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gavel

Amanda Knox and Bad Maths in Courts

This post is inspired by the recent BBC News article, “Amanda Knox and Bad Maths in Courts.”   The article highlights the importance of good mathematical understanding when handling probabilities – and how mistakes by judges and juries can sometimes lead to miscarriages of justice.

A scenario to give to students:

A murder scene is found with two types of blood – that of the victim and that of the murderer.  As luck would have it, the unidentified blood has an incredibly rare blood disorder, only found in 1 in every million men.  The capital and surrounding areas have a population of 20 million – and the police are sure the murderer is from the capital.   The police have already started cataloging all citizens’ blood types for their new super crime-database.  They already have nearly 1 million male samples in there – and bingo – one man, Mr XY, is a match.  He is promptly marched off to trial, there is no other evidence, but the jury are told that the odds are 1 in a million that he is innocent.  He is duly convicted.   The question is, how likely is it that he did not commit this crime? 

Answer:

We can be around 90% confident that he did not commit this crime.  Assuming that there are approximately 10 million men in the capital, then were everyone cataloged on the database we would have on average 10 positive matches.  Given that there is no other evidence, it is therefore likely that he is only a 1 in 10 chance of being guilty.  Even though P(Fail Test/Innocent) = 1/1,000,000,  P(Innocent/Fail test) = 9/10.

Amanda Knox

Eighteen months ago, Amanda Knox and Raffaele Sollecito, who were previously convicted of the murder of British exchange student Meredith Kercher, were acquitted.  The judge at the time ruled out re-testing a tiny DNA sample found at the scene, stating that, “The sum of the two results, both unreliable… cannot give a reliable result.”

This logic however, whilst intuitive is not mathematically correct.   As explained by mathematician Coralie Colmez in the BBC News article, by repeating relatively unreliable tests we can make them more reliable – the larger the pooled sample size, the more confident we can be in the result.

sally clark

Sally Clark

One of the most (in)famous examples of bad maths in the court room is that of Sally Clark – who was convicted of the murder of her two sons in 1999.  It has been described as, “one of the great miscarriages of justice in modern British legal history.”  Both of Sally Clark’s children died from cot-death whilst still babies.  Soon afterwards she was arrested for murder.  The case was based on a seemingly incontrovertible statistic – that the chance of 2 children from the same family dying from cot-death was 1 in 73 million.  Experts testified to this, the jury were suitably convinced and she was convicted.

The crux of the prosecutor’s case was that it was so statistically unlikely that this had happened by chance, that she must have killed her children.  However, this was bad maths – which led to an innocent woman being jailed for four years before her eventual acquittal.

Independent Events

The 1 in 73 million figure was arrived at by simply looking at the probability of a single cot-death (1 in 8500 ) and then squaring it – because it had happened twice.  However, this method only works if both events are independent – and in this case they clearly weren’t.  Any biological or social factors which contribute to the death of a child due to cot-death will also mean that another sibling is also at elevated risk.

Prosecutor’s Fallacy

Additionally this figure was presented in a way which is known as the “prosecutor’s fallacy” – the 1 in 73 million figure (even if correct) didn’t represent the probability of Sally Clark’s innocence, because it should have been compared against the probability of guilt for a double homicide.   In other words, the probability of a false positive is not the same as the probability of innocence.  In mathematical language, P(Fail Test/Innocent) is not equal to P(Innocent/Fail test).

Subsequent analysis of the Sally Clark case by a mathematics professor concluded that rather than having a 1 in 73 million chance of being innocent, actually it was about 4-10 times more likely this was due to natural causes rather than murder.  Quite a big turnaround – and evidence of why understanding statistics is so important in the courts.

This topic has also been highlighted recently by the excellent ToK website, Lancaster School ToK.

