GMAT Club Olympics 2024 (Day 7): At the Olympic Games, 10 gymnasts

Question

At the Olympic Games, 10 gymnasts must perform two routines, compulsory and free, each gymnast receiving a separate score for each routine. If the standard deviation of the 10 scores for the compulsory routine was 0.7 points, what was the standard deviation of the 20 scores for both routines combined?

(1) The standard deviation of the 10 scores for the free routine was also 0.7 points.
(2) The range of scores for each of the compulsory routine and free routine was 3 points.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Bunuel · Accepted Answer

GMAT Club Official Explanation:

At the Olympic Games, 10 gymnasts must perform two routines, compulsory and free, each gymnast receiving a separate score for each routine. If the standard deviation of the 10 scores for the compulsory routine was 0.7 points, what was the standard deviation of the 20 scores for both routines combined?

(1) The standard deviation of the 10 scores for the free routine was also 0.7 points.

Having two lists with the same standard deviation does not mean that the combined list will have the same standard deviation. The lists might be very far apart. For example, {0, 1} and {100, 101} both have the same standard deviation. However, {0, 1, 100, 101} is far more widespread than either of the lists and thus will have a much higher standard deviation than the initial lists. Therefore, this information is not sufficient.

(2) The range of scores for each of the compulsory routine and free routine was 3 points.

The range alone does not help to determine the standard deviation. Therefore, this information is also not sufficient.

(1)+(2) Using the same example from (1): {0, 1} and {100, 101} both have the same standard deviation and range. However, {0, 1, 100, 101} is far more widespread than either of the lists and thus will have a much higher standard deviation than the initial lists. Therefore, we cannot determine the range of both routines combined. Not sufficient.

Asnwer: E.

KshtriyaNaveen · Answer

Given That,

10 gymnasts must perform two routines, compulsory and free, each gymnast receiving a separate score for each routine.

Compulsory  = { c1, c2 , ... c10 }, S.D ( C) = 0.7
  Free  = { f1,f2 , ... f10 }

what was the standard deviation of the 20 scores for both routines combined?

Statement 1: 
 The standard deviation of the 10 scores for the free routine was also 0.7 points.
  Even though S.D(F) = 0.7, we can't add 2 different S.D without knowing nature of the score. 
 SO Statement 1 is not sufficient

Statement 2: 
 The range of scores for each of the compulsory routine and free routine was also 3 points.
 Even if ranges are same , there min and max value may differ for each set, so we can't find S.D ( C & F) 
 SO Statement 2 is not sufficient

Even if we club both statment, we can find unique S.D Value 
 Therefor Option E is correct

jack5397 · Answer

IMO - E

St1 - The standard deviation of the 10 scores for the free routine was also 0.7 points.
 We don't know anything about individual values so we cannot know the SD for all 20 values - hence insufficient.

St2-The range of scores for each of the compulsory routine and free routine was 3 points.

Does not telling anything about individual values - hence insufficient

Combined also does not tell anything about the individual values and relation between them, hence insufficient

hence E

Kinshook · Answer

Given: At the Olympic Games, 10 gymnasts must perform two routines, compulsory and free, each gymnast receiving a separate score for each routine.
Asked: If the standard deviation of the 10 scores for the compulsory routine was 0.7 points, what was the standard deviation of the 20 scores for both routines combined?

(1) The standard deviation of the 10 scores for the free routine was also 0.7 points.
Since the arithmetic mean of 10 scores for the compulsory routine may be different from the arithmetic mean of 10 scores for the free routine. 
Case 1: The arithmetic mean of 10 scores for the compulsory routine = the arithmetic mean of 10 scores for the free routine. 
The standard deviation of the 20 scores for both routines combined = .7
Case 2: The arithmetic mean of 10 scores for the compulsory routine is different from the arithmetic mean of 10 scores for the free routine. 
The standard deviation of the 20 scores for both routines combined can not be determines from the information provided.
NOT SUFFICIENT

