Chris Martin was Not-out a remarkable 50% of the time. That is, 52 times out of 104 innings. Is this a record? I don’t know an answer, so I sent off an email to the gurus at Cricinfo to see if they do. Michael Jones replied that “Yes it is” for batsman with over 100 innings (see here)! Well done Chris! What this does raise is the possibility of working out whether it was better for an incoming batsman to swing and hope to score a few runs before Chris was out, or whether they should just play normally? For this we must first consider what to do with the innings in which both Chris and the other batsmen were Not out. In such circumstances the choice is to include the innings on both sides or to exclude. I’ve chosen to exclude as I think this has the least room for bias.
Now let us apply my Rule #1 and visualise the data (see previous Chris Martin post).
Plot A is a histogram in which I have grouped for each of the two sets of data (the partnership scores when Chris was Out and the scores when he was Not out) into bins. Each bin is 5 runs wide except for the first. That is the first bin is from 0 to 2.5 (really to 2), the second from 2.5 to 7.5 etc. What can be seen from this is that there appear to be more very low partnerships when Chris was Out than when the other batsman was Out. However, don’t be fooled by histograms like this. Remember, there were not the same number of innings in which he was out (52) compared to when the other batsman was Out (49). This may distort the graph.
Plot B is better, but harder to read. Each black or red dot is a score. The coloured boxes show the range called the “Interquartile range”. That is, 25% of the scores are below the box, and 25% are above. The line in the middle of the box is the median – that is 50% of score are below and 50% of scores are above. The “Whiskers” (lines above and below the box) show the range of scores.
Plot C is less often used in the medical literature (at least), but is really very useful. It plots cumulatively the percentage of scores below a particular score for each of the two sets of data. For example, we can read off the graph that about 27% of the partnership scores for when Chris Martin was out were zero. If we look a the dashed line at 50% and where it intersects the blue line, then we see that 50% of the scores for when Chris Martin was out were 2 or below. This is a bit more informative than plot B.
What all the plots show is that the distribution of scores in both data sets is highly skewed. That is, there are many more scores at one end of plot A than the other, or the lines in plot C are not straight lines. This is very important because it tells us what tests we can not use and how we should not present data. Quite often when I referee papers, and in papers I read I see the averages (means) presented for data like this. This is wrong. They are presented like:
Chris Out: 8.4±13.9
Chris Not Out: 10.8±11.8
The first number is the mean (ie add all the scores and divide by the number of innings). The second number after the “plus-minus” symbol is called the standard deviation. It is a measure of the spread of the numbers around the mean. In this case the standard deviation is large compared to the mean. Indeed anything more than half the size of the mean is a bit of a give away that the distribution is highly skewed and that presenting the numbers this way is totally meaningless. We should me able to look at the mean and standard deviation and conclude that about 95% of the scores are between two standard deviations below the mean and two above. However two below (8.4 – 2*13.9) is a negative score! Not possible.
What should be presented is the medians with interquartile range (ie the range from where 25% are below and 75% are below).
Chris Out: 2.0 (0-12.8)
Chris Not Out: 8 (1-16.5)
We are now ready to apply a statistical test found in most statistical packages to see if Chris being out or the other batsmen being out was better for the partnership. The test we apply is called the Mann-Whitney U test (or Kruskall-Wallis test if we were comparing 3 or more data sets). Some people say this is comparing the medians – it is not, it is comparing the whole of the two data sets. If you don’t believe me, see http://udel.edu/~mcdonald/statkruskalwallis.html.
So, I apply the test and it gives me the number p=0.12. What does this mean? It means that if Chris Martin were to bat in another 104 innings, and another, and another etc, then 12% of the time we would see the difference (or greater) between the Outs and Not Out partnerships that we do actually see (see significantly p’d for more explanation of p). 12% for a statistician is quite large and so we would suggest that there is no overall difference in partnerships whether Chris Martin was Out or was Not Out. Alas, Chris Martin’s playing days are over and we have the entire “population” of his scores to assess his batting prowess. The kind of statistical test I’ve presented is only really useful when we are looking at a sample from a much greater population. However, in the hope that Chris may make a return to Test cricket one day, then what is presented here should give pause for thought for the next batsman who goes out to bat with him… perhaps there is not a lot to gain by swinging wildly, and thereby increasing their chances of getting out; they are probably not improving the chances of the team.