How long does it take conversion rate to stabilize?

Conversion rate (goals per shot attempt) is known to be a highly volatile aspect of performance that can fluctuate randomly within and between seasons. Conversion rate also tends to converge to a typical value over time (often called, regression to the mean).

This does not mean that conversion rate is soley luck based however. But the skill component may only be evident over a relatively long period of time. In other words, we may have to observe a lot of shot attempts before the skill component of conversion rate emerges. The question is how many?

In this post I will attempt to discover how many shots are needed before we can be confident that the skill component of conversion rate is evident. I will examine both conversion rate for and against using data from football-data.co.uk for the top division of Scottish football (SPL/SPFL) over ten seasons (2004-2014). My analysis is limited to six teams who played in the top division every season during this period, these are: Aberdeen, Celtic, Dundee United, Motherwell, Kilmarnock, Hearts, and Hibs.

My basic approach will be to plot cumulative conversion rate (for/against) vs cumulative shot totals at the match level, and to visually inspect these plots for a point or region where the volatililty seems to subside.

Click to Enlarge

Click to Enlarge

The take home message from these charts is that it takes a looooooooong time for conversion rate to stablize to a point where we can reasonably argue for team differences in skill. We are talking about shot numbers in the thousands, not hundreds (2000-3000, depending on whether the shot is for or against).

The exception to this rule is Celtic's conversion rate for. Becaue they are the biggest club in Scotland with the biggest budget, they generally have much higher quality players. As such, their conversion rate for is distinctly high, even at relatively low shot numbers. This is clear evidence of a skill-based difference in conversion rate.

Celtic aside, these charts highlight the huge importance of random variance (luck) in short term football results. Given that the typical Scottish top flight team will attempt/concede 300-400 shots per season, it is possible that a team could be lucky/unlucky for several seasons in a row, despite common wisdom to the contrary. Also, we should not necessarily expect teams to fully regress to the mean within a single season. Regression to the mean may take many seasons.

Lastly, conversion rate against appears to take much longer to stabilze than conversion rate for, which may explain why goal-keeper performance is so hard to predict.