## Thursday, December 1, 2016

### Analyzing AP Calculus Exam Results

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?

 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(X-E(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 % (X-E(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(X-E(X))3/SD3

(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 (X-E(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.