A contender would be the Normal Distribution curve, above, which encompasses so much of the natural and statistical world.

Or perhaps the more humble straight line: y = mx + c gets your vote

To simple? Well the quadratic curve crops up a lot and has the right level of difficulty to make it non-trivial, but is within the scope of all.

And then there is the sine curve, an understanding of which opens up a huge area of mathematics, not just a curve, but a wave.

But none of the above are my favourites, although they all stake a good claim.

No, my favourite curve was first plotted by a somewhat obscure 19th Century German Physcologist, Hermann Ebbinghaus, his “Forgetting Curve”

I first came across his curve a few years ago, and it made instant sense to me. In essence, we quickly forget what we have learned, no matter how well we may have been taught. However, if we revisit the work we soon remember but once again, we soon forget, although the “speed” of our forgetting is diminished, and we don’t forget quite as much as we did before.

If we continue this process, each time we forget less and retain more.

It confirms what experience has taught me – that we need to constantly revisit and revise work, it is not enough to master a topic in a week, because in a month we’ll have forgotten most of it.

The idea is being given a modern spin (retrieval practice and spaced learning being a few new buzz words that use Ebbinghaus’ work) and, whilst I *think* I’ve always done this in my teaching, since discovering Ebbinghaus I’ve made a more conscious effort to resit the relentless pressure to plough on with the syllabus and revisit work covered earlier in the term and year.

For example, every few weeks we have unit tests – so we revise a few weeks work for the test (the first revisit), but then after the test I also devote a lesson or two to revisit unit tests from earlier in the year, thereby revisiting topics again and again as the year unfolds.

To find out more about the Ebbinghaus Forgetting curve, this Wikipedia page is a good starting point.

And, as an aside, I’ve often thought tha investigating the Ebbinghaus Curve could make for a good EPQ project – plenty to research, plus the opportunity to devise one’s own test to try and replicate the curve, thereby creating primary data of their own.

(Now having read this blog post, make sure you came back next week, next month, next year … to ensure you don’t forget it!!)

]]>I’ve spent much of the last week helping students with “Hypothesis Testing” as they prepare for their A level exams in the next few months.

Fed up of wading through connived examples, and upon stumbling across perhaps the best headline I’ve read in sometime (“Man United regress to the mean after Solskjaer bounce“) I thought I’d use a bit of A level maths to see if the Solskjaer bounce was real or just another Norse myth.

(I’m about to walk through how this may be presented as an A level question, followed by my worked solution, so my less mathematically minded readers may want to skip the next few lines.)

Since taking over as manager of Manchester United in December 2018, Ole Gunnar Solskjaer (OGS) has transformed the club, returning it to winning ways.

In the season to date, in all competitions, Man Utd have won 24 out of 43 games.

Since OGS was appointed manager, Man Utd have won 15 of their 21 games.

Does the data support the theory that OGS has transformed the club at the 5% significance level?

Let p be the probability that Man Utd win a match. They have won 24 out of 43, so the probability of winning is 24/43 = 0.558

H_{o}:p=0.558

H_{1}:p>0.558

The null hypothesis is that the probability is 0.558, the alternative hypothesis is that the probability of a win (under OGS) is greater

Let X be the number of wins in a sample of 21 games (the games that OGS was manager)

If H_{o} is correct then X~B(21,0.558)

We’re modelling the data as a Binomial distribution as there are two outcomes: win, don’t win.

Test statistic: X = 15 (the number of games OGS won)

P(X>=15) = 1 – P(X<=14)

= 1 – 0.8905 = 0.1095

We use our calculator in Binomial CD mode to find the cumulative probability of up to, and including 14, then take that away from 1 to get the probability that Man Utd would win 15 or more of their games under the null hypothesis (i.e OGS has made no difference), which works out to be 0.1095 or 10.95%)

0.1095 is not less than 0.05

Sometimes easier to think in percentages, even if we give answers as decimals. 10.95% is not less than 5% (our significance level)

So there is insufficient evidence to reject H_{o}, our null hypothesis

**(Non-mathematicians, start reading from here)**

At the 5% significance level, there is no evidence to support the theory that OGS has transformed the club.

Or, in other words, the Solskjaer bounce is probably just a myth.

We have shown that even if there had been no change of manager there was a 10.95% chance Man Utd would have won 15 out of the next 21 games.

