During games, the process gets slightly more complicated. First, we need to modify Winston’s formula to account for the diminishing amount of time remaining in the game. To quote Winston again:
“If we assume that the changes in margins during different parts of the game are independent and follow the same distribution (the technical term is identically distributed), then the standard deviation of the margin during n minutes of [a] game is:
(game standard deviation of margin) / sqrt(fraction of game that n minutes is)”
Using the 13.45 standard deviation we derived earlier, that formula is as follows for NFL games:
STDEV = (13.45 / SQRT((60 / minutes_remaining)))
So after 1 quarter, the expected standard deviation of scoring margin goes from 13.45 at pregame to 11.65, etc.
In addition to modifying the standard deviation about the mean, we also need to adjust the mean (the Vegas line) itself to account for the reduced amount of time remaining in the game. Though he doesn’t address this issue directly in Mathletics, in an email exchange with P-F-R, Winston suggested to scale down the Vegas line linearly based on how much time had elapsed. For instance, if the pregame mean is +3 for 60 minutes, then after a quarter (for the remaining 45 minutes) it would be 0.75 * 3 = +2.25.
This means the home team’s probability of winning after a quarter of play -- assuming perfectly neutral possession, down, distance, and field-position conditions -- can be computed using the following Excel function:
(1-NORMDIST(((away_margin)+0.5),(-home_vegas_line*(45/60)),(13.45/SQRT((60/45))),TRUE))+(0.5*(NORMDIST(((away_margin)+0.5),(-home_vegas_line*(45/60)),(13.45/SQRT((60/45))),TRUE)-NORMDIST(((away_margin)-0.5),(-home_vegas_line*(45/60)),(13.45/SQRT((60/45))),TRUE)))