(2) The range of scores for each of the compulsory routine and free routine was 3 points.
Range of scores is not an indicator of standard deviation. 
Case 1: Scores = {6, 7, 7, 7, 7, 8, 8, 8, 8, 9} : Range of scores = 3; Arithmetic mean = 7.5; Standard deviation = 2.54
Case 2: Scores = {6, 6, 6, 6, 6, 9, 9, 9, 9, 9} : Range of scores = 3; Arithmetic mean = 7.5; Standard deviation = 4.74
Range of scores are same but standard deviations are different in the above example.
NOT SUFFICIENT

(1) + (2) 
(1) The standard deviation of the 10 scores for the free routine was also 0.7 points.
(2) The range of scores for each of the compulsory routine and free routine was 3 points.
Since the arithmetic mean of 10 scores for the compulsory routine may be different from the arithmetic mean of 10 scores for the free routine. The range of scores for each of the compulsory routine and free routine was 3 points but means may still be different.
Case 1: The arithmetic mean of 10 scores for the compulsory routine = the arithmetic mean of 10 scores for the free routine. 
The standard deviation of the 20 scores for both routines combined = .7
Case 2: The arithmetic mean of 10 scores for the compulsory routine is different from the arithmetic mean of 10 scores for the free routine. 
The standard deviation of the 20 scores for both routines combined can not be determines from the information provided.
NOT SUFFICIENT

IMO E

OmerKor · Answer

After the opening day of the week, 
We're back to business.
I feel the luck is going to shift to our Green team as we get closer to the end of the Champions.
Let's get started with our explanation for this topic:

Glance - the Question:  
 Question:  We have here a Standard deviation problem. 
Pay attention to the rules of SD.

Rephrase - Reading and Understanding the question:  
 Given: 
The first routine is 0.7 SD.
scores are 1-10 ("the 10 scores")
  :  The SD combined. We need to find 1 value, if we have two value = Insufficient.

Solve:  
 (1)  SD of the 2nd routine is 0.7
now we have two SD of the 2 routines.
but we don't know the average of each of them.
suppose we in 1 routine average of 9 and the other average of 2. so if we add them together we can't know for sure the SD combined.
let's consider this scenario:
e.g: 6 6 6 6 6 6 6 6 6 9    - we have same SD but when we combined them we have much bigger SD.
       2 2 2 2 2 2 2 2 2 5
but if we have exactly the same number somehow. -> we have SD equal to 0.7 combined.
 Insufficient.

(2)  Range of 2 routines were 3
As we know of SD. Range doesn't give us information about SD. Unless the range is 0 and then we can infer that the SD is 0.
but here we can't use this information to make the statement sufficient.
We can use the same example of statement (1) to prove it wrong:
e.g: 6 6 6 6 6 6 6 6 6 9    - we have same SD but when we combined them we have much bigger SD.
       2 2 2 2 2 2 2 2 2 5
but if we have exactly the same number somehow.
 Insufficient

(1)+(2) 
As we can see, clearly, it is insufficient too.
because we already combined our example to both statements (1) and (2).
so if we have the same example we can see that we can come up with 2 possible value of the combined SD with our given statements.
Therefore - >  Insufficient.    Our Answer is E

THE END  
I hope you liked the explanation, I have tried my best here.
Let me know if you have any questions about this question or my explanation.
   https://cdn.gmatclub.com/cdn/files/forum/images/smilies/1f44f-1f3fb.png

AthulSasi · Answer

Statement 1
If the standard deviations of both routines are the same (0.7) but we don't know the relation between the two sets of scores, we cannot directly calculate the standard deviation of the combined. The combined series spread can be anything. For eg is the C routine scores are around 10 and of F ruoutines scores are around 100, then the standard deviation of the combined series will be very big.

Statement 1
Similar to statment 1, we don't know the relation between the two sets of scores, we cannot directly calculate the standard deviation of the combined series. The combined series spread can be anything. For eg is the C routine scores are around 10 and of F ruoutines scores are around 100, then the standard deviation of the combined series will be very big.

Even with 1&2 put together we does not have enough idea about the combined series. There can be mutiple series of 10 numbers with range 3 and std deviation 0.7 (there is no need of citing examples here as the result is clearly evident)

Hence correct asnwer E

Lizaza · Answer

This one is pretty tricky. 
In order to find the standard deviation of the combined set, let's look at each of the points given.

