Brendon McCullum and the algebra of batting risk

The batting average for a player contains within it the sum total of every factor imaginable - the quality of the bowling, the match situation, the condition of the pitch, the quality of the other batsmen, luck, the batsman's form, and so on. The reason records over a career are meaningful is that the influence of extreme individual occurrences on the overall record is minimised. This is a rough description of the law of large numbers.

As a measurement, though, the batting average presents obvious problems. Consider five innings - 0, 0, 0, 0, 100* - four dismissals, 100 runs, average 25. Consider another set of five innings - 19, 17, 20*, 24, 20 - four dismissals, 100 runs. You might say that the first group of five scores is unlikely, and you would be right. 0, 3, 11, 4 and 82* are perhaps more realistic numbers. A score of 81 is more likely to prevent defeat than a score of 24. The following chart briefly illustrates this idea. It shows how often scores in a given range (shown on the vertical axis) occur in wins, losses and draws or ties.

Bigger individual innings reduce the likelihood of defeat in Test cricket. This is not a controversial claim. But does getting to 20 make it more likely that a player will get to 40? This question has become pertinent in recent times because run-scoring patterns in Test cricket appear to be changing. The McCullum Effect, if you will, has meant that teams are willing to allow their batsmen to play attacking innings from the outset.

For about 130 years, the orthodoxy in Test cricket has been that a batsman should build innings patiently and not take chances; develop batting technique and use it to keep the good balls out early in the innings. The logic behind this is sound. Playing an attacking shot always involves risk. The surest way of playing a ball safely is to watch it as carefully as possible for the longest possible time, and play it as late as possible, as close to the body as possible. Playing this way, attacking strokes are not possible. Dead defense is. Attacking strokes are almost always played further away from the body than defensive strokes. The ball is met earlier, with greater bat speed. There are, very simply, more moving parts, and therefore more uncertainties involved in playing an attacking stroke. This is also why bowling short, allowing batsmen to stay back and square-cut or play off the hips, is considered bad bowling - because it does not require batsmen to face the uncertainties of playing far out in front of the body to score runs. Bowling short, unless it is a bouncer (which can create specific uncertainties of its own), therefore amounts to giving the batsman relatively risk-free runs. Line and length matter for this reason - some lines and lengths force great risks for batsmen who wish to score off them.

The upshot of this algebra of risk is that bowlers try to force batsmen to defend and not give easy runs away, while batsmen are constantly trying to take advantage of the smallest risk. Great batsmen are better at snatching every last available run within this algebra than good batsmen. They also tend to be better than others at defending the good balls.

The longer a batsman stays undefeated, the more tired bowlers get, the less advantage bowlers get from the uncertainties of the conditions. The batsman learns about the pace at which the ball is coming off the pitch, how much it is bouncing or turning.The chances of a batsman getting a delivery he can't play safely reduce with time. Of course batsmen can also get tired and lose concentration. This is why a large part of the art of batting in Test cricket is to do with managing time.

Brendon McCullum appears to want to upend this logic. Virender Sehwag did it too. When certain kinds of uncertainties were systematically absent (the moving ball to name just one), Sehwag was spectacularly successful. In England, South Africa and New Zealand, where the tracks offer seam movement and the cool weather ensures that the pitches do not dry out and the movement persists, Sehwag averaged 25 in 19 Tests and reached 50 only four times in 34 innings. He was much happier with the hotter weather and drier pitches of Asia and Australia, where he scored 50 or more 49 times in 136 innings. Sehwag's kryptonite was not pace, bounce or turn but movement.

As Sehwag's example shows, McCullum's approach to batting in Test cricket is not new. What makes McCullum unique is his leadership position. McCullum seems to have prompted his team-mates to bat the same way. Sehwag was a one-off genius. He was given the licence to play his way because it was evident that he could pull it off regularly. The other batsmen batted their own way. There was no suggestion with Sehwag that he was blazing a trail for his team-mates. McCullum is.

Will it work? Here is the evidence McCullum is working against. A middle-order batsman (Nos. 3, 4, 5 and 6 in the order) averages 38 in Test cricket. For innings in which such a batsman reaches 10, the average is 55. This includes the first 10 runs, so let us remove those. The average for a batsman who reaches 10 is 45. In other words, if you take a batsman batting on 10, and assume that this is the start of his Test innings at Nos. 3, 4, 5 or 6, such a batsman is equivalent to one who averages 45 in Test cricket. Getting through the first 10 runs improves a middle-order batsman's average by seven runs.

I worked such a figure for every run from 0 to 100, for middle-order batsmen and openers. Middle-order batsmen benefit from not-outs, which are more likely at higher scores than lower ones. The not-out is a significant idea in Test cricket - beyond the way it inflates batting averages. It suggests that a batsman was undefeated at the end. The end occurs either when the bowling side has dismissed 10 other batsmen, or when the batting side decides that it has scored enough runs, or has won, or saved the match. It is the logical, triumphant conclusion (from the batsman's point of view) of the algebra of risk described above.

The average batsman (overall average 38) who gets to 50 in the middle order is equivalent to a batsman who averages 57 in Test cricket. At the start of his innings the average middle-order Test batsman is Mudassar Nazar or Brendon McCullum. When he gets to 40, the average batsman is Tendulkar or Greg Chappell. That's the value of getting set in the orthodox way. This also shows where Chappell or Tendulkar are better than the player who averages 38-40.

The evidence unequivocally says that playing carefully for the first few runs and spending time getting set early in the innings makes a Test batsman more valuable to his team. McCullum seems to challenge this idea. His preference for attacking nearly every delivery early in his innings either suggests that he lacks confidence in his defensive ability or that he does not think much of the idea that backing one's technique to get set is a good idea.

Steve Waugh's Australians tried something similar. Waugh's stated aim was to try and score at 4 an over. But this was more a function of having Adam Gilchrist in their side than otherwise. Under Ricky Ponting's captaincy, when Gilchrist was playing in the XI, Australia scored at 3.7 runs per over. When Gilchrist was absent, from 2008 to 2012, they scored at 3.3 runs per over. The scoring rate had less to do with strategy or any systematic approach than it did with the serendipity of a specific type of talent coming together in the same side.

Batting poorly recently cost McCullum's team the Lord's Test. That match turned when McCullum played his bizarre innings with New Zealand on top in their first innings. The approach was further epitomised by Corey Anderson's astonishing batting on the last day, when he kept trying to hook (unsuccessfully), though Alastair Cook had three men on the leg-side boundary and New Zealand were trying only to save the Test. This had less to do with approach than with confidence in defensive ability.

McCullum is clearly too good a player to rely on slogging to make runs in Tests. In any case, he bats too high up in New Zealand's order to play that way. There are times when he plays in the orthodox way, as he did in New Zealand's second innings at Headingley, or during his epic match-saving triple-hundred against India. At Headingley he held New Zealand's innings together for nearly two and a half hours and played a careful captain's hand (55 off 98 balls) in a century stand with BJ Watling.

Will McCullum's team blaze a new trail in Test match batting tactics and change the orthodoxy of the game? Will he successfully change the algebra of risk? Or will their success simply mean that they had special batsmen who were good enough to play a certain way? Time will tell. But for now, one prefers the latter explanation.