# Maybe a decline in pace explain a reduction in foul outs, after all

#### By Dan Feldman

My initial post on foul outs concluded foul outs are down, even beyond what a decline in pace indicates. Unfortunately, I couldn’t find an actual reason for such a sharp decline in foul outs, so I offered a couple half-hearted explanations and solicited ideas.

It turns out, a decline in pace might actually fully explain the decline in foul outs.

A reader, Kenny, sent me the following very smart e-mail:

"I don’t think you can compare percentage changes of fouls with foul-outs and expect them to be proportional. This is because there’s a limit on fouls. Take one single player, call him Kaun Shemp. If Kaun by himself averages 3 fouls per game, I suspect he’d foul rarely, let’s say 2% of the time to just pick a number out of the blue, the tail end of the normal distribution of fouls per game. If he moved that average up to 5 fouls per game, the variance would cause many more foul-outs. An average of 5 fouls per game might mean he’d foul out 30% of the time. So the fouling increased by 66% but the foul-outs by %1300. That’s an extreme case, and somebody can probably do the math for the real case and see how responsible this part of it is. But I’m sure that as you increase in fouls, the number of foul-outs would not increase at the same or even any constant rate."

In my initial post, I failed to account for the fouls limit (six) holding steady the last 25 years. I watched pace and fouls per game decline during the last 25 years and expected foul outs to shrink at the same rate. But the fouls limit didn’t decline, and that obviously affects foul outs.

I ran the numbers Kenny suggested, and here here’s my new conclusion (if you want to see just the chart that goes with it, jump to the final section):

**Foul outs are down, but the league’s decline in pace appears to explain the reduction in foul outs.**

## Review

Here’s the key portion of my previous post on fouls that incorrectly led me to believe pace doesn’t explain foul outs:

"As I said above, foul outs per game have declined 43.3 percent in the last 25 years.In that same span, the league’s pace has declined a mere 8.7 percent, and fouls per game have declined just 11.8 percent.To illustrate this, I created foul outs per game plus, fouls per game plus and pace plus. I set the average foul outs per game, fouls per game and pace for the last 25 years each at 100. Then, I adjusted each season’s value for all three measures accordingly.**This model is similar to OPS-plus and ERA-plus for baseball.Foul outs per game plus is in gray. Fouls per game plus is in red. Pace plus is in blue. Solid lines are actual numbers, and dotted lines are the linear trend lines."

Sheet 4

## Finding odds of fouling out

The first thing I needed to do was find the odds of a player fouling out based on his fouls per game.

So, I plotted everyone who played in the last 25 years by their fouls per game and the percentage of their games they fouled out.

*A black dot represents each player. The vertical blue lines represent the average number of fouls a player committed per game in 1986-1987 and 2010-11. The red line is a best-fit regression line (which is statistically significant.*

Sheet 1

## Applying odds of fouling out to each season

The next step was applying the formula for the regression line* found in the previous chart to each season, based on the average number of fouls a player committed per game.

**y=0.00700494x^2+-0.00813879x+0.0021663*

Sheet 4

## Players per game

Then, I needed to find the number of players who played per game each year.

This step was also useful in analyzing one idea for why foul outs have declined – because of expansion, teams have less depth and are therefore less willing to allow a player to foul out rather that bench him with early fouls.

The data doesn’t support that theory. Players per game has has fluctuated slightly during the last 25 years, but not in any particular direction.*

**I wouldn’t read much into the spike this year. It’s the only partial season included.*

Sheet 5

## Expected vs. actual foul outs per game

Based on Kenny’s theory, I found the expected foul outs per game (an individual’s likelihood of fouling out each game multiplied by the number of players per game) for each season. Then, I charted that against foul outs.

Voila: we have what appears to be an explanation.

*Blue is expected foul outs per game, and red is actual foul outs per game.*

Sheet 3

As you can see, both measures match up fairly well. That indicates the decline in foul outs is based on pace after all.

But since 2008, actual foul outs per game has been below the expected value. Is that just statistical noise, or is there more to it?