In fact, even if we widened the significance level to 10%, the data still wouldn’t have supported a Solskjaer bounce. Winning 15 out of the 21 games since taking charge whilst unlikely was not so unlikely that it couldn’t have been due to chance rather than the genius that is OGS.

My theory: if we’d looked for the bounce a little earlier, we may have found evidence for it – perhaps the bounce peaked at around 15 games and Man Utd are, indeed, now regressing to the mean.

As ever, statistics raises as many questions as they answer, but it is good to be able to apply some A level mathematics to answer a “real” question.

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What chance Brexit?

As the deadline date of March 29th looms ever closer, are we any closer to finding out what will happen?

On his blog Jon Worth has produced an excellent flow chart where can follow all the possible twists and turns before ending at one of five possible outcomes. In addition to suggesting possible outcomes, he has added probabilities at each decision point.

Using nothing more than GCSE Maths*, it is then possible to calculate the probabilities of the various outcomes.

I’ve done the maths for you, the results are below (rounded to 2 decimal places)

**No Deal Brexit 0.25 (or 25% chance)****General Election 0.22 (or 22% chance)****Never ending spiral/May brings deal back to Commons for 3rd time 0.17 (or 17% chance)****Peoples Vote 0.15 (or 15%)****May’s Deal 0.2 (or 20%)**

So, not much to pick between any of those.

Once again, many thanks for Jon Worth producing his flowcharts and sharing them (under creative commons sharealike license): Do visit his site : Brexit – Where now? The flowcharts and follow him on Twitter

*Although looking a little more complicated, this is no different from the decision tree questions when Bob & Linda play badminiton, familiar to many a Year 11 student. To find the probability of following a route to its conclusion, multiply all the probabilities along the path. Then add all the probabilities of the routes to that end to find the total probability of reaching that outcome. If unsure, ask a GCSE student – they’ll be able to explain!

]]>If you look at the Premier League table this evening you will see that **L**iverpool sit proudly atop the football pyramid.

No great surprise there, you may think.

But then see who tops the Championship tonight: **L**eeds United.

League One? **L**uton Town, and League Two? Yes, you’ve guessed it, **L**incoln City.

So all four of the top leagues are crowned by a team beginning with the letter L. All the more surprising as the only other team in the 92 that begin with L are Leicester City (who, coincidentally, play Liverpool this evening.)

But gets even better – if we go down to the next tier, The National League, we find that **L**eyton Orient lead that divsion!

What are the chances of that – having a team beginning with L at the of the top all first five leagues in English football?

Less than one in three million, I reckon.

(One in 3,317,760 to be precise)

So how did I arrive at this answer? Here’s how:

I assumed each side had the same chance of topping their table. With two out of twenty teams in the Premier League beginning with L, the probability of one of those teams topping the table is 2/20, which simpilfies to 1/10. By no means a certainty, but not improbable, either.

The Championship, League One, League Two and National League all have twenty four teams, with only one beginning with an L. So the probability is 1/24 that an L will top, say, the Championship.

But for all five leagues to be topped by a team beginning with L, we need The Prem, and The Championship, and League One, League Two and National League to all have L’s at the top, so we need to multiply those probabilities together:

1/10 x 1/24 x 1/24 x 1/24 x 1/24 =

1/3,317,760

Not sure if its ever happened before, but take a moment to enjoy it whilst it lasts – not an everyday occurrence.

I know who I’ll be supporting between now and the end of the season!

]]>Back in the summer, we were all struck with World Cup fever and, in this post, I shared some stats and charts looking at the heights of players in the tournament.

In a few days time the Six Nations rugby tournament kicks off for another year, so I thought it appropriate to have a look at the stats of those involved.

Above you can see a box plot illustrating the weights of the six squads.

[In a box plot, the line through the box is the median – or middle – value: half the players are heavier than this value, half lighter; the top of the box the upper quartile – 75% of data values (in this case, player weights) are below this line; the bottom the lower quartile – 25% of data values are above this line; the upper and whiskers lower whiskers denote data values that fall outside the middle 50%.]

The plot above suggests to me that England have the heaviest squad, France the lightest, and also the team with the greatest range of weights.

Another way to look at the weight of the squads is using density plot, below:

Having plotted the above, it looked like there were a couple of peaks in some of the distributions, particularly noticeable in the England, Ireland and Scotland squads. If we took a random selection of the population we world expect to see a much more smooth “normal” distribution, or bell curve.

