 similarly sized reservoirs in State X and found that the standard deviation of their volumes at the start of the year was 4 million cubic gallons. Was the standard deviation of those 10 volume measurements lower at the end of the year?

(1) At the end of the year, the average volume of the water in the 10 reservoirs had decreased by 20%.

(2) The percent decrease in the volume of the water in each reservoir during the year was the same.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed

I have issues solving such problems.
Can someone please let me know how do we solve such problems.

Here is quick summary for SD:

TIPS:
1. |Median-Mean| <= SD.

2. Variance is the square of the standard deviation.

3. If Range or SD of a list is 0, then the list will contain all identical elements. And vise versa: if a list contains all identical elements then the range and SD of a list is 0. If the list contains 1 element: Range is zero and SD is zero.

4. SD is always >=0. SD is 0 only when the list contains all identical elements (or which is same only 1 element).

5. Symmetric about the mean means that the shape of the distribution on the right and left side of the curve are mirror-images of each other.

6. 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.

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.

8. Changing the signs of the element of a set (multiplying by -1) has no effect on SD.

9. The SD of any list is not dependent on the average, but on the deviation of the numbers from the average. So just by knowing that two lists having different averages doesn't say anything about their standard deviation - different averages can have the same SD.

The above is from Math Expert Bunuel. It's very handy.
For more on this topic refer to

math-standard-deviation-87905.html
and

Coming back to the question

We need to find whether SD lower than what it was in the beginning of the year

St 1 says average vol of water had decreased by 20%. So if Total volume of water for 10 resevoirs was 100lts then at the end of the year Volume of 10 resevoirs was 80 lts.

Case 1 :Now Imagine if 9 resevoirs see an increase of 2 lts each and 1 resevoirs sees a decrease of 38 Lts then volume of water is still 80 lts (Average vol of water is 8 lts) but SD will be different (lower or more is not important)
Case 2:Also another case where each resevoir has volume of 10 lts each and each sees uniform decrease of 20% ie. at the end of the year each resevoir has 8 lts and Total volume is 80 lts and average volume is 8 ltr--------> In this case we can safely say SD is lower but not in Case 1

So ST 1 is not sufficient

Consider St 2 points out that each resevoir saw percent decrease and is pointing to Case 2 of St 1 and hence it is sufficient. It is in line with Pt 7
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The Department of Environmental Protection measured the volu  [#permalink]

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19 May 2014, 01:39
akhil911 wrote:
The Department of Environmental Protection measured the volume of water in 10 similarly sized reservoirs in State X and found that the standard deviation of their volumes at the start of the year was 4 million cubic gallons. Was the standard deviation of those 10 volume measurements lower at the end of the year?

(1) At the end of the year, the average volume of the water in the 10 reservoirs had decreased by 20%.

(2) The percent decrease in the volume of the water in each reservoir during the year was the same.

I have issues solving such problems.
Can someone please let me know how do we solve such problems.

Almost similar question to practice: during-an-experiment-some-water-was-removed-from-each-of-94166.html (GMAT Prep)

For more on that topic 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 DS questions on SD: ds-questions-about-standard-deviation-85896.html

Hope this helps.
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Re: The Department of Environmental Protection measured the volu  [#permalink]

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08 Jul 2014, 02:55
I think on the GMAT it's not necessary to calculate the SD, just to basically know what it is. The SD gives us somehow the "distance" (deviation) between the mean of some data and each data point. Thus, if you only know something about the overall change and nothing about each data point, you can't tell whats happening with the SD. But if you know about each data point, you can answer if increase, decrease or stays same.

To the question:

(1) Here you only get to know the overall change of all data points. Wont help you to determine the SD. IS.
(2) Here they tell you that each reservoir as decreased in the same amount. This is enough to tell you that the SD will have decreased too.

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Re: The Department of Environmental Protection measured the volu  [#permalink]

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09 Aug 2015, 07:52
What if the decrease is 0%? Then the standard deviation does not change.
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Re: The Department of Environmental Protection measured the volu  [#permalink]

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09 Aug 2015, 08:09
steamtrain wrote:
What if the decrease is 0%? Then the standard deviation does not change.

Hello steamtrain
In this case they can not call it "decrease" and statement should be "The volume of the water in each reservoir during the year does not change"
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The Department of Environmental Protection measured the volu  [#permalink]

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09 Aug 2015, 12:04
Harley1980 wrote:
steamtrain wrote:
What if the decrease is 0%? Then the standard deviation does not change.

Hello steamtrain
In this case they can not call it "decrease" and statement should be "The volume of the water in each reservoir during the year does not change"

Calling it a decrease of 0% is not wrong. A 'decrease' can be of a 0 value of of 100 units. Calling a decrease =0 means that the net effect of the decrease was 0 or the quantity is question did not undergo any change.
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Re: The Department of Environmental Protection measured the volu  [#permalink]

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09 Aug 2015, 12:28
Engr2012 wrote:
Harley1980 wrote:
steamtrain wrote:
What if the decrease is 0%? Then the standard deviation does not change.

Hello steamtrain
In this case they can not call it "decrease" and statement should be "The volume of the water in each reservoir during the year does not change"

Calling it a decrease of 0% is not wrong. A 'decrease' can be of a 0 value of of 100 units. Calling a decrease =0 means that the net effect of the decrease was 0 or the quantity is question did not undergo any change.

Hello Engr2012
I will not argue that this is correct in real life (actually I hear about this first time, but I believe you because my learning of math begins from GMAT)

But as I know in GMAT each word of quantitative task should has some sense and in good task you will never meet information that has a purpose to throw you off.
So I think if we meet word decrease we shouldn't take into account possibility of 0.

Maybe you have some example of task in which decrease means 0?
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Re: The Department of Environmental Protection measured the volu  [#permalink]

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28 Aug 2019, 18:14
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Re: The Department of Environmental Protection measured the volu   [#permalink] 28 Aug 2019, 18:14
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