## Saturday, May 18, 2019

This post analyzes the tail size and skewness of outcomes for a basketball player shooting fouls while in the single bonus and the double bonus.   It is a good problem to help students understand the concept of skewness.

Question:   The chart below has the probability distribution of free throw makes for a player shooting free throws in the single bonus and in the double bonus.   The symbol p denotes the probability the player makes a free throw on a single shot.

 Number of Made Free Throws in Single and Double Bonus Number of Made Free Throws Single Bonus Double  Bonus 0 1-p (1-p)2 1 p*(1-p) 2*p*1-p) 2 p2 p2 Total 1 1

What is the skewness in the single bonus and in the double bonus for a player with a 90 percent free throw success rate?

Set up a spreadsheet so that you can input the probability of making a free throw and instantly obtain the skewness statistic.

Formula For Skewness:  The formula for skewness is

Skewness = E(O-E(O))3/Var(O)1.5

Here is the spreadsheet calculating Skewness of free throw outcomes for a 90 percent free throw shooter in the single bonus.

Single Bonus Skewness Calculation:

 Free throw success probability 0.9 Number of Made Free Throws Single Bonus  Outcome Probability Estimate Squared Deviation From Expected Value Cubed Deviation from Expected Value 0 0.1 2.9241 -5.0002 1 0.09 0.5041 -0.3579 2 0.81 0.0841 0.0244 Total 1 Expected Value 1.71 Variance 0.6723 Standard Deviation 0.8199 "E(X-E(X))3 -0.3091 Skewness -0.5608

Here is the spreadsheet calculating skewness of free throw outcomes for a 90 percent free throw shooter in the double bonus.

Double Bonus Skewness Calculation

Free Throw Success Probability
0.9
Single Bonus  Outcome Probability Estimate
Squared Deviation From Expected Value
Cubed Deviation from Expected Value
0
0.01
3.2400
-5.8320
1
0.18
0.6400
-0.5120
2
0.81
0.0400
0.0080
Total
1
Expected Value
1.8
Variance
0.7200
Standard Deviation
0.8485
E(X-E(X))3
-0.4960
Skewness
-0.8119

A final comment:   The skewness statistic presented here is sort of strange.   Its value is determined more by the value of outcomes than by the probability of each outcome.  I am often actually more interested in the percent of observations in each tail than in the value of observations in the tails.   More on this point will eventually follow.