I don't really know how to quantify Howard's arm in terms of run value, but I am going to try to put his arm in perspective with a few simple invented statistics. I went on FanGraphs and used the standard fielding section to pull these numbers. I should premise this by saying these are not meant to be absolute perfect measures of throwing ability. I guess there is no real way to quantify throwing ability, even for pitchers. We can measure pitches with PITCHf/x and infer certain things, but there's no way to measure intended target vs actual location. We have even less data for position players, but I decided to take a stab at it anyway.
The first of these fake statistics is cleverly titled Throwing Error Percentage (TE%). It's the number of throwing errors divided by the total number of errors. Using errors to derive any valuable insight into defense is a fool's errand, so feel free to take these numbers with a grain of salt, or even a portion of a grain of salt, or even a piece of a portion of a grain of salt. But as I said before, these aren't going to be perfect, or even good, but I'm writing it, and you're reading it, so let's go to the graph:
So Howard apparently had no throwing errors in 2005 and has no throwing errors so far this year. But in between he really did some damage. Overall, from 2005-2013, 34.4% of Howard's errors were of the throwing variety, well above the league average of 24.0% for the time frame. And this is not a result of Howard having very few errors, which could cause a small number of throwing errors to show as a large percentage. Howard has averaged an error roughly every 105 innings of his career, while the league average first baseman averaged an error every 150 innings. Ryan Howard is not a good defensive first baseman.
Another fake statistic I wanted to look at was something I call Double Plays Started Percentage (DPS%) which, as its name would imply, is the number double plays a player started divided by the total number of double plays in which he was involved.
Ryan Howard does not start a lot of double plays, and this could be for a variety of reasons, but one of them, as you'll see momentarily, is his terrible arm. On average, Ryan Howard has started double plays at just 3/4 the rate of the average first baseman. Again, this is not just a mirage of the Phillies turning more 6-4-3 double plays than average, as Howard has started a double play once every 175 innings, while the average first baseman does so once every 130 innnings.
Now that I've done my best to show statistically how bad Howard's arm really is, I'm going to turn to Sunday's game against the Brewers. The Phillies were cruising through seven innings, ahead 7-0 off a near-cycle by Domonic Brown and a masterful pitching performance by Cliff Lee. In the eighth, Lee seemed to fall victim to the BABIP fairies that have been haunting Cole Hamels all season, and the bullpen couldn't close the door on the Brewers once he departed.
So Jean Segura stepped up to the plate with no outs in the ninth and runners on first and second, the Phillies up just 7-5. Antonio Bastardo got Segura to hit a ground ball to first on an inside fastball, which set Ryan Howard up for a perfect double play ball:
Ryan Howard catches the ball with ample time to gun down Jeff Bianchi heading for second, as Bianchi is only about three steps out of his lead when Howard catches the ball. In this situation, too, it's key to get this lead runner because, remember, the Phillies are only up two. If Howard can get this guy at second, it'll take a double to tie the game, instead of a single. Assuming Bastardo covers first quickly enough, the Phillies might even be looking at a double play in this critical situation. If that were to happen, it would take a home run to tie the game.
But Howard chickened out.
Now before you get all worked up; don't worry, the Phillies still won. Bastardo was able to induce two weak fly balls to end the game. But this bad defense could have cost the Phillies the game. It's gotten to the point where Howard's weak arm is making him gun-shy, and he's unable to get the ever-important lead runner in an critical situation.
This is the exact reason why an error doesn't tell the whole story. In this situation there was no error because there was no attempted throw because the potential thrower is a chicken. It wasn't an error, but it surely was not the ideal play. At some point, the benefit from aggressively pursuing the ideal play becomes significant enough to negate the risk of the error. When does that happen? Is it when a player is accurate 75% of the time? 90% of the time? Well I dunno. I guess for Howard, it never happens.