12 Days of Christmas GMAT Competition - Day 3: Leonard had each comic

(1) x = 2
Standard deviation is to calculate the spread between each item and the mean. If we know the relative position of each item vs. the mean, we would be able to calculate the STD.

(2) The lowest grade assigned to the comic books was 35.
Since we don't know x, we can't calculate STD

3 -> Lowest, 4 -> 2nd Lowest, 5 -> 2nd Highest, x -> Highest
Grades are consecutive multiples of 5 between 0 to 100 (inclusive)

(A) is the correct answer.

We need only one thing to find the S.D. The number of comic books that got graded by the professional service. Usually, the grades would help as well but in case the values are consecutive multiple of 5, the Standard deviation will be the same no matter what.

(1) x = 2 (2 comics got the highest grade)
Total comics = 14

We can assume the rating to be any value between 0 and 100 that are consecutive multiples of 5 and the Standard Deviation will remain the same. The value that we would get upon solving is 4.8

Statement (1) is sufficient.

(2) Lowest rating is 35. This means that the ratings were 35, 40, 45, and 50.

But we still do not know the number of comics that have been given the highest grade. Since we do not know the total number of comics, we cannot determine the Standard deviation.

Statement (2) is not sufficient.

Let be grades be a, b, c, d where d>c>b>a
So for n=3, grades = a
n = 5, grades = b
n = 4, grades = c
n = x, grades = d

i) x = 2
Since, we need SD & we know that the grades are consecutive multiples of 5 - we can take any multiples as SD doesn't change when a constant is multiplied with the set (i.e. grades in this case)
Let a,b,c,d be 5,10,15,20 & since we have the frequency (i.e. the number of books) as 3, 5, 4, 2 respectively; SD can be found

Sufficient

ii) This only provides us with the lowest grade but not with x. Without that, we cannot find the SD

Not sufficient

IMO A

Leonard had each comic book in his collection graded by a professional service. Each comic book received one of four different grades. 3 comic books received the lowest grade, 5 comic books received the second-lowest grade, 4 comic books received the second-highest grade, and x comic books received the highest grade. If the four grades were consecutive multiples of 5 between 0 and 100, inclusive, what was the standard deviation of the grades of Leonard's comic books?
(1) x = 2
(2) The lowest grade assigned to the comic books was 35.

To find the standard deviation (SD) of some values, we need to know how the values are spaced and how many elements. From the question, we already know the following,
Lowest grade = 3 values = Let assume---> 20, 20, 20
Second-lowest = 5 values ----------------> 25, 25, 25, 25, 25
Second-highest = 4 values ----------------> 30, 30, 30, 30
Highest = x values -------------------------> x number of 35s.

As spacing is always same, we just need x value alone to find the SD. With this idea let's look into the statement.

Imp. note: Only spacing is required, not actual values. For e.g.: SD of 1,2,3,4,5 is equal to SD of 100,101,102,103,104.

St 1: SUFFICIENT. A and D are possible answers.
St 2: NOT SUFFICIENT. Because we don't need the actual values. So answer is A.

Question: what was the standard deviation of the grades of Leonard's comic books?

Standard deviation = E(x-xbar)/Ef

X is each item in the observation

X-bar is the mean

E is the Greek letter for summation

f is the frequency of each item

Ef is the number of observation


Each item is listed thus with their frequency

Comic book with the lowest grade- 3

Comic book with the second lowest grade- 5

Comic book with the second highest grade- 4

Comic book with the highest grade- x

The four grades were consecutive multiples of 5 between 0 and 100

Statement 1: x=2

the grades are unknown, although the frequency of the highest grade has been provided

Statement 1 is insufficient

Eliminating option A and D


Statement 2: The lowest grade assigned to the comic books was 35.

This provides us information on the values of all 4 grades

It would be the array 35,40,45, 50

However, the frequency of the highest grade is unknown

Statement 2 is insufficient, eliminating option B


Together, we have information on all the frequency of the grades and the figures of each grade. Hence, together the information is sufficient to get the standard deviation of the grades of the comic book

Tricky wording but when you pay attention it becomes easy to deduce.

Statement 1 does not give consecutive value for a multiple of 5 so we can not deduce standard deviation.

Statement 2 does not tell the value of x so again, it can not solve for standard deviation.

But when we combine the statements we get x=2 and consecutive values of multiples of 5. i.e. 35,40,45,50

Therefore we can solve this by combining both statements.

Hi everyone

This is SD question.
we know there are 4 grades and we know how many books in each grade.
In order to calculate the SD we need "x" and the scores of the grades.

(1) x=2
it's not enough for us to determine the SD.
the grades could be: 5,10,15,20 and also 10,40,80,90
Insufficient

(2) the lowest score is 35
so the grades are: 35,40,45,50
still we don't know how many books got the 50 score so we can't calculate the SD.
Insufficient

(1)+(2)
x=2 and the grades are: 35,40,45,50.
we can calculate the SD now.
Sufficient- Answer is C.

No. of books	Grades
3	lowest
5	2nd lowest
4	2nd highest
x	highest

Q: What was the standard deviation of the grades?

