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Mathematics of Cricket
We don't like Cricket - We love It !
Taking a rest from really intense facts about mathematics, let us now have a bit of fun. Moving on from the previous Hubs, Mathematics - the Science of Patterns and More on the Patterns of Maths, where we looked at relationships between numerical identities, it is time to consider the analysis of the Statistical side to the bat and ball summer sport of most of the British Commonwealth, which is Cricket. This has little to do with that insect named Jiminy™, of Disney movie fame, but more to do with finding formulas for some of the requirements of winning games decided by run rates. Although a lot of the World does not know this sport, it is still a multi billion dollar industry, with millions of fans world wide, and has over the years been graced by now legendary international stars such as Scott Styris, Glenn McGrath, Rahul Dravid, Muttiah Muralitharan, and Freddie Flintoff, just to name a few.
Now this Hub will assume You might have a basic idea of what Cricket is, and meaning of the Cricket related terms I shall present here, but I would not wish anyone to miss out, so please follow the Wikipedia™ link Cricket, and learn up on one of the best Summer sports in the World.
Five Greats of the Recent Game - Photo Wikipedia, edited on Paintbox
Run Rates to begin with
If you’ve watched as much Cricket as I have, and are mathematically minded, you will notice all the statistics, numbers and formulae the television production staff come up with while you observe all this physical action going on at venues such as Alexandra, Mohali, Dambulla, and Edgbaston.
If we look at one of the basics, ( which is especially relevant in one day, limited overs cricket, ) the run rate, we see that it is just a simple average, or mean, of the total number of runs scored so far, divided by the total number of overs in which they have been scored. The required run rate refers to the total number of runs required for victory, divided by the total number of overs available to get them, without regard to wickets in hand.
Some of the International Venues where we play the Game - Photo Wikipedia, edited on Paintbox
So, imagine England, batting first at The Oval, scores 299 - 7 in its fifty overs, thus setting India 300 runs for victory in their 50 overs, with a required run rate of exactly 6.00. This is worked out simply by dividing 300 by 50, or, what I do, is double 300, then divide by 100 - the second part I do simply by moving the decimal point two places to the left. Now I used to watch all of this kind of thing, and as a match progresses, the chasing team may score at above, below or right on, the required run rate.
England vs. India, and the Oval, Kensington, courtesy Wikipedia, edited on Paintbox
Now, to begin With . . .
Let us say that after ten overs India has reached 71 - 1. Obviously they are scoring at ( an average of ) 7.1 runs per over ( not that they ever scored exactly 7.1 runs in any one over ), so then the question is, after this, what do they now require ?
To work this out, we would normally subtract the 71 they do have from the total runs required, to give us 229 runs left to get, which they now need to achieve off the remaining 40 overs. 229 ÷ 40 = 5.725. So we see, because they have gone at more than they initially needed, their current required run rate has gone down from 6.00 to 5.725, an overall drop of .275 runs per over, although they had been scoring at 1.1 runs per over more than first required.
. . . then the Game continues . . .
Let’s see what is going on. Had India been going at just six runs per over, they would now have exactly 60 runs. What this means is that the 11 extra runs, spread over the remaining 40 overs, equal the .275 difference in required run rate. Realise that they had indeed been scoring 1.1 more runs per over than initially needed, but did so in just ten overs. It might take more effort to keep this kind of rate up over a greater period of overs. Now the required run rate, however, only goes down by this .275, as this is the equivalent of those 11 extra runs, but now there are 40 overs left to cover this difference.
We could clarify this by noting that if India continue to go at 7.1 runs per over right up to the end of 25 overs, which is half the allotted overs they had, then for the remaining 25 overs, the required rate would drop down by this 1.1 runs per over difference in what was initially required, as it is exactly half and half, and from then on therefore, only 4.9 runs per over would be needed.
This can be confirmed by working out that if India did score at 7.1 runs per over for 25 overs, they would have made it to 177.5, which could not be true, as we cannot halve runs, but that only means that one cannot average exactly 7.1 runs in 25 overs. But just say we could have half a run, then subtracting that total from the original 300 required gives us 122.5 left in the second lot of 25 overs, and this does average out to exactly 4.9 runs per over, as predicted.
Now, imagine instead, going on from 71 runs in the first ten, that after a further ten overs, India has slumped to 112 - 4, with Virender Sehwag and Mahendra Singh Dhoni desperately trying to hold it together. 112 after 20 is a run rate of 5.6, ( which isn’t all that bad a run rate to slump to ), but now, they are .4 runs short of what was initially required, in terms of run rate, and eight runs in total behind. This means that if they had been going at exactly six for those twenty overs, they would have been at 120.
