ratio of standard deviations (RSD), skew, and kurtosis of the six different panels of winning percentages. RSD, founded by Noll (1988) and Scully (1989), and more rigorously established in Quirk & Fort (1992) and Fort & Quirk (1995) can be visualized as follows. Let ASD be the actual sample standard deviation. Let ISD be the “idealized” standard deviation of a league where the probability that any team in the league beats any other is 0.5. For the binomial without ties, Fort and Quirk (1995) show that , where G is the number of games in the season. Then RSD = ASD/ISD. As , the league is as balanced as the league with equal win probability in all games. The larger is , the less competitively balanced the league becomes. The third moment, skew, measures asymmetry of the distribution and its measurement can identify which side of win distribution is a primarily responsible for competitive imbalance. Negative skew (longer left tail) would suggest imbalance is primarily due to weaker teams. Positive skew (longer right tail) would suggest imbalance is primarily due to stronger teams. Kurtosis is the fourth moment and measures both whether the distribution has a high peak and excess observatins in the tail or a low peak and deficiency in the tail. The behavior in the tails has been used in past analysis of competitive balance, primarily in terms of “tail likelihood” (Fort and Quirk, 1995) or excess (positive or negative) tail frequency (Lee, 2004). Tail likelihood focuses a certain number of teams that is determined priori. Tail frequency counts a discrete number of teams outside two standard deviations. While it has yet to see any use to this end, Kurtosis is a likely addition to this list of competitive balance variables since it measures the tails of the distribution directly.
ratio of standard deviations (RSD), skew, and kurtosis of the six different panels of winning percentages. RSD, founded by Noll (1988) and Scully (1989), and more rigorously established in Quirk & Fort (1992) and Fort & Quirk (1995) can be visualized as follows. Let ASD be the actual sample standard deviation. Let ISD be the “idealized” standard deviation of a league where the probability that any team in the league beats any other is 0.5. For the binomial without ties, Fort and Quirk (1995) show that , where G is the number of games in the season. Then RSD = ASD/ISD. As , the league is as balanced as the league with equal win probability in all games. The larger is , the less competitively balanced the league becomes. The third moment, skew, measures asymmetry of the distribution and its measurement can identify which side of win distribution is a primarily responsible for competitive imbalance. Negative skew (longer left tail) would suggest imbalance is primarily due to weaker teams. Positive skew (longer right tail) would suggest imbalance is primarily due to stronger teams. Kurtosis is the fourth moment and measures both whether the distribution has a high peak and excess observatins in the tail or a low peak and deficiency in the tail. The behavior in the tails has been used in past analysis of competitive balance, primarily in terms of “tail likelihood” (Fort and Quirk, 1995) or excess (positive or negative) tail frequency (Lee, 2004). Tail likelihood focuses a certain number of teams that is determined priori. Tail frequency counts a discrete number of teams outside two standard deviations. While it has yet to see any use to this end, Kurtosis is a likely addition to this list of competitive balance variables since it measures the tails of the distribution directly.
HistoricalMomentsYHHayley.docx