Statement 1:
Given: x = 2 - This is insufficient as the grades could be of any value between 0 and 100

Statement 2:
Given: The lowest grade assigned to the comic books was 35. We can determine the grades but we don't know the value of x - Insufficient

Combining Statements 1 and 2: We know both x and the grades, so we can calculate standard deviation - SUFFICIENT

No. of books	Grades	Grades
3	lowest	35
5	2nd lowest	40
4	2nd highest	45
2	highest	50

Ans: C. Both statements together are sufficient, but neither statement alone is sufficient

We know that each comic book received a grade and the grades are consecutive multiples of 5 between 0 to 100
Additionally, it is mentioned that:
x comic books received the highest grade
4 comic books received the 2nd highest grade
5 comic books received the 2nd lowest grade
3 comic books received the lowest grade

Since we have to calculate the standard deviation, we have to know the mean and atleast one of the grades since between 0 to 100 there are 20 multiples of 5

Statement 1 mentions the value of x which is one of the factors needed to calculate the mean grade but we cannot calculate the mean as nothing is given about the grades

Statement 2 mentions that the lowest grade is 35 i.e., the grades are 35,40,45 and 50 going from the lowest to the highest but since we do not know how many books received the highest grade, we cannot calculate the mean

Combining both statements will help us calculate the mean and we will have all values necessary to calculate the deviation.
Therefore both statements together are sufficient, but neither statement alone is sufficient

Standard Deviation requires how far each term is from each other , or the relative distance of the terms from the mean.

1) The difference between the each term is constant. For example the difference between the lowest and the highest term is always 15, no matter what the values are.

Let's the lowest grade = p

Lowest Grade = p

Second Lowest Grade = p + 5

Second Highest Grade = p + 10

Highest Grade = p + 15

The four grades are in AP. We just need to know the number of terms in order to find the standard deviation. As x is given, this statement is sufficient to find the answer.

Eliminate B, C and E.

2) The lowest grade assigned to the comic books was 35.

Knowing the lowest grade doesn't help as we need to know the number of terms as well. Hence, statement 2 alone is not sufficient.

Option A

Let's consider the first statement:

x=2 ... ok but I don't know the grades of the book, I know they are consecutive multiples of 5, but this information alone will lead me to different standard deviations.

Hence Statement 1 alone is not sufficient.

The second one neither is sufficient. Now I know the grades, but I don't know how many books will receive the highest one, and that is a piece of important information to the std dev. calculation.

Hence Statement 2 alone is not sufficient.

If I combine the two, however, I will be 100% sure to be able to calculate the std dev without any problem.

Hence Option C is correct IMHO.

Standard deviation = Root of Mean of Squares of differences between average grade and each grade

Total number of books = 12 + x

We will need to know the grades to be able to find the average and standard deviation

Neither Statement 1 or 2 tell us the actual grades (OPTION E)

To answer the question we just need the total number of books which we can find from statement 1 hence choice A. For the grades we can use either set as long as the difference is 5 across all the grades

Step 1: Understanding the problem
There are four grades:
G1 ,G2 ,G3, G4
, where
G1 <G2 <G3 <G4
These grades are consecutive multiples of 5:
G1 ,G2 =G1 +5,G3 =G1 +10,G4 =G1 +15.
The number of comic books assigned to each grade is as follows:
3 comic books received G1
5 comic books received G2
4 comic books received G3
x comic books received G4
The formula for the standard deviation is:
σ= ∑fi(Gi −μ)^2 / N
,
where:
fi : frequency of grade
Gi: grade
μ: mean of the grades,
N: total number of comic books.

Step 2: Analyze each statement.
Statement (1):
x=2
If x=2, the total number of comic books is:
N=3+5+4+2=14.
However, the values of the grades
G1 ,G2 ,G3 , G4 are not given. Without knowing the specific grades, we cannot calculate the mean (μ) or the deviations (Gi−μ).

Thus, Statement (1) is insufficient.

Statement (2): The lowest grade assigned to the comic books was G1=35.
From this, the grades are:
G1=35, G2 =40, G3=45, G4 =50.
However, the value of x (the number of comic books that received G4 ) is not given.
Without x, we cannot determine the total number of comic books N or compute the mean (μ).

Thus, Statement (2) is insufficient.

Step 3: Combine Statements (1) and (2).
From Statement (2), the grades are
35,40,45, and 50.

From Statement (1),
x=2, so the frequencies are:
f1=3,f2=5,f3=4 ,f4 =2.
The total number of comic books is:N=3+5+4+2=14.

The mean (μ) is:
μ= {∑fi * Gi}/N = {(3×35)+(5×40)+(4×45)+(2×50)}/14
​
μ= (105+200+180+100)/14 = 585/14
μ =41.7857(approximately).

The variance (σ^2) is:
σ^2 = {∑ fi (Gi −μ)^2} / N

Substitute
Gi , fi , and μ to compute σ^2
, then take the square root to find σ.
Both statements together provide all necessary information to calculate the standard deviation.

Final Answer: C (Both statements together are sufficient).

Let the grades be a, a+5, a + 10, a + 15
No of comic books with each grade be 3,5,4,x

1-> x =2

No of comic books with each grade be 3,5,4,2

2-> The lowest grade assigned to the comic books was 35

a = 35 => Grades = 35,40,45,50

Now, both statement alone not enough,
But together we have the no of books and the grades assigned which allows us to get the standard deviation

C. both statements together are sufficient, but neither alone are enough

Statement (1):
Given x=2,
the grades are assigned as 3 books at a, 5 at a+5, 4 at a+10, and 2 at a+15.

This gives one equation but insufficient information to calculate the standard deviation.

Statement (2):
If the lowest grade is 35,
the grades are 35, 40, 45, and 50.
We can't calculate standard deviation without x.


Combining Statements:
With a=35 and x=2,
we calculate the total sum of the grades, the mean, and the standard deviations.

Thus, both statements together are sufficient.

That's why they say "Be very mindful of carrying your knowledge of statement 1 to statement 2 in DS" you will end up choosing the wrong answer..