by This is a bit of a technical post, so drop out now if you are worried
about head-spins.
When we do any sort of statistical study (e.g. how many cars does it
take to clog up the freeway on average?) the results (if they are done
properly) include a margin for error. For example, it takes an
average of 784 cars to clog the freeway, with a error margin of 41
cars either way.
The 'error' can be interpreted in various ways. One way is that the
number of cars required to block the freeway can actually vary from
day to day, depending on weather, size of cars, skill of drivers etc,
so the error reflects the range of values you are likely to come
across (strictly this error is different to the error on the average
but you get the idea).
Another way of interpreting the 'error' is to say that there really is
an 'actual' average (if we watched the freeway for an infinite number
of days) and the average we measure is only an approximation to this
actual average. That is, the 'true' average is probably somewhere
between 743 cars (784-41) and 825 cars (784+41).
So, when you have a limited set of observations of a particular thing
the average of the observations is only ever an approximation to the
'true' average. That is, even if we accept that batting and bowling
averages are the best way of measuring skill, these averages are only
ever approximations of the 'actual' skill of the player.
The key question is: how big is the 'error' on a batting or bowling
average? With a few simple assumptions I think we can get a pretty
good idea.
Let's assume that a bowler has an 'actual' average of 30 runs per
wicket. Let's say that over his career he bowls 12000 balls and
concedes one run every two balls. He would concede 6000 runs in his
career. If his bowling truly reflected his average then he would take
200 wickets.
An average of 30 means that on any ball he has a probability of
200/12000 = 1/60 of taking a wicket.
The error for the result from a simple probability distribution can be
represented by the standard deviation, which is given by:
s.d. = square root of p*(1-p)*n
where p = probability of a wicket (1/60) and n = number of
observations (12000).
Therefore: s.d. = sqrt of 1/60*59/60*12000 = 14
Now, we expect that 95% of all cases the actual result will fall
within two standard deviations of the average result.
So, the number of wickets this bowler will take in his career,
assuming that his 'true' bowling average is 30, will be somewhere
between 172 (200-28) and 228 (200+28).
Now, this means that the bowler's career average will be somewhere
between 26.3 and 34.9. This is quite a wide range.
It is worth repeating in a different way:
Two bowlers, both equally as skilled, could both take around 200
wickets and have very different averages!
Or to make an even stronger point: there is virtually no
justification to rate one bowler as better than another on the basis
of a few points difference in average: the 'error bands' on bowling
averages are too high for them to be of any use.
That is, raw bowling averages provide almost no justification for
separating, say, Lillee (23.92), Holding (23.68), Miller (22.97),
Lindwall (23.03) and Hadlee (22.33).
I would go further and say that once you also take into account
opposition, pitch standard, laws and other such stuff, bowling
averages are pretty much useless as a comparison tool except if
comparing, say, a 20 average with a 26+ average bowler.
If you have got this far then I am interested in reponses!
John Clark