## Sunday, March 5, 2017

### The shape of 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.

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%