An out is an out is an out
Overall, an out is an out. The run value of a strikeout is roughly the same as any other out. On average. There are individual cases where a strikeout is much worse than another type of out: runner on 3B and less than two outs. In this case, a strikeout is a killer, as was detailed in The Book. There are situations where a strikeout is actually preferable than another out: runner on 1B and less than 2 outs, since you won't ground into a double play on a strikeout. Overall, there's not much to choose between a strikeout or a regular out. The difference is around .01 or .02 runs per out. Being 100 strikeouts worse than average means that your strikeouts have a additional direct cost of 1 or 2 runs.
But, that's not what we talk about when we talk about cutting down your K rate. We are talking about changing your hitting approach to lower strikeouts (and presumably walks and homeruns), and increase contact. Overall, we would hope that the production of the hitter increases. But, who are we to tell Adam Dunn or Alfonso Soriano that they need to change their approach to hitting? They must be doing something right.
The study
The K rates for the Kleskoyan dropped from 20% to 14%, while the Jedmonds increased from 14% to 20%. Prorated to 600 PA, that's a change of 36 strikeouts. The other 5250 players in the pool were flat at 13%.
What else was effected?
First thing I want to know is about their walk rates. The other 5250 players' walk rate went from 9.5% to 9.8%, a reflection that as you get older, you get wiser. The Kleskoyan went from 9.8% to 10.1%, keeping pace with the wise man approach. The Jedmonds went from 10.0% to 10.8%. Their K rates shot up, but not much of a jump for walks, especially after you consider the wise man effect. In short, walks were not dependent on the change in strikeouts! I'm surprised.
Now, the important one: homeruns. The 5250 players kept their HR rate at 2.8%. The Kleskoyan dropped from 3.6% to 3.4%, which on over 600 PA is just 1.2 HR. But, that was balanced against an increase in doubles and triples (almost 2 per 600 PA). The Jedmonds went from 3.5% to 3.9%, which is 2.4 HR increased per 600 PA. But, that increase in HR was tempered against the 2.6 decrease in doubles and triples.
We see therefore that a change in 36 strikeouts is accompanied with a change of around 1.8 HR in the same direction, but a slightly larger decrease in doubles and triples.
Their batting average on balls in play was virtually unchanged. That is, once they made contact with the ball, and it stayed in the field of play, all the groups' average stayed at around .300.
And the answer is...
However, the Kleskoyan had more chance to put the ball in play, and since they had less strikeouts, they didn't just trade a strikeout for another kind of out. Instead, they traded a guaranteed out (the strikeout) for a chance at getting a hit. Their batting average soared from .263 to .280. The Jedmonds did the reverse, dropping from .280 to .263.
And how about overall? The wOBA of the 5250 went from .345 to .344. The Kleskoyan increased from .340 to .354, while the Jedmonds dropped from .354 to .347. So, even though the Kleskoyan actually hit less HR, their lowering of strikeouts actually increased their overall production. And even though Jedmonds hit more HR, their increase in strikeouts lowered their overall production.
Regression toward the mean would just be a blip at this point. We've got over 1000 PA in each group, meaning regression would be around 16%.
The final result: The change is around 10 points in wOBA, which is equivalent to 5 runs per 600 PA. And this was because of a change of 36 strikeouts in the other direction. In terms of free agent dollars, 5 runs is almost half a win, or almost 2 million$. Basically, every 20 strikeouts you can reduce is worth an extra one million dollars in your pocket.
While I wouldn't propose a star hitter like Dunn or Soriano change his approach, there are plenty of guys struggling to make it who would find it worthwhile to change their approach to hitting.