Four Ways of Thinking: Statistical, Interactive, Chaotic and Complex - David Sumpter

Summary

In this enlightening talk, the speaker, an applied mathematician, delves into the power and limitations of mathematical thinking in understanding the world. He explores four distinct modes of thinking: statistical, interactive, chaotic, and complex. Through engaging stories from football, science, and everyday life, he illustrates how these approaches can provide insight but also emphasizes the importance of recognizing when statistical significance does not equate to practical importance. The speaker highlights the beauty of chaos theory and the concept of complexity, as defined by Kolmogorov, suggesting that true understanding comes from finding succinct descriptions of complex phenomena.

Takeaways

• 📚 The speaker is an applied mathematician whose motivation stems from a desire to understand the world around them, rather than a love for calculations themselves.
• 🔢 The talk emphasizes four different ways of thinking: statistical, interactive, chaotic, and complex, using stories from football, science, and personal life to illustrate each.
• 👨‍💼 The speaker discusses the importance of experimental design and the contributions of Ronald Fisher, highlighting the power of statistics in understanding phenomena but also its limitations.
• 🏈 Football is used as a medium to demonstrate how statistics can measure aspects of a player's performance and attitude, challenging the notion that some elements are immeasurable.
• 🧬 The speaker introduces Alfred J. Lotka and his work on differential equations to model ecological interactions, such as predator-prey dynamics, showcasing interactive thinking.
• 🦋 The concept of chaos theory is introduced through the story of Margaret Hamilton and Edward Lorenz, illustrating how small differences can lead to vastly different outcomes, as in weather prediction.
• 🤖 An experiment is conducted during the talk to demonstrate the divergence of outcomes from small initial differences, reflecting the butterfly effect in chaos theory.
• 👨‍🚀 Margaret Hamilton's story is highlighted to show how a deep understanding of chaos led to meticulous control in critical situations, such as the Apollo moon mission.
• 🌐 The speaker discusses the balance between order (Yang) and chaos (Yin), suggesting that while we can control some aspects of life, we must accept the randomness in others.
• 🌟 The final point touches on complexity theory, suggesting that finding simple rules that capture the essence of complex systems can lead to a deeper understanding of the world.

Q & A

What is the main theme of the speaker's talk?

The main theme of the speaker's talk is to provide insight into their thinking process in four different stages: statistical, interactive, chaotic, and complex thinking, using examples from football, science, and personal life.

Why did the speaker choose to discuss Ronald Fisher in their talk?

The speaker chose to discuss Ronald Fisher because he was a significant figure in the development of applied mathematics and experimental design, and his work exemplifies the application of statistical thinking.

What is the significance of the milk-first or tea-first experiment mentioned in the talk?

The milk-first or tea-first experiment is significant because it demonstrates the application of combinatorics and experimental design to solve a seemingly trivial problem, highlighting the power of mathematical thinking in everyday situations.

How does the speaker use football as an example to illustrate statistical thinking?

The speaker uses football to illustrate statistical thinking by analyzing player performance metrics, such as how a player's performance changes when their team concedes a goal, and comparing these metrics to evaluate player attitudes and impact on team dynamics.

What is the 'Gary Neville statistic' referred to in the talk?

The 'Gary Neville statistic' is a measurable statistic developed by the speaker to quantify a player's performance change after their team concedes a goal, which was initially thought to be immeasurable by Gary Neville.

What is the limitation of using statistics to measure concepts like 'grit'?

The limitation of using statistics to measure concepts like 'grit' is that while it can provide some predictive power, it often only explains a small percentage of the variance in outcomes, indicating that many other factors contribute to success.

What is the role of Alfred J. Lotka in the development of interactive thinking?

Alfred J. Lotka played a crucial role in the development of interactive thinking by introducing the concept of unbalanced chemical equations to model ecological interactions, such as predator-prey dynamics, which laid the foundation for understanding complex systems.

How does the speaker use the applause experiment to illustrate interactive thinking?

The speaker uses the applause experiment to illustrate interactive thinking by showing how social behaviors, like clapping, can be modeled using the same principles as ecological interactions, demonstrating the spread of a 'social epidemic'.

What is the concept of chaos theory and how does it relate to the speaker's talk?

Chaos theory is the concept that small changes in initial conditions can lead to vastly different outcomes in complex systems, making long-term prediction impossible. It relates to the speaker's talk by illustrating the limitations of control and predictability, even with advanced mathematical models.

How does the speaker's personal life example of moving a sofa with friends relate to interactive thinking?

The speaker's personal life example of moving a sofa relates to interactive thinking by demonstrating how social interactions and cooperation can lead to a desired outcome, which is a simple model of how collective behavior can be understood and predicted.

What is the significance of the experiment involving doubling numbers and subtracting from 100 in the context of chaos theory?

The significance of the experiment involving doubling numbers and subtracting from 100 is to demonstrate how small differences in initial conditions can quickly lead to divergent outcomes, illustrating the concept of sensitive dependence on initial conditions in chaos theory.

What is the 'butterfly effect' in chaos theory and how was it discovered?

The 'butterfly effect' in chaos theory refers to the idea that the flap of a butterfly's wings in Brazil could set off a chain of events leading to a tornado in Texas. It was discovered by Edward Lorenz in the context of weather prediction models, where small differences in initial data led to vastly different forecasts.

Who is Margaret Hamilton and what is her contribution to the field of mathematics?

Margaret Hamilton was a mathematician who worked on the Apollo moon mission, creating the software that helped navigate the spacecraft and control the thrusters. She is known for her attention to detail and her work in ensuring error-free computation in critical systems.

