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

1p

(1p)2

1

p*(1p)

2*p*1p)

2

p2

p2

Total

1

1

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(OE(O))^{3/}Var(O)^{1.5}
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(XE(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
Number
of Made Free Throws
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(XE(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.
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