Standard Deviation vs Mean Deviation to Measure Batting Consistency
It was one of the greatest contemporary philosophers of our generation, J Cole, who once said “No such thing as a life that’s better than yours”. This -baffled me the first time I saw it on someone’s Instagram bio(Do we really have a better place to find more profound words of wisdom?). I pondered on it for a reasonable amount of time and came to a conclusion which was rather dissatisfying. It’s quite easy to find the amount of sugar in our evening tea and the amount of fat in that greasy meal we love. But, the most important things in life(I’ve been told), are not quantifiable. For instance, there are many meaningful things that makes our existence worthwhile, which we cannot put a number on: how close we lived to the purpose of our lives, the value we added to the society and not even the amount you loved, even though, 3000 — at least according to Morgan Stark— seems like a solid number to aim at. But, if the truly important things in life are simply immeasurable, what metrics or proxies do our left brains use to measure ourselves against the society? That’s a topic to be discussed by an expert in the field of neuroscience or even Matthew McConaughey. But, not to get disheartened, there is something I can introduce— quantifiable metric to our fortune— that might be of interest to you.
It’s a Sunday afternoon. You turn the TV on to see your favorite team batting at 15 for 3. As the new batter comes up to the crease and takes his guard, a bunch of numbers appear on the screen. Your little brother or your clueless sister who’s curious and annoying as any other sibling would be, asks you what each number means.
Just when you feel like Harsha Bogle after all the explanation you just gave, your brother/sister hits you up with “Can we trust him to build an inning here? How reliable is this guy anyway?”. Deer. Headlights. Does the average really show how reliable a batsman can be? Average of the batsman is what we can expect from a batsman on a given day. But do we have a metric to show how consistently he has scored around the number he is expected to hit every inning?
Let’s take Bob and Todd over here for instance. If you were asked to only pick one for your village cricket team, who would you choose given the above stats? More runs, higher average, two centuries against none and a high score twice the size of the next: It’s a no brainer to go with Todd over Bob. But, let me give you an inning by inning runs sheet of the two to see if you’d change your mind.
Given the new information presented above, if the score read 15 for 3, would you still choose Todd over Bob? Looking at the score sheet, I’d rather have a guy with six 30+ scores to save a collapse than a ’hit-or-miss' guy in my middle order. But, do we have a number — from the usual numbers shown when a batsman comes out to bat — to put Bob above Todd in such a scenario? It was easy to make a decision by eye since it was just ten innings. In the case of 180 innings, we wouldn’t have the luxury of making such a decision by eye. This is where Standard Deviation and Mean Deviation come into play.
Standard Deviation & Mean Deviation
Standard Deviation(SD) vs Mean Deviation(MD)
1914 was the year when Eddington pointed out that a simple mean residual error (i.e. Mean Deviation) is preferred over a mean square residual (i.e. Standard Deviation) when working with empirical data. Fisher countered Eddington’s empirical evidence with a mathematical proof to show that SD is more efficient under ideal conditions than MD. I chose to go with the mean deviation for our cricket data set due to two reasons: Non-normality of the distribution and interpretability.
Non- normal distribution
As you can see, the distributions for the top four batmen of our generation looks more chi-square than normal. Fisher himself pointed out that Mean Deviation is better for use with non-normal distributions. Since we square the deviations from average to produce SD, longer-tailed distributions — such as this one — tend to explode the variation in SD. The act of squaring makes each unit of distance from the mean exponentially (contrast to additive from MD) greater, and the act of square-rooting the sum of squares does not completely eliminate this bias. Therefore, MD seems like a better bet over SD in the face of non-normal distributions.
Interpretability
If broadcasters ever decide to add a number to show a batman’s consistency, which one would you prefer to be on that screen? It doesn’t take a Stat major to explain standard deviation. But, in the context of cricket, it would not have a meaning that’s comprehensible to the average viewer. Median Deviation on the other hand can be simply explained as “the range that a batsman usually scores around his average”. In the case of Bob and Todd, Bob usually scores in the range of 10 from his average of 35.1 and Todd, 39 from his average of 40.5. Which means that Todd has a tendency to either hit an 80 or a duck when he comes in at 5 and Bob to hit somewhere between 25 and 45. I’d rather try my luck with Bob on this one, thank you very much.
By analyzing the FAB4 of cricket, the mean deviations for the top four batsmen in our generation seems quite large. But, don’t forget that we are only measuring consistency and not an overall measurement to judge a batsman’s ability. The downfall of using mean deviation as a measurement of consistency is that you’re punished for scoring 40 above your average the same way you’re punished for scoring the same amount, below. The challenge here is to come up with a metric that weighs in several other factors — such as the % of innings the batsman crossed his average — without loosing too much interpretability.
Even though, mean deviation is a good metric to measure consistency, there is a lot of scope for improvement. Mean Deviation as a measure of consistency can be good place to start for a more sophisticated measurement with a good balance of accuracy and interpretability.
“An imperfect something is better than a perfect nothing”-Marcus V. Calvert