But a rugby squad is not normal! I suspected that the weights of those who play in the scrum would be more than those in the backs. So I let the data do the talking, comparing all players taking part in the tournament by position: scrum or backs (I combined the data from all nations as to do this on a country by country basis would result in sample sizes that were too small)

The individual distributions for the backs and scrum are fairly “normal”, but it is clear that backs tend to be lighter than those who ply their trade up front in the scrum.

So what about player height? Below you can see some charts that map this data.

I also had a look at the age of each squad:

I’m not sure if any of the above can help predict the eventual tournament winners, but you might find it interesting reading as we eagerly await Friday night’s kick off.

]]>The “10 Year Challenge” is the current rage on social media, whereby you are encouraged to post a picture of yourself from 10 years ago, alongside one from now, to see how you’ve aged.

I thought it would be interesting to see how the last ten years have been for the nation’s football teams.

I found their league position at the end of the 2008/09 season and compared it to their current* league position.

*before kick off, Saturday 19th January 2019.

I’ve plotted the results above – those above the blue diagonal line are in a better place now than ten years ago, those below are now lower down the football league structure. The further from the line the better (or worse) a team has performed over those years.

I’ve reversed the direction of the x and y axis – it seemed more intuitive that way: the higher up the y axis, the higher your current league position (although a lower number e.g. Liverpool, who top the table on 19th January 2019, have a ranking of 1).

**Winners and losers:**

Bournemouth are the success story, a full 77 places above where they were ten years ago. Brighton are 47 places better off, all of which must be somewhat galling to their south coast cousins, Portsmouth who, along with Sunderland, have had the toughest ten years, slipping 31 places down the league ladder.

**Same old, same old:**

Four teams, Newcastle Utd., QPR, Sheffield Utd., and West Ham are in exactly the same place now, as they were ten years ago.

**Gone, but not forgotten:**

The observant amongst you will have spotted that there are not 92 clubs on the plot above. In that time ten clubs have been relugated from the league:

Aldershot Town

Barnet

Chester City

Chesterfield

Dagenham & Redbridge

Darlington

Hartlepool United

Hereford United

Leyton Orient

Stockport County

To be replaced by:

Burton Albion

Cambridge

Crawley

Fleetwood

Forest Green

Mansfield Town

Newport

Oxford Utd

Stevenage

Wimbledon

As ever, statistics only tell us so much, and end up posing more questions than they answer. What is the secret Bournemouth, Brighton, Luton & Rotherham’s success? I think it would be interesting to see if there is any correlation between length of tenure/number of managers and how well a team has fared? And if so, which is the cause and which the causation? Do managers stay because the team is successful, or is a team successful over the long term because the manager stays? Does geography have a part to play? What happens if we look at trends over twenty years, not ten?

So how has your team done over the last ten years?

Below are some screen shots of the spreadsheet I used in my calculations. The number is the amount of places they have moved, positive is good, a negative number is how many places they have dropped.

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I recently discovered the video above that ranks the top ten countries by GDP from 1960 to 2017.

It is quite mesmeric watching it (I am reminded of the great Hans Rosling and how he presented data) and got me wondering: is it Economics, History or Statistics? (It is, of course, all three)

And, as I’ve said before, good statistics always prompt more questions than they answer. The first of which may be:

What about GDP per capita?

Well here’s a video that answers that question:

… which prompts the question:

so what happened to Monaco in 2012 and Lichtenstein in 2016?

… for which I don’t (yet) have an answer.

Whatever you take from the above videos, however you use them, I hope you enjoyed them, and I hope they’ve raised some questions of your own.

The videos were made by WawamuStas with the original data coming from the World Bank, a great source of Large Data Sets

]]>A student asked me a difficult question the other day.

I’m normally pretty confident with my subject knowledge, and am rarely stumped when quizzed out of the blue. Sometimes a tricky question from Further Maths, or a more esoteric A level problem may leave me scratching my head for a minute or two. Worst case scenario, I may need to ponder the problem for ten minutes in the calm, peace and quite of break time or lunch time, when I can focus on it without distraction, but, typically, I’ll get there in the end and give the pupil the answer they were seeking.

But not this time. As soon as the question was asked, I knew I could not give a definitive answer.

I tried flanneling and digressing, diverting and avoiding, but this Year 11 student was having none of it (perhaps a future career as a “Paxman” on Newsnight beckons?)