If you enjoyed this topic you might also like:

Benford’s Law – Using Maths to Catch Fraudsters

The Mathematics of Cons – Pyramid Selling

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IB Maths Revision

I’d strongly recommend starting your revision of topics from Y12 – certainly if you want to target a top grade in Y13.  My favourite revision site is Revision Village – which has a huge amount of great resources – questions graded by level, full video solutions, practice tests, and even exam predictions.  Standard Level students and Higher Level students have their own revision areas.  Have a look!

Does it Pay to be Nice?  Game Theory and Evolution

Golden Balls, hosted by Jasper Carrot, is based on a version of the Prisoner’s Dilemma. For added interest, try and predict what the 2 contestants are going to do. Any psychological cues to pick up on?

Game theory is an interesting branch of mathematics with links across a large number of disciplines – from politics to economics to biology and psychology.  The most well known example is that of the Prisoner’s Dilemma.  (Illustrated below).  Two prisoners are taken into custody and held in separate rooms.  During interrogation they are told that if they testify to everything (ie betray their partner) then they will go free and their partner will get 10 years.  However, if they both testify they will both get 5 years, and if they both remain silent then they will both get 6 months in jail.

prisoner dilemma

So, what is the optimum strategy for prisoner A?  In this version he should testify – because whichever strategy his partner chooses this gives prisoner A the best possible outcome.  Looking at it in reverse, if prisoner B testifies, then prisoner A would have been best testifying (gets 5 years rather than 10).   If prisoner B remains silent, then prisoner A would have been best testifying (goes free rather than 6 months).

This brings in an interesting moral dilemma – ie. even if the prisoner and his partner are innocent they are is placed in a situation where it is in his best interest to testify against their partner – thus increasing the likelihood of an innocent man being sent to jail.  This situation represents a form of plea bargaining – which is more common in America than Europe.

Part of the dilemma arises because if both men know that the optimum strategy is to testify, then they both end up with lengthy 5 year jail sentences.  If only they can trust each other to be altruistic rather than selfish – and both remain silent, then they get away with only 6 months each.   So does mathematics provide an amoral framework?  i.e. in this case mathematically optimum strategies are not “nice,” but selfish.

MAD

Game theory became quite popular during the Cold War, as the matrix above represented the state of the nuclear stand-off.  The threat of Mutually Assured Destruction (MAD) meant that neither the Americans or the Russians had any incentive to strike, because that would inevitably lead to a retaliatory strike – with catastrophic consequences.  The above matrix uses negative infinity to represent the worst possible outcome, whilst both sides not striking leads to a positive pay off.  Such a game has a very strong Nash Equilibrium – ie. there is no incentive to deviate from the non strike policy.  Could the optimal maths strategy here be said to be responsible for saving the world?

selfish gene

Game theory can be extended to evolutionary biology – and is covered in Richard Dawkin’s The Selfish Gene in some detail.  Basically whilst it is an optimum strategy to be selfish in a single round of the prisoner’s dilemma, any iterated games (ie repeated a number of times) actually tend towards a co-operative strategy.  If someone is nasty to you on round one (ie by testifying) then you can punish them the next time.  So with the threat of punishment, a mutually co-operative strategy is superior.

You can actually play the iterated Prisoner Dilemma game as an applet on the website Game Theory. Alternatively pairs within a class can play against each other.

An interesting extension is this applet, also on Game Theory, which models the evolution of 2 populations – residents and invaders.  You can set different responses – and then see what happens to the respective populations.  This is a good reflection of interactions in real life – where species can choose to live co-cooperatively, or to fight for the same resources.

The first stop for anyone interested in more information about Game Theory should be the Maths Illuminated website – which has an entire teacher unit on the subject – complete with different sections,a video and pdf documents.  There’s also a great article on Plus Maths – Does it Pay to be Nice? all about this topic.  There are a lot of different games which can be modeled using game theory – and many are listed here . These include the Stag Hunt, Hawk/ Dove and the Peace War game.  Some of these have direct applicability to population dynamics, and to the geo-politics of war versus peace.