We can work out the new required run rate simply by subtracting the total runs achieved so far from the total required, then dividing by the remaining 30 overs now left, to give us 6.267, which is of course .267 runs more per over needed now, than at the start. Even Ramanujan would be upset at that, and as for another Indian, the Great Khali ( Dalip Singh ), I shudder to think how he would react. Still, as I say, it’s still not too bad, but they do need to preserve wickets.
Some of the supposed great fans of the Game - Photo Wikipedia, edited on Paintbox
. . . and after it is ended, my musings on the Road to a Cricket Run Rate Formula
As noted, the Indians have scored eight less runs than needed by this time to maintain six an over. This eight extra runs required therefore gets spread over the remaining 30 overs, and added to the original required run rate. 8 ÷ 30 = .267, which indeed is the added difference in run rate now required, so the idea does work.
You see, in all my years watching cricket, and looking at how they worked out run rates and such, I often wondered whether there was a formula available for working out required run rates as a game progressed, based on initial run rate, new current run rate, and overs remaining. When I finally attempted to work this out about 7 or 8 years ago, I actually didn’t believe there was, so it was more like I was trying to disprove my theory, rather than come up with one that really worked.
How to spot the potential for a Formula
The thing is, if you watch cricket, you may see for instance seventeen runs scored off a single over. Now this does not reduce the required run rate by 17 runs because of course, those runs occurred in only one over, and may not be representative of the whole situation. If after exactly halfway, half the runs needed are scored, you know instinctively that exactly as many runs are required from the second half, and the current and required run rates would be both the same. I have even seen games in which teams are just keeping up with the required rate, whether very little behind, or slightly above, meaning their current rate is very close to the rate needed to win.
The key was to find a formula that locates the discrepancy between required and achieved run rates up to a certain time, then spreads them over the remaining overs. The first formula I worked out is to find what is the current run rate now, after a given passage of play of so many overs and runs, when added on to previous play.
A useful Example
For the use of this one, imagine we had first played twenty overs, scoring at fives, ( meaning a total so far of 100 runs ), then in a five over burst, we amassed an extra 79 runs - in itself a run rate of 15.8. But we cannot average it like that unless the 15.8 runs per over had occurred during an additional 20, just as the previous 5 runs per over had. Then we would add the two, then divide by two to give us an overall rate of 10.4 after forty. No. Because the different run rate occurred in an amount of overs not identical to how many we had amassed our previous run rate of five in, we have to do it another way.
Using [(r - cn) ÷ t ] + c , looking at the illustration above to work out the significance of each letter, we find : [( 79 - 5 × 5) ÷ 25 ] + 5 = 7.16.
We can work out this run rate by totalling all the runs ( 179 ), and dividing by total overs gone ( 25 ), to give us a total run rate of 7.16, since, as we have scored at a greater rate in these extra five overs as before, our overall run rate will be higher.
An Explanation leading up to the Second Formula
Now, 7.16 is obviously not the average of five and 15.8, but how it is achieved, is that, if you look at the 5 over burst, which had an average run rate of 15.8, you can take from that the 5 runs an over you would score to maintain the current run rate up to then of 5, so that you are left with 10.8 runs per over extra, scored in those additional five overs, over and above what you initially had.
Now this 10.8 has to be "averaged" by dividing it by a number we can work out. To find this number, we take the number total overs to date, of which there have been 25, and calculate that this is five times the 5 overs in which the extra runs were scored. This gives us 10.8 ÷ 5 = 2.16, which is indeed how many extra runs per over we have now scored above our initial run rate of five.
The second formula is to be used to do what we had initially been doing ; trying to work out new required run rates from existing information.
Using the New Formula
So let’s put our facts in from the England vs. India game, and see what happens.
Which is indeed what we had. We see here, if we analyse the formula, that all it is doing, is subtracting from the original run rate the difference in run rates, multiplied by the fraction that the overs to date are of the overs remaining, which is basically what we were doing in the first place without a formula. Now don’t get concerned about the first minus sign, since, if our current run rate is less than the initial one, then we end up subtracting a negative, which is the equivalent of adding a positive, which just indicates a higher run rate required than before. Let’s see :
A running Commentary
And once again, we get the correct answer, so both the formulae work. These can even be adapted for the practice of Statistics, which also deals with averages of many sorts.
One other thing to consider, especially if you do follow Cricket and are aware of it, is that you can also find the run rate during an over. I prefer to do this by multiplying the number of overs by six, then adding to that number the number of extra balls on top of whole overs bowled, then expressing that number as being over six. For example, imagine 23 overs and two balls have been bowled. This is 23⅓ overs in total, but I refuse to round it to 23.33, because the rounding error will affect your answer. Instead I express it as ( 140 ÷ 6 ) overs, which is the exact number I would put into the formula, especially if it had been pre programmed into my calculator. It is not so bad if three balls had been bowled out of the six in the over, since this can be expressed accurately as .5.
