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18 Dec 2024, 09:00
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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.
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.
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18 Dec 2024, 10:00
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Bunuel
GMAT Club's Official Explanation:
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?
Let the lowest grade be g. The distribution of grades is as follows:
- 3 comic books received g.
- 5 comic books received g + 5.
- 4 comic books received g + 10.
- x comic books received g + 15.
(1) x = 2
With x = 2, the distribution becomes:
{g, g, g, g + 5, g + 5, g + 5, g + 5, g + 5, g + 10, g + 10, g + 10, g + 10, g + 15, g + 15}.
An important property of standard deviation is that adding or subtracting the same constant to every value in a list does not change the standard deviation.
Subtracting g from each term gives:
{0, 0, 0, 5, 5, 5, 5, 5, 10, 10, 10, 10, 15, 15}.
The standard deviation can now be calculated directly using this adjusted list. Sufficient.
(2) The lowest grade assigned to the comic books was 35.
If the lowest grade is 35, then the grades are 35, 40, 45, and 50. However, without knowing the value of x, the number of comic books that received the grade 50, the distribution is incomplete. Different values of x will result in different standard deviations. Not sufficient.
Answer: A.
General Discussion
18 Dec 2024, 09:35
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Let lowest grade be g, then,
To calculate standard deviation we need to know the mean of all these numbers and then calculate the distance of these numbers from the mean
Statement 1 : x = 2
This gives us total number of books = 14, but doesn't allow us to calculate the mean as we don't know the value of g.
Insufficient.
Statement 2 : g = 35
Just knowing g, but not the total number of books will not allow us to calculate the mean.
Insufficient.
But combining both these statements will give us values for both the unknown variables and allow us to calculate the mean and hence the resulting standard deviation, so both together should be sufficient.
Answer: C
| Grade | Number of books |
| g | 3 |
| g + 5 | 5 |
| g + 10 | 4 |
| g + 15 | x |
To calculate standard deviation we need to know the mean of all these numbers and then calculate the distance of these numbers from the mean
Statement 1 : x = 2
This gives us total number of books = 14, but doesn't allow us to calculate the mean as we don't know the value of g.
Insufficient.
Statement 2 : g = 35
Just knowing g, but not the total number of books will not allow us to calculate the mean.
Insufficient.
But combining both these statements will give us values for both the unknown variables and allow us to calculate the mean and hence the resulting standard deviation, so both together should be sufficient.
Answer: C