If you enjoyed this post you might also like:

Simulations -Traffic Jams and Asteroid Impacts

Langton’s Ant – Order out of Chaos

IB Revision

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If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

Screen Shot 2018-03-19 at 4.42.05 PM.pngThe Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

Screen Shot 2019-07-27 at 10.02.40 AM

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

Is God a Mathematician?

That’s the provocative question posed by American Physicist Michio Kaku in this fascinating 5 minute interview which takes in the ideas of Newton, Einstein and modern ideas on String Theory.  It addresses the fundamental questions in maths ToK – is mathematics invented or discovered?  What explains the “unreasonable effectiveness” of mathematics in the universe?

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.

Mathematician Eugene Wigner (quoted above) has called this seemingly fundamental relationship between mathematics and our world as nothing short of miraculous.  Why should the motion of planets, the interaction of subatomic particles, the energy of a Black Hole, the growth of populations, the energy in a spring all  be described by mathematical equations?

Since the invention of quantum mechanics in the 1920s, mathematicians have been searching for a grand unified theory – a so called Theory of Everything – which can describe the entire universe through a single mathematical equation.  We have quantum mechanics which describes the sub-atomic world with remarkable accuracy, and general relativity which does equally well with describing the macroscopic world.  However the 2 theories break down when objects are both massive and subatomic (such as the singularity in a Black Hole or the conditions at the birth of the Universe) – hence the need for a single unified theory that can bridge this gap.

Another Michio Kaku video talking through the problem with describing the singularity at the heart of a Black Hole

For a more in-depth look at this topic, read, “Is Maths Invented or Discovered?” 

This blog was inspired by a video posted on Larry Ferlazzo’s ToK site – which has hundreds of links and ideas for ToK maths articles in the news – well worth reading!

premier league net debt

Premier League Finances – Debt and Wages

This is a great article from the Guardian DataBlog analysing the finances for last season’s Premier League clubs. As the Guardian says, “More than two thirds of the Premier League’s record £2.4bn income in 2011-12 was paid out in wages, according to the most recently published accounts of all 20 clubs. The Guardian’s annual special report of Premier League clubs’ finances shows they spent £1.6bn on wages last season, most of it going to players.”

The first graph (above) shows the net debt levels for different clubs.

The second graph shows the total turnover:

turnover premier league

The third graph shows wages as a proportion of turnover:

wages as a proportion of turnover

and the last one is particularly interesting – as it ranks clubs on their wage bills and their league position. This would be an interesting piece of data to test for the strength of correlation:

wage bill to league position

I’ve used an online scatter plot to calculate both the regression line and the correlation coefficient:

scatter plot correlation coefficient

Which clearly shows a strong positive correlation.  This would be an interesting exercise for both IGCSE or  IB students (especially Maths Studies).

For even more data, a club by club full breakdown is also provided by the Guardian here.  I have also made the data above into a word document to be used as a some A4 posters – and you can download that here: Premier League Debt

If you enjoyed this post you might also like:

Which Times Tables do Students Find Difficult? An Investigation.

Why Study Maths? Careers Inspiration

IB Revision

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If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

Screen Shot 2018-03-19 at 4.42.05 PM.pngThe Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

Screen Shot 2019-07-27 at 10.02.40 AM

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

maths careers

This is inspired by a fantastic website – we use math – which has a massive amount of information about different careers using mathematics in a really well laid out format.

According to a comprehensive careers survey by Careers Cast – which looked at over 200 different jobs and ranked them for stress, pay, job stability and work environment – 6 of the top 10 jobs require or strongly prefer maths graduates or those with a good mathematical background.

maths jobs jobs2 jobs3

Meanwhile, a 2009 Survey by the National Association of Colleges and Employers – which looked at American students’ job prospects after leaving university found that – “The top 15 highest-earning college degrees all have one thing in common — math skills.”

