# Lee looked at the result of an exam he had given to his grade 2 class. He saw that Lousie had a score of 95%. Also, the standard deviation for the test was 0. If Arif was also in his grade 2 math...

Lee looked at the result of an exam he had given to his grade 2 class. He saw that Lousie had a score of 95%. Also, the standard deviation for the test was 0. If Arif was also in his grade 2 math class, what marks did Arif get?

mlehuzzah | Certified Educator

Intuitively, standard deviation measures how far apart the scores were.  The only way the standard deviation can be zero is if every score is the same.  So Arif also scored 95%.

More mathematically, think of the formula for standard deviation:

sqrt{[(x1-m)^2 + (x2-m)^2 + ... + (xn-m)^2]/n}

where m is the mean
n is the number of exams

(x1-m)^2 = 0 if x1=m
(x1-m)^2 > 0 if x1 doesn't = m

So, if person #1 scored anything other than the mean, then (x1-m)^2 will be positive.

Same with person 2:  if person #2 scored anything other than the mean (higher or lower) then (x2-m)^2 will be positive.

So the formula adds up a bunch of positive numbers:
(x1-m)^2 + (x2-m)^2 + ...

then divides by n, a positive number
then takes the square root, still giving a positive number

The only way to get 0 is if x1=m and x2=m and x3=m and ...
So everyone gets the exact same score (which is also the mean).

(side note: the formula for "sample standard deviation" instead of "standard deviation of the sample" is slightly different.  you divide by n-1 instead of n.  For this question, it doesn't actually matter: to get a standard deviation of 0, you still need all the scores to be the same)

justaguide | Certified Educator

When Lee saw the result of an exam he had given to his grade 2 class, Louise's score was 95%. Also, the standard deviation for the test was 0. As the standard deviation is 0, the square root of the sum of the square of the difference between all the marks and 95% was equal to 0.

It is not possible to estimate the marks of Arif based only on this information.