Now, it could be said that a lot of what these formulae do can be done by other means, and a bit quicker. Yes, but what I really wanted to show, was a mathematical exercise that can work. One thing about the second formula, is that you no longer need to subtract runs to date from total runs required, and, if you have the kind of calculator into which you can programme things, then such a formula will in this way save time.
The last Cricket formula I have here is indeed what could be called a waste of time, since it is certainly quicker to work it out the normal way, but once again, for me it makes a good exercise in working out viable formulae, and trying to figure out something like this, such as the summation formulae we did before, does give you valuable practice and insight.
The formula following the Whitlam Dismissal photo just below can be used to calculate a batsman’s batting average. Now, we know that all we need to do for this, is to add up all their runs, then divide by the number of their dismissals (which, if you were Gough Whitlam, was only once), and there you have it. But let’s check what I have come up with.
Platinum Duck, and sadly, recently dismissed for well earned 98. Images courtesy Wikipedia, Sydney Morning Herald via Google, edited on Paintbox
Batting Averages Formula
I have a Dream
So, let’s say that a few years ago, the Blackcaps had a first drop, who had amassed in fifty five tests a total of 4500 runs in 100 innings, having 10 not outs, for an average of exactly 50.00. ( We can dream, can’t we ? )
Now in our first scenario, this plonker goes out onto the M.C.G. and amasses a well compiled duck. Were we to calculate his average the usual way, he would have had 101 innings, still ten not outs, and still 4500 test runs, but now his average would be 4500 ÷ 91 = 49.45, so of course it has gone down slightly.
So let’s plug these figures into my behemoth of a formula, to see if it agrees with the truth according to Wisden © :
Batting Formula Example
which is indeed, if I hadn’t rounded either total, exactly the same answer ! Mama mia, it works ! ( and I’m Irish, not Italian) So far, so good, but let’s continue.
The Second Dig
Now say, having retained the selectors’ favour, our Number Three shuffles out on Day Four, his head lowered in shame - still smarting from his previous act of stupidity, and this time, he makes exactly 100, before being taken in the slips by a very safe Ricky Ponting, one of the best in the World back in those days. If we work out his new average, we quickly realise that Mr. Fantastic has 4600 runs from 92 outs, giving him an average of 50.00 again. Well, he’s back where he started. Stands to reason. Pretty good, since most of the time New Zealand ends up going backwards ( to the Pavilion, mostly ). And if the Blackcaps have a problem with this comment, wake up, get out there, and make runs, instead of excuses. We know you can. After all, have we not suffered enough, and is it not simply that we want You to do well , ( as You seem to be doing at this time in Dec 2013 - keep it up ! ) For our batsman, this is the same as if he scored 50 and out off each innings. Let’s see if the formula agrees :
So again, it works ! You also see how I gave the previous average as the exact fraction it is, rather than the rounded decimal version, for more accuracy.
Another Five Days, another City, another Country
Let’s try another. Say that in the next series, wheret this time we are hosting Australia, for the First Test, our hero is put down the order to give him less pressure so he can come in once an innings has been built. So, at a respectable 78 - 4, Danger Mouse strides out to the middle of the Basin Reserve, and whacks up a match saving 250*, so that, if we calculate this turn of events the normal way, we see he has 4850 test runs for still 92 dismissals, giving him an impressive average of 52.717.
Let us see, shall we ?
Possible Image of Wellington Test in Question, Images from Wikipedia, edited on Paintbox
An interesting Foray into the World of Space and Time
There is a rather curious thing about a team trying to keep up with a run rate that gives a useful analogy to the idea of two craft travelling in outer space. Now imagine one is moving at great speed, away from the first, and achieving about, say, for argument's sake, 100,000 miles per hour. The second craft is following behind the first, but at only 75,000 miles per hour. This means, that although the second is following the first, and following behind it in the same direction, for every hour that passes, relatively speaking, it is as if the first is still, and the second is moving away at 25,000 miles per hour, for that is in effect how they are relating to each other. The second is getting further away from the first, but the difference is, the space they are travelling in, since the first being still, having the second moving away at that difference in their speed, is not ultimately the same as the second at 75,000 following the first at 100,000, but does have the same effect.
A similar thing can occur in Cricket. Imagine the team batting second needs 9.4 runs per hour at a given time, to win. Now say the batsman takes a single. Now sure, he is getting closer to the total, but now the required runs per over has just increased, so in terms of that, he is now a bit further away, and if he continues to get singles, even though he is getting closer to the total, he will run out of balls before he can get there. Each scoring shot is not bad in itself, and by doing so he may even have increased the team's overall run rate up to now, but he could still do that and yet also increase the run rate required, so that even though he is getting his team closer to the total, it is as if they are moving further away, although that is obviously not possible. He could even score so as to decrease his team's overall run rate, and yet that one score could still be more than the now required runs per over. This observation comes from many years of watching teams not keeping up with run rates, at least of individual balls.