(1) 
Theoretically, two sets with the same standard deviation of 0.7, should give the combined standard deviation of 0.7 as well. However, this is  only  the case if   the means are also identical   for these sets.
As we don't know this here, this is  insufficient.

(2) 
There's an interesting empirical rule for range:
A  very rough  (!) estimation of standard deviation is   range divided by 4.  
In our case, st.d. will be ~ 3/4 = 0.75
However, as you already see, this only gives us the same thing that point 1 - it's not very accurate (firstly), but it's around the same 0.7 st.d., which means without the average values we can't calculate the answer.
Thsi is  insufficient .

And finally, there isn't much sense in combining the conditions, because they are effectively fiving us the same thing from two different angles.
  The right answer is E.

GuadalupeAntonia · Answer

SD of Compulsory routine= .7
number of member in a team=10
SQUARE ROOT=(((sum(average of compulsory score-individual compulsory score))^2)/(10-1))=.7
(sum(average of compulsory score-individual compulsory score))^2=.49*9=4.41

(1) The standard deviation of the 10 scores for the free routine was also 0.7 points.
SD of Free routine= .7
number of member in a team=10
SQUARE ROOT=(((sum(average of free score-individual free score))^2)/(10-1))=.7
(sum(average of free score-individual free score))^2=.49*9=4.41

NOT SUFFICIENT, COULD BE THE CASE THAT BOTH ROUTINE SCORES BE THE SAME SO IT WILL BE .7 THE STANDAR DEVIATION OR COULD BE DIFFERENT NUMBER THAT GIVES THIS .7 EACH STANDAR DEVIATION THAT WILL AFECT THE 20 SCORES STANDAR DEVIATION.

(2) The range of scores for each of the compulsory routine and free routine was 3 points.

MAX-MIN=3
MAX= MIN +3

MIN _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ MIN + 3

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 6
3 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 3
3 3 3 3 3 3 3 3 3 3 3 6 6 6 6 6 6 6 6 6 6

3.5 3.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 .5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 6.5

NOT SUFFICIENT
we need to know the average of both compulsory and free scores.

ANS
LETTER  E

BGbogoss · Answer

At the Olympic Games, 10 gymnasts must perform two routines, compulsory and free, each gymnast receiving a separate score for each routine. If the standard deviation of the 10 scores for the compulsory routine was 0.7 points, what was the standard deviation of the 20 scores for both routines combined?

In order to compute the standar deviation of the 20 scores for both routines combined we need the value of the mean of the 20 scores combined which is derived from the value of the mean of each set and the standard deviation for the free routine.

(1) The standard deviation of the 10 scores for the free routine was also 0.7 points.
This gives us the standard deviation for the free routine but not the mean of any of the sets 
So statement (1) alone is not sufficient.
(2) The range of scores for each of the compulsory routine and free routine was 3 points.
This gives us the range of scores but not the mean of any set nor the standard deviation for the free routine. 
So statement (2) alone is not sufficient.

Statement (1) and (2) combined 
do not contain enough information for us to compute the combined standar deviation.

Answer : E

bomberjack · Answer

SD (C) = 0.7 , SD (C + F) = ?

(1) SD(F) = 0.7, Insuffiecient, the range of numbers could be increased or decreased and hence SD could be anything based upon that when we combine two series of numbers.

(2) Range(F) = 3, Range(C) = 3, Insufficient, the combined ranges could be very spread out or, smaller, based upon individual numbers.

Combinding both together, can't give a definitive answer for SD of combined list, due to different possible spreads of numbers.  Hence, (E) is the answer

Suboopc · Answer

To calculate the combined standard deviation, we need to mean of all 20 values and the deviations of each value from the mean.
Statement 1:
Since we cannot calculate the mean of all 20 scores, this information is not useful to calculate the standard deviation.
Depending on the value of combined mean, standard deviation of all 20 values will keep on changing.
Insufficient.

Statement 2:
There are estimates to calculate standard deviation from range. But in this case, we are given the individual ranges. So we have no clue about the combined range of all numbers and thus cannot calculate the standard deviation.
Insufficient.

Both together.
Even with this information the previous problems persist.

E

pranjalshah · Answer

Answer: E

Unless we don't know the values or the mean of the two sets, we cannot find out the Standard Deviation.

(1) The standard deviation of the 10 scores for the free routine was also 0.7 points.

