The Shape of Grade Distributions
Below are three
problems and an answer to one of the problems on the shape of grade
distributions for different AP exams. I
have included the answer for problem three on the shape of the AP government
and the AP World History exam grade distributions. These problems are a good way to teach the
concepts of expected value, variance, skew, and kurtosis of a distribution.
The data behind the
problems are from the 2016 and 2009 grade results from different AP subject
exams. The implicit assumption used in
these problems is that the estimated proportions of getting a grade (1, 2, 3,
4, and 5) are the probabilities of getting each grade.
I have provided the
answer to question one. It would be
nice if readers provided answers to questions two and three.
Question One: Below are the score results for the
Government and World History AP Exams in 2016.
What are the expected value, median, variance, skew and
kurtosis of the government and politics and world history grade distribution
functions?
Discuss the meaning of these statistics.
Government &
Politics

World History


Grade

# Students

% Students

# Students

% Students

5

4,519

0.205

18,953

0.066

4

4,663

0.212

44,424

0.156

3

4,493

0.204

83,774

0.294

2

4,728

0.215

81,877

0.287

1

3,598

0.164

56,323

0.197

Total

22,001

1.000

285,351

1.000

Note: These articles may help on the skew and
kurtosis concept.
Answer to Question
One:
The calculations for the two grade PDFs are presented below.
Statistics for
Government and World History Grade PDFs


Government &
Politics

World History


Expected Value

3.08

2.61

Median

3

3

Variance

1.89

1.34

Skew

0.049

0.996

Kurtosis

1.248

0.713

Observations:
Expected grade for Government and Politics is higher than
expected grade for World History.
Variance of grade for G&P is higher than variance of
grade for WH.
Median grades for the two courses are identical. (To get the median of the two grade PDFs
just observer where 50% of observations are below the grade.)
The negative skew is greater for world history than for
government and politics. This is
occurring because of the very small likelihood of getting a 5 on the WH exam. Note also that for government and politics
the mean is actually slightly greater than the median. There are however, three grade groups below
the mean for the G&P exam, causing the slight negative skew,
The kurtosis of government and politics is more negative
than the kurtosis of world history. This
is because the government and politics grade PDF has two large tails while the
world history grade pdf has one large tail.
Note: The discussion of skew and kurtosis in this
article has direct relevance to this problem.
I agree with the author’s view that kurtosis often is a better measure
of what is happening in the tails of the distribution than in the middle of the
distribution.
The AP exam results for multiple years can be the source of
a lot of statistics problems. Go here
for additional data.
Below are two more questions.
Question Two: Below are the score results for the Calculus
AB exam in 2009 and 2016.
Describe how the shape of the grade distributions for the
Calculus AB exam changed in the two years?
Calculate the mean, variance, skew, and kurtosis of the
grade distributions for the two years.
Based on the change in these statistics what can we say
about the change in the shape of the grade distributions?
Distribution of Calculus
AB Exam Scores in 2009 and 2016


2009

2016


Grade

# Students

% Students

# Students

% Students

N

% At

N

% At


5

52,921

23.0%

76,486

24.8%

4

43,140

18.7%

53,467

17.3%

3

41,204

17.9%

53,533

17.4%

2

35,843

15.5%

30,017

9.7%

1

57,480

24.9%

94,712

30.7%

230,588

100.0%

308,215

100.0%

For help on question one and two look at the following
document.
Question Three: Below are the score results for Calculus AB
and Calculus BC in 2016.
Describe differences in the shape of the two grade
distributions.
Calculate the mean, variance, skew and kurtosis of the grade
distributions for the two math exams.
Based on these statistics what can we say about differences
in the population of people taking the Calculus AB exam and the population of
people taking the Calculus BC exam?
Distribution of
Calculus AB and Calculus BC Grades in 2016


Calculus AB

Calculus BC


Grade

# Students

% Students

# Students

% Students

5

76,486

24.8%

60,632

48.5%

4

53,467

17.3%

19,191

15.4%

3

53,533

17.4%

21,441

17.2%

2

30,017

9.7%

7,212

5.8%

1

94,712

30.7%

16,455

13.2%

Total

308,215

100.0%

124,931

100.0%

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