Well, you guys think I'm a nut... So here it is, the data you've all been waiting so patiently for. I've included the Excel Workbook I used. the 20##P tabs are punting stats, the 20##D are defensive stats with correlations to punting, and the 20##WL tabs are W/L records with correlations to punting. I even made a sheet with all the glorious data in one spot.
There is no strong correlation between having a good punter and how good an NFL defense is and there is no strong correlation between having a good punter and win percentage. So, next time you want to defend a contract to a punter for millions per year, I'm going to tell you that you're an idiot. So long as a punter is NFL caliber, he's simply good enough.
Below are the correlation coefficients. I'll leave it to you folks to pour through the rest of it on your own.