It's been a whirlwind ride for Mookie Betts. Just one year ago he was in A ball, and now he's spending some substantial time playing at the major league level. He seems to be doing okay over the 22 games that he's played in. I was curious as to how well he's really doing overall, and also how he's been doing since becoming an everyday player for the first time, on August 19.
Here are his numbers over those two time frames, as compared to league averages on the season:
Betts vs. League ave. 
BA 
OBP 
SLG 
Season 
0.254 
0.341 
0.394 
Since Aug 19 
0.267 
0.421 
0.433 
MLB ave. through 8/26 
0.252 
0.312 
0.388 
Not bad. Looks like he's at or above leage averages all around. Also appears that in the last 9 games over which he's been playing every day, he's above league averages across the board. But with a sample size as small as this is, looks can be very deceiving. Let's take a closer look at that.
Especially deceptive is Betts' batting average since August 19. At .267, it appears to be significantly above the league average of .252. But had he had one fewer hit in his 30 at bats in that time, his average would be .0333 lower; that's the value of each hit when you've had only 30 AB. This would put his batting average down to .233, even further below league average than he is above it now. So as it turns out, he can't possibly be closer to the league average BA of .252 than he is now. He's not substantially above the league average; he's exactly at it.
To take a similar examination of his other numbers, it'll be easier if we express this in a different way. For Betts to be hitting a league average of .252 since August 19 would require him to have 7.55 hits. His 8 hits, 0.45 more than that, are as close as he can be to that.
Below is a chart that shows where he's at when viewed in this way. My acronyms will need explanation:
HAMBA = Hits above mean BA
BR = Base reaches (number of times got on base)
BRAMOBP = Base reaches above mean OBP
TBAMSLG = Total bases above mean SLG
For example, to get TBAMSLG, multiply the leagueaverage SLG by Betts' number of AB to get the number of total bases that would put Betts at leagueaverage SLG. Then subtract this from Betts' actual number of total bases.
The results:
H 
HAMBA 
BR 
BRAMOBP 
TB 
TBAMSLG 

Season 
18 
0.14 
28 
2.38 
28 
0.44 
Since Aug 19 
8 
0.45 
16 
4.13 
13 
1.35 
This puts a good dose of perspective onto his BA, OBP, and SLG numbers. Both his recent and season batting averages are exactly league average, as the differences for those are all between 0.5 and 0.5. His recent slugging percentage is only 1.35 total bases above league average; his recent home run vaulted the number to this height. For such a volatile statistic as SLG, that's a negligible difference.
The only statistic for which we can really say he has any difference from league average is OBP. But even though these numbers demonstrate a difference, there is still the question of whether the difference is significant. Someone whose chance of getting on base exactly matches the league on base percentage won't necessarily hit at the league OBP, due to the randomness of getting a success or a failure in any given trip to the plate. The binomial distribution tells us exactly how big a deviation this randomness is likely to cause. In Betts' case, with his 82 PA on the season, a leagueaverage hitter has a 32.3% chance of having an actual OBP equal to Betts', or higher. Of course a "fluctuation from the mean" can happen both up or down, so that means there's about a 64.5% chance of getting a random fluctuation that big out of a leagueaverage hitter (an equallyprobable downward fluctuation would actually be smaller than the upward fluctuation due to the asymmetry of the distribution, but I'm concerned about the probabilstic size of the fluctuation, not the fluctuation in value). While this is not the same thing as the probability that Betts is a leagueaverage hitter, it does tell us that his .341 OBP on the season is not significantly different from the .312 league average, given the small sample size.
Betts' recent OBP is a different story. The binomial distribution tells us that there's only a 10.4% chance that a leagueaverage hitter would have an OBP this high over Betts' 38 plate appearances. So with only approximately a 21% chance of having a fluctuation this big due to randomness, we can say it's a significant difference.
To sum up, Betts has been a leagueaverage hitter this season, and with regular playing time is becoming aboveaverage at getting on base. Add in his speed and his high success rate at stealing bases, and you've got a guy who profiles to be exactly what you want in a majorleague leadoff hitter. Not bad for a guy one year removed from A ball.
But will it last? After a certain amount of time in the bigs, rookies will have their weaknesses identified and exploited, and will have to adjust. This is true today moreso than ever. In the scouting evaluations I've heard on Betts, though, it doesn't sound like there's much of a weakness to exploit. Look for him to continue being a productive leadofftype hitter in the big leagues going forward.