“Math is at the crux of who gets paid,” said Ed Koc, director of research at NACE. “If you have those skills, you are an extremely valuable asset.”

maths jobs

From the we use math site there is information (with interviews, maths required, job skills etc) on about 50 maths related jobs – as well as some pretty impressive statistics about the usefulness of mathematics:

These are the A-E jobs included on the site:

maths careers

There is also an accompanying video with interviews from a large range of mathematicians talking about the jobs they have gone into:

Along with the equally brilliant Maths Careers website there should be enough ammunition to never be stumped by the perennial, “When are we ever going to use this in real life?”

I’ve used the data above to make a poster for display – it’s downloaded from here: Want to be an astronaut?

Michio Kaku – American Professor of Theoretical Physics and fantastic populariser of mathematics and physics takes us through a 40 minute journey on the importance of physics in explaining the universe:

Great stuff – well worth watching!

 

grahams number

Graham’s Number – literally big enough to collapse your head into a black hole

Graham’s Number is a number so big that it would literally collapse your head into a black hole were you fully able to comprehend it. And that’s not hyperbole – the informational content of Graham’s Number is so astronomically large that it exceeds the maximum amount of entropy that could be stored in a brain sized piece of space – i.e. a black hole would form prior to fully processing all the data content. This is a great introduction to notation for really big numbers. Numberphile have produced a fantastic video on the topic:

Graham’s Number makes use of Kuth’s up arrow notation (explanation from wikipedia:)

In the series of hyper-operations we have

1) Multiplication:

   \begin{matrix}    a\times b & = & \underbrace{a+a+\dots+a} \\    & & b\mbox{ copies of }a   \end{matrix}

For example,

   \begin{matrix}   4\times 3 & = & \underbrace{4+4+4} & = & 12\\    & & 3\mbox{ copies of }4   \end{matrix}

2) Exponentiation:

   \begin{matrix}    a\uparrow b= a^b = & \underbrace{a\times a\times\dots\times a}\\    & b\mbox{ copies of }a   \end{matrix}

For example,

   \begin{matrix}    4\uparrow 3= 4^3 = & \underbrace{4\times 4\times 4} & = & 64\\    & 3\mbox{ copies of }4   \end{matrix}

3) Tetration:

   \begin{matrix}    a\uparrow\uparrow b & = {\ ^{b}a}  = & \underbrace{a^{a^{{}^{.\,^{.\,^{.\,^a}}}}}} &     = & \underbrace{a\uparrow (a\uparrow(\dots\uparrow a))}  \\       & & b\mbox{ copies of }a     & & b\mbox{ copies of }a   \end{matrix}

For example,

   \begin{matrix}    4\uparrow\uparrow 3 & = {\ ^{3}4}  = & \underbrace{4^{4^4}} &     = & \underbrace{4\uparrow (4\uparrow 4)} & = & 4^{256} & \approx & 1.34078079\times 10^{154}& \\       & & 3\mbox{ copies of }4     & & 3\mbox{ copies of }4   \end{matrix}
3\uparrow\uparrow 2=3^3=27
3\uparrow\uparrow 3=3^{3^3}=3^{27}=7625597484987
3\uparrow\uparrow 4=3^{3^{3^3}}=3^{3^{27}}=3^{7625597484987}
3\uparrow\uparrow 5=3^{3^{3^{3^3}}}=3^{3^{3^{27}}}=3^{3^{7625597484987}}
etc.

4) Pentation:

   \begin{matrix}    a\uparrow\uparrow\uparrow b= &     \underbrace{a_{}\uparrow\uparrow (a\uparrow\uparrow(\dots\uparrow\uparrow a))}\\     & b\mbox{ copies of }a   \end{matrix}

and so on.

