# Is 100 the maximum possible batting average?

I know nothing about cricket, I'm just trying to understand what "batting average" means. By checking this table, I noticed that the batting average can be calculated by `Ave = Runs/(Innings - N.O.)`. The histogram bellow is scaled from 0 to 100, with Bradman at 99.94:

I also read here the following:

An England batting collapse resulted in an innings defeat, denying Bradman the opportunity to bat again and so his career average finished at 99.94; if he had scored just four runs in his last innings, it would have been 100.

All of that made me think that 100 was the maximum possible batting average.. until I read this here:

In Australia's first innings, Bradman was bowled for a duck by Eric Hollies, causing his Test average to fall from 101.39 to 99.94; had he scored just four runs, his average would have been 100.

So, is 100 the maximum possible batting average?

Theoretically, the maximum batting average is unlimited, but I would also like to explain the concept of batting average a bit more in detail.

Batting average, as the name suggests, is the average score a batsman makes per innings during his career, and often serves as a measure of "how good" of a batsman he is, especially in comparison to others. (Hang on, I will come to the 'not out' in a moment.)

For example, suppose a batsman makes following scores in 10 innings: 71, 142, 0, 7, 31, 94, 26, 75, 113, 57. His aggregate score over 10 innings is 616, and the batting average is 61.6. Making big scores (typically 400 or more in an innings) increases a team's chances of winning, so a batsman who can score about 61.6 runs per innings will definitely contribute to the team's wins more often.

However, there is a little complication, by way of 'not out'. Normally, a batsman can continue batting for as long as he likes until he gets out. However, a team's innings can end without all batsmen being dismissed.

1. An innings ends when 10 batsmen of the team get out, leaving the 11th batsman as 'not out'.
2. The captain of the batting side declares the innings closed in order to give his bowlers enough time to get the opposing side's batsmen out. In such cases, both batsmen at the crease would be considered 'not out' (unless the declaration happens on the fall of a wicket, in which case, the only batsman remaining at the crease is 'not out').
3. The batting team achieves its target, in which case, both batsmen at the crease are 'not out'.
4. The game is interrupted for any reason, such as rain, and never resumes. Again, the one or two batsmen at the crease are 'not out'.

In such cases, the batsman's innings is not considered as "complete" and so it was deemed "unfair" to bring down their batting average, by counting such innings in the denominator of batting average calculations. For example, if a batsman stays not out on 3, 1, 2, 4 in 4 innings, his batting average would be 2.50, which hardly tells us how good (or bad) a batsman he is. As some sort of compromise, it was decided to exclude such 'not out' scores from the calculations.

As a result of this, batsmen who have played only a handful of innings could end up with a "ridiculous" average. For example, have a look at this page on Cricinfo Statsguru, where a few players have average close to the Don, and a couple of them even better than him.

Such edge cases tend to make the statistical data less meaningful, so statisticians (and fans) typically include an additional criteria, such as batsman should score minimum 1000 runs or play at least 20 innings to be "considered", and that's how, the Don's 99.94 is considered the greatest batting average.

As you've already worked out, batting average in cricket isn't like that. As defined by Wikipedia:

a player's batting average is the total number of runs he has scored divided by the number of times he has been out.

Taking AB de Villiers as an example, he has (as of 19th December 2014) scored 7296 runs in Test matches in 159 innings. In those innings, he's been not out 16 times, so his Test match average is 7296 / (159 - 16) = 7296 / 143 = 51.02.

Note that this means there is actually no maximum batting average - if a player is not out in every innings he's ever played (and has scored at least one run!), then his batting average would be infinite - and as I write this, this is actually the case for Stiaan van Zyl, who scored 101* (* after a score in cricket denotes not out) in his first and currently only test innings. Even ignoring that case, Andy Ganteaume scored 112 in his only test innings in 1948, thus formally giving him a test batting average of 112. The small number statistics here are why there's normally a minimum number of innings cut-off applied to things like this, after which Bradman has, by far and away, the best Test match average of all time.

No, 100 is not the limit, because there is no maximum average. The batting average is calculated as:

Ave = Runs/(Innings - N.O.)

Runs: There is no fixed limit of runs in an innings, and scores over 100 are possible. The current record in a test match is 400 runs by Brian Lara. So even discounting not outs, the average could be over 100.

Innings - N.O. : This boosts the average if the player is not out. So if a batsman scored 50, then 60 Not Out, his average would be

(50+60) / (2-1) = 110

So either way, by scoring centuries, or accumulating not outs, your average can be over 100. As you note, Bradman's average going into the final match was above 100, and other players have averaged over 100 for a year , but nobody who has played multiple test matches has got close to matching Bradman's career test average