We don't know the score distribution or the mean. Thus we cannot really find out the SD of the combined set.

For example, {4,4,4} has SD = 0. {5,5,5} has SD = 0, Combined set C1 is {4,4,4,5,5,5}
and, {4,4,4} has SD = 0, {6,6,6} has SD = 0, Combined set C2 is {4,4,4,6,6,6}.
From the above example we can clearly see that the SD of both combined sets are different. (mean of C1 is 4.5, mean of C2 is 5, from a quick glance we can say that SD of C1 is 0.5 and that of C2 is 1)

(2) The range of scores for each of the compulsory routine and free routine was 3 points.

Lets take small sets of 3 variables each. S1 = {3,4,6} and S2 = {5,6,8} Combined C1 = {3,4,5,6,6,8}
And S11 = {3,4,6} and S12 = {7,8,10} Combined C2 = {3,4,6,7,8,10}.
We see SD of C1 is not equal to SD of C2, C2 must be greater than C1. We don't know the values but it can be deduced from the way C1 and C2 are distributed.

(1) and (2) combined.

SD of S1 = 0.7, SD of S2 = 0.7 and range of S1 and S2 is 3 points.

Let's take smaller sets of 3 variables. S1 = {2,2,5} S2 = {3,3,6} Mean of S1 is 3 and S2 is 4. Thus we can see SD of S1 = S2.
C1 = {2,2,3,3,5,6} which has some SD, let's say SD1.

Now we take another set S3 = {2,2,5} S4 = {7,7,10}. SD of S3 = S4
C2 = {2,2,5,7,7,10}

We see from above that S1 and S2 satisfy both conditions and give C1 as the combined set. And S3 and S4 also satisfy both conditions and give C2 as the combined set. We can clearly see SD of C2 > SD of C1.

Thus, unless we don't know the actual values or mean, we cannot find out the SD.

smile2 · Answer

Given the 10 gymnasts perform two routines,  compulsory and free, each gymnast receiving a separate score for each routine, and the standard deviation of the 10 scores for the compulsory routine was 0.7 points.

From statement 1, we know the standard deviation of the 10 scores for the free routine was also 0.7 points.
If we compute the standard deviation of the 20 scores for both routines combined, we will get different values depending on the different data sets considered. 
Not sufficient.

From statement 2, the range of scores for each of the compulsory routine and free routine was 3 points
If we compute the standard deviation of the 20 scores for both routines combined, we will get different values depending on the different data sets considered.

We will still get different values depending on the different data sets considered when we combine both the statements. So it depends on which data set we consider.

Therefore, the right ans choice is E.

DG1989 · Answer

At the Olympic Games, 10 gymnasts must perform two routines, compulsory and free, each gymnast receiving a separate score for each routine. If the standard deviation of the 10 scores for the compulsory routine was 0.7 points, what was the standard deviation of the 20 scores for both routines combined?

(1) The standard deviation of the 10 scores for the free routine was also 0.7 points.
(2) The range of scores for each of the compulsory routine and free routine was 3 points.

(1) The standard deviation of the 10 scores for the free routine was also 0.7 points.
(2) The range of scores for each of the compulsory routine and free routine was 3 points.

Solution:  Given that,
 
 Each gymnast receives a separate score for each routine. 
 The standard deviation of the 10 scores for the compulsory routine is 0.7 points.

We need to find the standard deviation of 20 scores for both routines

Statement 1:    The standard deviation of the 10 scores for the free routine was also 0.7 points 
This means that the variability of scores for the free routine is the same as that for the compulsory routine. However, knowing the standard deviation of each routine separately does not tell us anything about the combined standard deviation, as the combined standard deviation also depends on the covariance between the scores of the two routines.
  INSUFFICIENT

Statement 2:    The range of scores for each of the compulsory routine and free routine was 3 points 
The range gives us the spread of scores but does not provide enough information about the standard deviation. Knowing the range alone does not allow us to determine the combined standard deviation.
  INSUFFICIENT

Combining statements 1 and 2  
If the two data sets are independent and identically distributed, their combined standard deviation will be the same as the individual standard deviations. If the data sets are dependent or have different distributions, the combined standard deviation can be different from the individual standard deviations. Since from combined statements, we cannot determine the distribution of two routines and their covariance, we cannot determine the SD of 20 scores for the two routines.
  INSUFFICIENT