Examples:

3\uparrow\uparrow\uparrow2 = 3\uparrow\uparrow3 = 3^{3^3} = 3^{27}=7,625,597,484,987
   \begin{matrix}     3\uparrow\uparrow\uparrow3 = 3\uparrow\uparrow3\uparrow\uparrow3 = 3\uparrow\uparrow(3\uparrow3\uparrow3) = &     \underbrace{3_{}\uparrow 3\uparrow\dots\uparrow 3} \\    & 3\uparrow3\uparrow3\mbox{ copies of }3   \end{matrix}   \begin{matrix}    = & \underbrace{3_{}\uparrow 3\uparrow\dots\uparrow 3} \\    & \mbox{7,625,597,484,987 copies of 3}   \end{matrix}=\underbrace{3^{3^{3^{3^{\cdot^{\cdot^{\cdot^{\cdot^{3}}}}}}}}}_{7,625,597,484,987}

Which clearly can lead to some absolutely huge numbers very quickly. Graham’s Number – which was arrived at mathematically as an upper bound for a problem relating to vertices on hypercubes is (explanation from Wikipedia)

grahams number

where the number of arrows in each layer, starting at the top layer, is specified by the value of the next layer below it; that is,

G = g_{64},\text{ where }g_1=3\uparrow\uparrow\uparrow\uparrow 3,\  g_n = 3\uparrow^{g_{n-1}}3,

and where a superscript on an up-arrow indicates how many arrows are there. In other words, G is calculated in 64 steps: the first step is to calculate g1 with four up-arrows between 3s; the second step is to calculate g2 with g1 up-arrows between 3s; the third step is to calculate g3 with g2 up-arrows between 3s; and so on, until finally calculating G = g64 with g63 up-arrows between 3s.

So a number so big it can’t be fully processed by the human brain.  This raises some interesting questions about maths and knowledge – Graham’s Number is an example of a number that exists but is beyond full human comprehension – it therefore is an example of a upper bound of human knowledge.  Therefore will there always be things in the Universe which are beyond full human understanding?  Or can mathematics provide a shortcut to knowledge that would otherwise be inaccessible?

If you enjoyed this post you might also like:

How Are Prime Numbers Distributed? Twin Primes Conjecture – a discussion about the amazing world of prime numbers.

Wau: The Most Amazing Number in the World? – a post which looks at the amazing properties of Wau

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IB Revision

Screen Shot 2018-03-19 at 4.35.19 PM

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

Screen Shot 2018-03-19 at 4.42.05 PM.pngThe Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

Screen Shot 2019-07-27 at 10.02.40 AM

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

tegemark

Plus Maths has a large number of great podcasts which look at maths ToK topics:

1) An interview with Max Tegmark (pictured above) about why he thinks that the universe is itself a mathematical structure.

2) An interview with physicists David Berman about how many dimensions exist.

3) A talk with cosmologist John Barrow about infinity.

4) A discussion with Roger Penrose about the puzzle of time.

And many more.  Well worth a listen!

5) There’s also a good lecture by Professor Ray Monk on the University of Southampton page (see “useful downloads”) looking at the link between philosophy and mathematics – which takes a fascinating journey through the history of maths and the great ideas of great men.

6) Maths for Primates is a fantastic source of podcasts – 14 and counting, on fractals, Zeno, Hilbert’s hotel and more.

jeopardy

One of my favourite resources is the Jeopardy quizzes. For those not familiar with the game (I think it’s American), it’s a gameshow, where you get to choose questions of different levels of difficulty, from a range of categories. I downloaded the template from TES – it’s a ready-made powerpoint which you can click on to take you to relevant questions, and then another click returns you to the home screen. The class can be split into teams, each team given a whiteboard and (say) 2 minutes to answer a question. Teams with the correct answer get those points for their teams. The challenge round adds a bit more excitement – and you can add any general questions or puzzles – such as dingbats or memory challenges (memorise pi to 10 places etc).

