Sure enough, in England's first innings at Lord's Mitch caused considerable damage to England, despite the placid pitch, taking 3-53.
So, is this a general pattern in Johnson's career?
I must admit I was sceptical that there would be any evidence for this but it turns out that there is a little bit, but it seems to be a short term effect. By that I mean that when Johnson has a good innings with the bat he often does do well with the ball in the next innings. However, when averaged over the course of a series there is no particular relationship between him batting well and bowling well.
The short term effect:
Mitchell Johnson has scored 1 century and 11 half-centuries in test cricket. I looked at his performance in the bowling innings immediately following these performances. I excluded the case of his 123* South Africa which was right at the end of a series, so there was no immediately following bowling innings, so that leaves his 11 half centuries.
In these 11 bowling innings following half centuries Mitch has taken 36 wickets at an average 21.33.
Comparing this to his career bowling average of 27.90 it seems that his bowling performance does pick up slightly in the immediate aftermath of a good batting performance.
Over the course of a series:
The graph below plots Mitchell Johnson's bowling average against his batting average for all the completed test series he has played in.
Looking at it by eye there's no clear pattern to associate a good batting average in a series with a good bowling average, suggesting that when averaged over the course of a series Johnson's batting and bowling averages behave more or less independently.
Getting a bit more Mathsy about it, I evaluated Spearman's rank correlation coefficient which is a way of determining how much two datasets are correlated. The answer was pretty close to zero (0.09 if you want to know) suggesting once again that Johnson's series batting and bowling averages are pretty uncorrelated.
So there you have it: the commentators were right- a good innings with the bat does seem to help Mitchell Johnson bowl well next time out. However, if you look at it over a course of a series his batting and bowling averages have pretty little to do with each other.