# 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

## Just Imagine

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**.

## Batting Averages

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.

## Conclusion

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.

## Disclaimer

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.^{[8]} 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 :

Bartholomew Webb , They Came and The Great New Zealand Flag

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.