M45-05

Question

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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Official Answer

A

(1)  x = 2 
 
(2) The lowest grade assigned to the comic books was 35.

Bunuel · Answer

Official Solution:

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

benbil · Answer

This is a great question that’s helpful for learning.

adarsh2209 · Answer

I like the solution - it’s helpful.

Robo_123 · Answer

I did not quite understand the solution. Why will standard dev change for different values of x, when the grades will still be consecutive?

Bunuel · Answer

Even though the grades themselves (35, 40, 45, 50) remain consecutive, the  distribution  of how many comics got each grade changes with different x. Standard deviation depends not only on the values but also on how frequently each value occurs. So if more comics get the highest grade, the spread increases; if fewer get it, the spread decreases.

For example:

If x = 1, the scores will be {35, 35, 35, 40, 40, 40, 40, 40, 45, 45, 45, 45, 50}, which has the standard deviation of about 4.5

If x = 10, the score will be {35, 35, 35, 40, 40, 40, 40, 40, 45, 45, 45, 45, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50}, which has the standard deviation of about 5.5

M45-05

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