Naturally if the batsman scores a boundary when need only 9.4, then the required will go down, and another interesting thing is that, the closer we get to the end of the match, with very few balls remaining, the greater proportion a single ball is of the overs remaining, and any action or lack thereof at later stages has far greater effects on the current run rate and run rate required than it would at the start. A four on the first ball of the innings chasing will make a huge mark on the current run rate, since four runs per ball is the equivalent in run rate of 24, but it may have little effect on the run rate required, as it is only one ball out of 300 legal deliveries allowed, and the difference four runs per ball is to the original run rate required is, as shown in my formulas, spread out over the remaining 49.5 overs.
So again, we have worked it out right. The formula, sure, as cumbersome as it may seem, does indeed work, and as I have stated before, still serves as an example of how one can work out an accurate universal equation for a good number of things, whether or not it is quicker than doing it the old way. It may even have other applications where it can speed things up. I have certainly managed to use my other cricket formula to facilitate problems in Statistics.
But now, having seen enough of numbers, let us take a little side journey into another aspect of mathematics, not so much involving numbers as we normally know them, but in a rather different way, but You will have to go to the next Hub to do that, and it will be all about Shapes.
Even though some of this Hub contains certain Mathematical knowledge that can be accessed in the public domain, and therefore is not subject to any Copyright, other information has been drawn from textbooks which themselves are Copyright, but only in the sense of how they deliver the information which itself is shared and sometimes ancient Mathematical knowledge. Other information has also been found on Wikipedia ( Copyright 2013, the Wikimedia Foundation ), which is a good source of information. Part of this is also my own discovery, but may also independently have been found by others.
Any quote or part of this material which seems to belong to any other author should be treated as such, and I claim no ownership of anything I did not myself invent or discover, nor of any obvious copyright, trade mark, or registered trade mark. A special mention goes to the band 10cc, and their song "Dreadlock Holiday" ( © 1978 ), where some lyrics from which have been paraphrased to create the title of this Hub. Reference to Jiminy Cricket, created by Carlo Collodi for his children's book The Adventures of Pinocchio, owned now by The Walt Disney Company.
Danger Mouse is a cartoon character produced by Cosgrove Hall Films for Thames Television. The Great Khali is currently working for WWE, but has worked for other wrestling and TV franchises. " I have a Dream " originally quoted publicly by Reverand Doctor Martin Luther King, Jr. on Wednesday, August 28, 1963. Delivered to over 250,000 civil rights supporters from the steps of the Lincoln Memorial during the March on Washington, the speech was a defining moment of the American Civil Rights Movement, as quoted in Wikipedia, reference given is Hansen, D, D. (2003). The Dream: Martin Luther King, Jr., and the Speech that Inspired a Nation. New York, NY: Harper Collins. p. 177. My use of this quote was in no way meant to make light of such a historical occasion, fifty years now having passed since this event.
Blackcaps logo owned by New Zealand Cricket, since the New Zealand cricket team became known as the Black Caps in January 1998, after its sponsor at the time, Clear Communications, held a competition to choose a name for the team. Cricket Australia owner of ACB logo. Wisden is the Cricket Almanac founded 150 years ago by the English cricketer John Wisden (1826–84) According to Wikipedia : Wisden was acquired and published by Robert Maxwell's publishing conglomerate, Macdonald, in the 1970s. Cricket fan Sir Paul Getty bought the company, John Wisden & Co, in 1993. And in December 2008, it was sold to A&C Black, which is owned by Bloomsbury.
The Adventure continues in the next Hub The Shape of Things to Come
If You are curious, then by all means do take a look at the other Hubs, The Maths They Never Taught Us - Part One, The Maths They Never Taught Us - Part Two , The Maths They Never Taught Us - Part Three, The Very Next Step - Squares and the Power of Two , And then there were Three - a Study on Cubes, Moving on to Higher Powers - a First look at Exponents, The Power of Many More - more on the Use of Exponents, Mathematics - the Science of Patterns , More on the Patterns of Maths Trigonometry to begin with, Pythagorean Theorem and Triplets, Things to do with Shapes, Pyramids - How to find their Height and Volume, How to find the Area of Regular Polygons, The Wonder and Amusement of Triangles - Part One, The Wonder and Amusement of Triangles - Part Two, the Law of Missing Lengths, The Wonder and Amusement of Triangles – Part Three : the Sine Rule, and The Wonder and Amusement of Triangles - Part Four : the Cosine Rule.
Also, feel free to check out my non Maths Hubs :
Just take a good look at it, and note how interesting it all is, then see if you can come up with anything else along the same lines. As usual, I would appreciate any comments, feedback and suggestions which would be given due credit, or indeed have a go and publishing Your ideas Yourselves, but firstly, by all means, add Your comments - it's a free Country.