Question

Three students were applying to the same graduate school. They came from schools with different grading systems.

Student GPA Scool Average GPA School Standard Deviation
Thuy 2.6 3.2 0.8
Vichet 82 75 20
Kamala 8.5 8 0.4

Which student had the best GPA when compared to his school? Explain how you determined your answer.

Answer 1

Answer:

$z=\frac{8.5-8}{0.4}=1.25$

Kamala had the best GPA compared to the other students at his school, since his GPA is 1.25 standard deviations above his school's average.

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution for the problem

Let X the random variable that represent the scores of a population, and for this case we can assume that the distribution for X is normal

And the best way to solve this problem is using the normal standard distribution and the z score given by:

$z=\frac{x-\mu}{\sigma}$

The case on which we have a z score better would represent the best GPA respect to the averag school.

Thuy case

$z=\frac{2.6-3.2}{0.8}=-0.75$

That means 0.75 deviations below the school mean

Vichet case

$z=\frac{82-75}{20}=0.35$

That means 0.35 deviations above the school mean

Kamala case

$z=\frac{8.5-8}{0.4}=1.25$

That means 1.25 deviations above the school mean

So for this case as we can see the higher z score is for the Kamala case, and would be on a higher percentile compared to the other groups.

Kamala had the best GPA compared to the other students at his school, since his GPA is 1.25 standard deviations above his school's average.