In the end I hand to come clean, I had to give an answer, so I did, but I still feel uneasy about it as I’m not sure I’d give the same answer today, as I did then (but I might do.)

So what was this question that floored me?

Sir, what is your best, ever, music track?

And you can’t answer that question as it constantly changes (but if you want to know what answer I gave when my resistance crumbled, then keep reading.)

I was reminded of this exchange as I’ve just seen my Spotify data for the year.

We live in the age of Big Data and understanding this, how its used and how it shapes our lives is an important lesson for us all to learn.

Fortunately, I love data and I didn’t just stop with what Spotify told me in their glossy end of year review of me, and my listening habits.

I was able to work out that it costs me less than a penny a minute to listen to Spotify, over the year I spent about 50p an hour listening to my music through their streaming site.

Good value? I think it is, I love my music and having so much on tap makes that a price I’m happy to pay (and, as I’m on a family membership, the cost per hour for all four of us in the household is significantly less.)

But the important thing is is that I was able to calculate that cost, and then decide if it was good value for money for me. Many of your students will have received a similar review from Spotify – why not get them to calculate what it costs them (or, more likely, their parents) for each hour they use the service? The maths is pretty simple, but the process and analysis is so important. I suspect that if I did a similar calculation for my gym membership it *may* not be such good value for money. Netflix – how much do a pay for each hour I watch?

Still with me? That’s probably because you want to know my “favourite track”.

Well this are my favourites based on my Spotify listening:

But how did I answer *the *question?

Well, the band in question – The Jam – is in the list above, but not the song.

So what is my all time favourite track? With the caveat that it changes, I can reveal it as “Thick as Thieves” by The Jam.

I came across the graph above and I was immediately struck by the stories it tells by forcing you, the reader, to ask the obvious questions.

Clearly something happened in the 1990’s.

The peace process was begun in Northern Ireland, culminating in the Good Friday Agreement of 1998. Surely this graph alone is enough to convince anyone of the importance and historical significance of the Good Friday Agreement? Why would anyone do anything, anything, to jeopardise its continued success? If anyone should need any convincing that we shouldn’t, we mustn’t, return to a hard border on the island of Ireland, then surely this graph must be all it takes.

86% of the deaths between 1970 and 1990 were in Northern Ireland

1988 – includes 271 deaths due to the Lockerbie bombing, when Pan Am flight 103 from Frankfurt to Detroit, via London & New York, was destroyed in the air over the Scottish town of Lockerbie by a terrorist bomb.

2005? The tragedy of the London bombings, or 7/7

A simple, sobering graph, but one that deserves – demands – to be viewed.

]]>As ever, the publication last week of A level results, and the imminent release of GCSE results later this week, signal the beginning of the end of the wonderful long summer holidays.

I hope you’ve had a fantastic time and, possibly, enjoyed the delights of foreign travel. If you did, I doubt you flew with an airline with such a relaxed seating plan as the one in the problem below …

On this particular flight, MathAir 314, there are 100 hundred passenger seats, and 100 passengers.

The first passenger to board has lost their boarding pass and doesn’t know their seat number. “No worries” declares the helpful steward, sit wherever you like.

The next (and all subsequent passengers) does have her boarding pass – if her seat is free, she sits in that seat. If it is occupied, the helpful steward allows them to chose any unoccupied seat they wish. This continues until all 100 seats are filled by the 100 passengers and the jet departs for its destination, Angle C (Anglesey – geddit?)

The question is:

**What is the probability that passenger 100 gets to sit in their own allocated seat?**

I’m indebted to Zoe Griffiths, @ZoeLGriffiths for this problem and you can see her video introducing the problem and, more importantly, her solution in the video below. But before you watch it, have a go at solving the problem yourself first. (You can start the video and the pause it after 1min 30 sec to see her intro to the problem.)

How might I use this problem? I might introduce it to a class towards the end of a lesson, and ask them to go away and think about it, reporting back with possible solutions, or even just approaches to a solution, next lesson.

Or I might offer the problem at the end of a weekly department meeting, inviting colleague to think about it before next week’s meeting. PE teachers regularly play their sports for fun, music teachers their instruments, its important for maths teachers to remain engaged with the subject and “do” some maths from time to time.

So, here’s the video with the solution, but give the problem some thought before you hit play.

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