The level of nitrates was measured and recorded for 10 water samples

Question

The level of nitrates was measured and recorded for 10 water samples taken at the bottom of a certain lake and for 10 water samples taken at the surface of the same lake. Was the standard deviation of the measurements from the bottom samples greater than the standard deviation of the measurements from the surface samples?

(1) The least of the measurements from the bottom samples exceeded the greatest of the measurements from the surface samples.

(2) The range of the measurements from the bottom samples was greater than the range of the measurements from the surface samples.

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Official Answer

E

gmatophobia · Accepted Answer

Standard deviation helps us understand how much spread the data points are within a set. A low standard deviation means most points are close to the average. Conversely, a high standard deviation indicates that points deviate more from the average.

Statement 1

(1) The least of the measurements from the bottom samples exceeded the greatest of the measurements from the surface samples

While this Statement provides us information on how the values of the measurements from the bottom samples relate to the values from the measurements from the surface samples, we have no information on how the data are spread. Hence, the statement alone is not sufficient to comment anything on the standard deviation of the samples. We can eliminate A and D.

Statement 2

(2) The range of the measurements from the bottom samples was greater than the range of the measurements from the surface samples.

The statement is not sufficient as the range relates to only the lowest and the greatest measurements. We do not have information on other measurements and their spread across the mean.

Hence, eliminate B.

Combined

The statements combined do not help either as the information provided in both Statements do not help us determine the spread of the elements with respect to mean.

Option E

Bunuel · Answer

Getting a YES answer.

The surface samples:

To minimize the SD, construct data for the surface sample with points close to the mean: {0, 5, 5, 5, 5, 5, 5, 5, 5, 10}. The SD is approximately √5.

The bottom samples:

To maximize the SD, construct data for the bottom sample with points far from the mean: {11, 11, 11, 11, 11, 11, 11, 11, 11, 23}. The SD is approximately 3.6.

Getting a NO answer.

The surface samples:

To maximize the SD, construct data for the surface sample with points far from the mean: {0, 0, 0, 0, 0, 0, 0, 0, 0, 10}. The SD is approximately 3.

The bottom samples:

To minimize the SD, construct data for the bottom sample with points close to the mean: {11, 17, 17, 17, 17, 17, 17, 17, 17, 23}. The SD is approximately 2.7.

stne · Answer

(1) The least of the measurements from the bottom samples exceeded the greatest of the measurements from the surface samples.

Since SD is deviation from mean, this statement does not give any idea of deviation from the mean of the two data sets.

INSUFF.

(2) The range of the measurements from the bottom samples was greater than the range of the measurements from the surface samples.

Lets consider a smaller sample to test this statement. The same logic can be applied to a data set containing  10   measurements.

Let the Bottom measurements be:  (1,1,2,4) , Range is =3  
Mean  = 8/4= 2 
Deviation from mean  = {1,1,0,2 } 
Squaring the deviations  = {1,1,0,4} 
Avg. of the squares  = 6/4= 3/2 
SD  = \sqrt{3/2}

Let the surface measurements be (5,5,5,5) Range =0 
SD=0

In this case SD surface measurements  <  SD Bottom measurements

If the surface measurements are =  (5,5,5,7)  Range is  2  
Mean  =22/4=11/2 =5.5 
Deviation from mean  = (.5,.5,.5,1.5) 
Squaring the deviations = (2.5,2.5,2.5,2.25) 
Avg. of the squares  =9.75/4 = 39/16 
SD  =\sqrt{39/16}

Since  39/16 >3/2 , therefore  \sqrt{39/16} > \sqrt{3/2}

In this case SD of Surface measurements  >  SD Bottom mesurements.

INSUFF.

1+2

If Bottom  :(1,1,2,4)  and surface  :(5,5,5,5) = YES to the prompt.
If Bottom  :(1,1,2,4)  and surface  :(5,5,5,7) = NO to the prompt.

As elaborated above even using both the statements we cannot answer the prompt.

INSUFF.

Ans E

Hope it helped.

Note:  
In the prompt it says that the least of the bottom measurements exceed the greatest of the surface measurements.But in my haste I have just taken the opposite. Either way the logic remains the same shows why these statements are INSUFF.

buitri84 · Answer

Can someone provide a definite EXAMPLE as to why (2) is not sufficient?

Appreciate stne example up there, but the calculation is wrong for this 2nd example:
>>If the surface measurements are = (5,5,5,7)

squaring the 0.5 deviation from mean should get us 0.25, not 2.5 as he calculated. The real numerical result should show that the sd from the higher-range sample is still bigger.
I've asked chatpgpt and it's hallucinating the same way the the 2 answers above are: insisting that a larger range is not sufficient to conclude that s.d is larger, while every numerical example it comes up with still result in the s.d associated with the larger-range sample being larger.

This is a cery tricky OG question, I think.

buitri84 · Answer

Wow finally an example that works, barely. In the 2nd example, the s.d of the bottom sample is 2.898 (vs. 3 for the surface sample).

Thks Bunuel. Human still beats chatgpt =))

KarishmaB · Answer

10 samples taken from top and SD-top calculated.
10 samples taken from bottom and SD-bottom calculated. 
The values of the samples from the top and from the bottom could be vastly different (say nitrates concentration could be very high at the bottom and low at the top). So comparing them is irrelevant. The point is how the 10 samples at the top compare with each other - how close they are to each other. And how 10 samples at the bottom are relative to each other. 
That will tell us which has a higher SD. SD does not depend on absolute values but only values relative to each other.

(1) The least of the measurements from the bottom samples exceeded the greatest of the measurements from the surface samples.

As we said, this is absolutely irrelevant to the question. Ignore it.

(2) The range of the measurements from the bottom samples was greater than the range of the measurements from the surface samples.

Range gives us Max - Min. It does not tell us how the values are spread. It is possible that even with higher range, all other values of the set are very close to the mean so SD is very low. It is possible 
that even with lower range, all other values are also at extremes so SD is high. Hence ranges cannot help us compare the SD except in cases where there are only say two elements in the sets. 
Not sufficient.

Using both also it is not sufficient since statement 1 anyway doesn't add any value.

Answer (E)

lawyeroffduty · Answer

i got this question wrong both in the mock, and while solving it too after the mock. but, now as i understood the concept of standard deviation in little bit more detailed, so can we say just by reading the statements 1 and 2, we can say select e as an answer as both the statements doesn't provide any data as to how the 10 sample are relative to each other and as we know, in order to find the standard deviation, we should know the mean difference (spread) so, only then we can tell whether the standard deviation is higher or not.

The level of nitrates was measured and recorded for 10 water samples

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GMAT Data Sufficiency (DS) Questions

The level of nitrates was measured and recorded for 10 water samples

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