The (Out) Law of Averages

A couple of years ago, as part of my statistics course, we were given an article on statistical fallacies in sports. We were then asked to come up with a different statistical fallacy we had found in a sport. It wasn't too difficult to find one. Here's what I wrote and the statistical evidence to back it up.

One of the biggest clichés in the sporting world is the Law of Averages. While there is a real theorem that a random variable will reflect its underlying probability over a large sample (The Law of Large Numbers), the law of averages typically assumes that unnatural short-term "balance" will occur. Television commentators, including former cricketers often claim a batsman is “due” as he hasn’t scored a “big one” for a while.

In order to test this “Law of Averages”, the stats of the 15 batsmen with the most number of innings were studied (Consider it roughly equivalent to the number of plate appearances in baseball). The reason for choosing these 15 was that they provided the largest samples for study. I calculated the number of innings each of them took to score a hundred and used it as a benchmark to test the law of average, rounding it up to N. For example, Allan Border scored 27 100’s in 267 innings, which roughly came out to be a hundred every 9.9 innings. Hence, according to the law of averages, if Border hasn’t scored a 100 in 9 consecutive innings, his tenth (Nth) inning should be a 100. I then calculated the probability of this event using the formula:

P’ = N₂/(N₁+ N₂), where

P’ = Probability of scoring a 100 in n^th inning after (N-1) consecutive scores of less than 100

N₁=No. of instances when batsman didn’t score a 100 in ‘N’ consecutive innings.

N₂=No. of instances when batsman scored a 100 after ‘N-1’ consecutive scores of less than 100.

The P-Factor is a measure of how frequently a batsman scored when he was “due”. A P-Factor of more than 1 indicates a batsman delivered more frequently when he was “due” as compared to his career record. Table 1 shows the analysis of the sample. Of the 15 batsmen sampled, 12 have a P-Factor of less than 1.1, 7 of which have a P-Factor less than 1! Seems to me that even some of the best batsmen don’t fare any better when they were “due”, isn’t it?

Player	Innings	100's	Innings/100's	N	N1	N2	P	P'	P-Factor
Allan Border	267	27	9.889	10	87	9	0.101	0.094	0.931
Steve Waugh	260	32	8.125	9	92	8	0.123	0.08	0.65
Sachin Tendulkar	237	39	6.077	7	65	13	0.165	0.167	1.012
Alec Stewart	235	15	15.667	16	86	6	0.064	0.065	1.016
Brian Lara	232	34	6.824	7	69	14	0.147	0.169	1.15
Graham Gooch	219	20	10.95	11	63	7	0.091	0.1	1.099
Sunil Gavaskar	214	34	6.294	7	58	11	0.159	0.159	1
Michael Atherton	212	16	13.25	14	62	4	0.075	0.061	0.813
Mark Waugh	209	20	10.45	11	46	8	0.096	0.148	1.542
Rahul Dravid	205	24	8.542	9	64	9	0.117	0.123	1.051
David Gower	204	18	11.333	12	69	6	0.088	0.08	0.909
Desmond Haynes	202	18	11.222	12	62	6	0.089	0.088	0.989
Inzamam-ul-Haq	200	24	8.333	9	49	6	0.12	0.109	0.908
Jacques Kallis	194	29	6.69	7	68	9	0.149	0.117	0.785
Geoffrey Boycott	193	22	8.773	9	55	8	0.114	0.127	1.114