Analyzing AP Calculus Exam Results
Question: Below is the frequency distribution of exam
results for all people who took the Advanced Placement AB Calculus Exam in
2015. What is the expected grade, the
variance of the grade and the skew of the grade on the exam?
Make some comments on the PDF of Advanced Placement Exam
Results
Grade Frequency on
AP AB Calculus Exam


Grade

%

5

24.40%

4

17.40%

3

17.60%

2

9.70%

1

30.90%

Analysis:
The expected grade is the sum of the products of the grade
and the corresponding probability. This
can be calculated with the SUMPRODUCT command in Excel.
If grades are in cells a1 to at and frequencies are in cells
b1 to b5 just go
SUMPRODUCT(a1:a5,b1:b5) and get 2.947.
I use the definition of the variance E(XE(x))^{2}
to calculate the variance. (I could
have used the computational formula but instead chose to use the
definition.) Calculation is laid out in
chart below.
Calculation of the
Variance of
AP Exam Results


Grade

%

(XE(X))^{2}

5

24.40%

4.21

4

17.40%

1.11

3

17.60%

0.00

2

9.70%

0.90

1

30.90%

3.79

2.947
is Expected
Grade

10.01

The skewnees of the random variable X is defined as E(XE(X))^{3}/SD^{3}
(Sorry I don’t know how to type the Greek parameter for random
variable standard deviation so I use symbol SD.)
My estimate of the variance is 10.01. The standard deviation is
I calculate the sum of (XE(X))^{3} for all X
(meaning grades 1 to 5). I get 1.591. I divide by the standard deviation is 3.16.
The Skewness is 0.502, slightly positive.
This is sort of a strange PDF. It is not normally distributed because the
mode the most likely outcome is a 1.0 failure.
The second highest likely outcome is a 5.0 or an A. The center category the CI is smaller than
the two extremes.
So a lot of students are highly prepared for this AP exam
and even more students bomb the
test.
The data appears to be bimodal. It almost seems as though we have two
different groups taking this exam. There
is no center in either AP exam results or American politics.
Readers may also be interested in my previous post comparing AP exam results to batting averages.
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