Question

Simpson's Paradox, Derek -vs- David: Averaging across categories can be misleading but this can be resolved with weighted averages.

In baseball, the batting average is defined as the number of hits divided by the number of times at bat. Below is a table for the batting average for two different players for two different years.

The number in parentheses gives the number of times at bat for each player for each year.

	Batting Average (# of times at bat)
	1995	1996
Derek	0.249 (45 times at bat)	0.313 (575 times at bat)
David	0.252 (415 times at bat)	0.322 (145 times at bat)
(a) What are the averages of the two batting averages for Derek(xDerek) and David(xDavid)? Do NOT use a weighted average, just take the mean of 1995 and 1996 batting averages. Round your answers to 3 decimal places. xDerek = xDavid = (b) Who had the higher average batting average using the non-weighted average?      (c) Using a weighted average, calculate the average batting averages for Derek(xDerek)and David (xDavid). Round your answers to 3 decimal places. xDerek= xDavid= (d) Who had the higher average batting average using the weighted average?      (e) What caused the discrepancy in average batting averages? Derek's higher average occurred with more times at bat (575). David's higher average occurred with fewer times at bat (145).      Derek's lower batting average was based on a small number of times at bat (45). All of these contributed to the discrepancy.

Answer 1

	1995			1996
Derek	0.249	45		0.313	575	620
David	0.252	415		0.322	145	560
a)	average	Derek	0.281
		David	0.287
b)	David has higher average
c)	Weigthed average		Derek	0.308355
			David	0.270125
d)
	Derek has higher average using the weighted average

	1995			1996
Derek	0.249	45		0.313	575	=SUM(C4,F4)
David	0.252	415		0.322	145	=SUM(C5,F5)
a)	average	Derek	=AVERAGE(B4,E4)
		David	=AVERAGE(B5,E5)
b)	David has higher average
c)	Weigthed average		Derek	=(B4*C4+E4*F4)/G4
			David	=(B5*C5+E5*F5)/G5
d)
	Derek has higher average using the weighted average