Since every team has now played at least eight games, one can now compare expectation to what is thus far observed. Using the average line and the point differential (expectation and observed respectively), I calculated the relative z-score of each team in regards to relative value, by first extrapolating a Pythagorean based against the spread record. The familiar “regression to the mean” immediately comes to mind, yet I contend this is vastly dissimilar. A team could be 8-1 against the spread yet have an expected ATS record (Pythagorean) of 9-0. Each team, essentially, possesses a unique value score, z-score, as a function of their performance, irregardless of league averages or average ATS record.
To calculate the z-score, first I found each team’s average line and per game point differential. Then I used the Pythagorean formula to deduce an expected against the spread record by dividing total points scored per game by two and subtracting (or adding) half of the average spread. I added (or subtracted) half of the per game point differential to extract average points above spread. Inserting the aforementioned into the Pythagorean formula, with an exponent of 2.37, results in a Pythagorean Against the Spread record. The basic principles of Mathematics allow one to factor out the 1/2.
The z-score is a descriptive measure that finds where an observation is situated in relation to the mean. Z is often integrated as a factor of measurement to use for normalizing purposes. For example, if team A has an offensive efficiency of 1.01 points per possession, finding a z-score for the aggregate defensive efficiencies of the teams they played normalizes the 1.01 to a defensive efficiency of 1. Any number multiplied by 1 is itself, and this is where a z-score has its benefits with drawing information from sports and betting data. One can simply take the aggregate z-scores and redistribute the data around a normal distribution. It is common in statistics and is calculated using the formula:
is the observed value
is the mean
is the standard deviation
After calculating each z-score based on the differential between expected and actual against the spread record (presented as a percentage), the teams are appropriated graphically like so:
The higher the z-score the greater the inherent value from here to the end of the season. Why a z-score? Z-scores are absolute, as opposed to the remarkably scant information contrived from a single differential calculation. If Team A has an Pythagorean ATS record 20% higher than the observed record, the content associated with one single observation of Team A is empty. Maybe every team in the league has a higher Pythagorean ATS than actual ATS. A z-score serves to remedy any false interpretations that arise from lack of relevance. As it happens, the expected ATS throughout the entire league correlates well with the actual data.