I’ve uploaded 20 of the quizzes onto TES here.  There’s 20 different ones –  KS3 Algebra, shape and space, fractions and general revision for different levels, GCSE and IGCSE topic specific quizzes on algebra, geometry, trigonometry, number, probability, matrices, functions, algebra 2 and linear graphs.  I have also done a small number of IB ones – which I’ll link to when I have a few more to share.


What is the sum of the infinite sequence 1, -1, 1, -1, 1…..?

This is a really interesting puzzle to study – which fits very well when studying geometric series, proof and the history of maths.

The two most intuitive answers are either that it has no sum or that it sums to zero.  If you group the pattern into pairs, then each pair (1, -1) = 0.  However if you group the pattern by first leaving the 1, then grouping pairs of (-1,1) you would end up with a sum of 1.

Firstly it’s worth seeing why we shouldn’t just use our formula for a geometric series:

with r as the multiplicative constant of -1.  This formula requires that the absolute value of r is less than 1 – otherwise the series will not converge.

The series 1,-1,1,-1…. is called Grandi’s series – after a 17th century Italian mathematician (pictured) – and sparked a few hundred years worth of heated mathematical debate as to what the correct summation was.

cesaro summation

Using the Cesaro method (explanation pasted from here )

If an = (−1)n+1 for n ≥ 1. That is, {an} is the sequence

1, -1, 1, -1, \ldots.\,

Then the sequence of partial sums {sn} is

1, 0, 1, 0, \ldots,\,

so whilst the series not converge, if we calculate the terms of the sequence {(s1 + … + sn)/n} we get:

\frac{1}{1}, \,\frac{1}{2}, \,\frac{2}{3}, \,\frac{2}{4}, \,\frac{3}{5}, \,\frac{3}{6}, \,\frac{4}{7}, \,\frac{4}{8}, \,\ldots,

so that

\lim_{n\to\infty} \frac{s_1 + \cdots + s_n}{n} = 1/2.

So, using different methods we have shown that this series “should” have a summation of 0 (grouping in pairs), or that it “should” have a sum of 1 (grouping in pairs after the first 1), or that it “should” have no sum as it simply oscillates, or that it “should”  have a Cesaro sum of 1/2 – no wonder it caused so much consternation amongst mathematicians!

This approach can be extended to the complex series, 1 + i + i^2 + i^3 + i^4 + i^5 + \cdots which is looked at in the blog  God Plays Dice

This is a really great example of how different proofs can sometimes lead to different (and unexpected) results. What does this say about the nature of proof?

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IB Maths Exploration Guide

IB Maths Exploration Guide

A comprehensive 63 page pdf guide to help you get excellent marks on your maths investigation. Includes:

  1. Investigation essentials,
  2. Marking criteria guidance,
  3. 70 hand picked interesting topics
  4. Useful websites for use in the exploration,
  5. A student checklist for top marks
  6. Avoiding common student mistakes
  7. A selection of detailed exploration ideas
  8. Advice on using Geogebra, Desmos and Tracker.

Available to download here.

IB HL Paper 3 Practice Questions (120 page pdf)

IB HL Paper 3 Practice Questions 

Seventeen full investigation questions – each one designed to last around 1 hour, and totaling around 40 pages and 600 marks worth of content.  There is also a fully typed up mark scheme.  Together this is around 120 pages of content.

Available to download here.

IB Exploration Modelling and Statistics Guide


IB Exploration Modelling and Statistics Guide

A 60 page pdf guide full of advice to help with modelling and statistics explorations – focusing in on non-calculator methods in order to show good understanding. Includes:

  1. Pearson’s Product: Height and arm span
  2. How to calculate standard deviation by hand
  3. Binomial investigation: ESP powers
  4. Paired t tests and 2 sample t tests: Reaction times
  5. Chi Squared: Efficiency of vaccines
  6. Spearman’s rank: Taste preference of cola
  7. Linear regression and log linearization.
  8. Quadratic regression and cubic regression.
  9. Exponential and trigonometric regression.

Available to download here.

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