The correct answer is Option E.

captain0612 · Answer

Choice E

Given:
1. 10 gymnasts performing 2 routines: Compulsory and Free
and each routine is separately rated.
2. SD (standard deviation) for the 10 compulsory routine is 0.7

Question: What is the combined standard deviation for the 20 scores (Compulsory and Free)

Statement 1:

Standard Deviation of 10 scores for the free routine = 0.7

1. Even though the SD are same, the arrangement of these numbers/ data could be different. And hence, the combined SD is different.
2. With same SD, the arrangement of these numbers/ data could be same. And hence, the combined SD is same as the individual SD

Hence, we cannot find the combined SD.  Insufficient

After a lengthy calculation,  I found out that SD could be different and same  (by taking simple values of different sets with similar standard deviation):
   Not very sure whether this is the correct approach   
 Case 1:  Set 1 = {1, 2} and Set 2 = {3, 4}

SD 1 = 0.5 and SD 2 = 0.5, but combined SD ~ 1.1

Case 2:  Set 1 = {1, 2} and Set 2 = {1, 2}

SD 1 = 0.5 and SD 2 = 0.5, but combined SD = 0.5. Tried the same approach with other sets and no correlation

Hence, we cannot solve it.  InSufficient

Statement 2:

Range of the scores of free routine = Range of the scores of Compulsory routine

based on above, the 2 set's range is same, but the arrangement of the values could be different, and hence their individual Standard Deviations.

Therefore, we need the SD of the Free routine scores as well to compute the combined Standard Deviation.  InSufficient

Statement 1 and Statemetn 2:

Same reasoning as Statement 1.  Insufficient

Milkyway_28 · Answer

It is important to understand what Standard Deviation is and how to calculate it in order to answer this question. To calculate Standard Deviation, you must find the average of the sum of the squares of the deviations, which gives you the variance; and then calculate the square root of the variance.

(1) "The standard deviation of the 10 scores for the free routine was also 0.7 points."

This is the first statement we are given. The problem with this is that Standard Deviation refers to the deviations from the mean. So you can have the same deviations from mean X for the free routine, and have the same deviations from mean Y for the compulsory routine. However, when you add the scores from these two routines, you might have a completely different mean. This means that all of the deviations will now be different.

(2) "The range of scores for each of the compulsory routine and free routine was 3 points."

In this statement we are told that the range of scores for each of the routines was 3 points. Again, this does not say much because we do not know what the averages are. The scores 4, 4, 4, 4, 4, 4, 4, 4, 4, 7 and 7, 7, 7, 4, 5 , 4, 4, 4, 6, 6 will have different means but both scores have the same range. Morevover, once we add the two scores together there could be a completely different mean and potentially a different Standard Deviation.

Combining the two statements also does not help much. Here is an example to prove my point.

Lets say you have two sets of two scores

Score 1: 1,4
Score 2: 3,6

The Standard Deviation of both scores is 1.5. Their range is also 3 points each.

If you were to add the scores you would have to calculate the Standard Deviation of 1,3,4,6 which would give you a value of approximately 1.8.

The two statements together are not sufficient. Answer is E

mvgkannan · Answer

If we need to combine two SD, we need to understand the definition of SD,  Standard deviation is  a statistical measure that describes the amount of variation or dispersion of a set of values around its  mean (average).   So the two separate sets of SD data whose knowledge of mean is not available cannot be combined into single set of SD data. Statement (1) & (2) gives no idea about the mean. So Option (E).

Keshav1404 · Answer

Hi  Bunuel

I have one query is there any situation in which the combined range of both score will be adequate enough to evaluate or Find the standard deviation of both the score combined?

Bunuel · Answer

For example, if statement (2) were:

"Each gymnast received the same score on the free routine as on the compulsory routine."

Then the 20 scores would simply be the 10 compulsory scores listed twice. Duplicating every value does not change the standard deviation, so the standard deviation of all 20 scores would still be 0.7, and the answer would be C.

GMAT Club Olympics 2024 (Day 7): At the Olympic Games, 10 gymnasts

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