What is the fourth way of thinking introduced by the speaker and how does it relate to complexity?

The fourth way of thinking introduced by the speaker is complex thinking, which is related to the concept of complexity as defined by Kolmogorov. It involves finding the shortest description that can produce a pattern, capturing the essence of complexity without losing detail or nuance.

What is the significance of cellular automata models in understanding complexity?

Cellular automata models, like the Game of Life, are significant in understanding complexity because they demonstrate how simple local interaction rules can lead to the emergence of complex patterns and behaviors, showcasing the self-organizing nature of complex systems.

Outlines

00:00

📚 A Mathematician's Journey and the Power of Statistics

The speaker, an applied mathematician, expresses gratitude for the invitation to speak and reflects on their previous work at Oxford University. They introduce the theme of the talk, which is to explore their thought process through four stages: statistical, interactive, chaotic, and complex thinking. The speaker emphasizes the importance of mathematics as a tool for understanding the world, rather than merely for calculations. They plan to illustrate their points with stories from football, science, and personal life, highlighting the multidisciplinary nature of their work.

05:00

🎲 The Birth of Applied Mathematics and Ronald Fisher's Legacy

This paragraph delves into the history of applied mathematics, highlighting the contributions of Ronald Fisher. Fisher, known for his arrogance and brilliance, sought to understand the real-world applications of mathematical theories. His work in experimental design and statistics, including the famous tea-tasting experiment with Dr. Muriel Bristol, demonstrated the power of statistical methods in practical scenarios. Fisher's innovative approaches to experimental design and his book on the subject have had a lasting impact, shaping the field of applied mathematics.

10:01

📊 Statistics in Football: Measuring Intangibles

The speaker discusses the application of statistical thinking in football, challenging the notion that certain aspects of the game, like a player's attitude, cannot be measured. They recount their experience with former footballer Gary Neville and present an analysis that quantifies the impact of a team conceding a goal on individual player performance. The 'Gary Neville statistic' is introduced as an example of how statistics can reveal the tangible effects of intangible aspects of the game.

15:01

🔍 The Limits of Statistics and the Dangers of Misinterpretation

The speaker acknowledges the limitations of statistical analysis, using the example of Angela Duckworth's 'grit' concept to illustrate how statistics can be misinterpreted. They explain that while 'grit' may predict some variance in success, it is only a small fraction of the whole picture. The speaker warns against confusing statistical significance with practical significance and emphasizes the importance of context and the difference between correlation and causation.

20:01

🧬 The Misuse of Statistics: Eugenics and Smoking

This paragraph explores the darker side of statistics, where they have been misused to support false theories, such as eugenics and the denial of the link between smoking and cancer. The speaker uses the example of Ronald Fisher, who, despite his significant contributions to statistics, used his skills to argue for scientifically unfounded and morally repugnant ideas, demonstrating the need for ethical considerations in the application of statistical methods.

25:01

🤖 Interactive Thinking and the Work of Alfred J. Lotka

The speaker introduces interactive thinking through the story of Alfred J. Lotka, who applied mathematical models to understand complex biological and ecological systems. Lotka's work on unbalanced chemical equations, such as the rabbit and fox population model, demonstrated how simple mathematical models could capture the dynamics of real-world interactions, leading to the development of new fields of study like mathematical biology.

30:03

🦊 The Spread of Social Phenomena: Modeling with Lotka's Equations

The speaker discusses the application of Lotka's equations to model social phenomena, such as the spread of applause among a group of people. They highlight the importance of understanding the cues and social dynamics that lead to collective behavior. The speaker also touches on the personal aspects of social interactions and the value of reflecting on these dynamics in everyday life.

35:03

🐟 Mathematical Modeling in Biology and Football

The speaker describes the process of creating mathematical models to understand complex systems, using the behavior of fish and football players as examples. They explain how simple rules of interaction can lead to complex collective behavior and how these models can be used to predict and analyze movements, escapes, and strategic decisions in both biology and sports.

40:04

🌀 The Emergence of Chaos Theory and its Implications

The speaker introduces chaos theory through the story of Margaret Hamilton and Edward Lorenz, who discovered the concept of the 'butterfly effect' while working on weather prediction models. They illustrate how small changes in initial conditions can lead to vastly different outcomes, emphasizing the inherent unpredictability in certain systems and the limitations of control and prediction.

45:05

🔄 The Experiment of Chaos: Diverging Numbers and Randomness

The speaker engages the audience in a simple numerical experiment to demonstrate the concept of chaos theory. By doubling numbers below 50 or subtracting from 100 and doubling for numbers above 50, the audience experiences how quickly numbers diverge from their initial values. This activity highlights the unpredictable nature of chaotic systems and the rapid spread of outcomes from similar starting points.

50:08

🌐 The Yin and Yang of Chaos and Order in Life

The speaker reflects on the personal implications of chaos theory, drawing a parallel between the concepts of Yin and Yang to represent order and disorder, respectively. They discuss the importance of finding a balance between controlling aspects of life that are important and embracing chaos for less critical matters. The speaker shares personal anecdotes to illustrate how understanding chaos theory has influenced their approach to life and relationships.

55:09

🛰️ Embracing Complexity and the Work of Kolmogorov

In the final paragraph, the speaker introduces the concept of complexity through the work of Kolmogorov, who defined a pattern's complexity as the length of the shortest description needed to produce it. The speaker suggests that finding concise yet detailed descriptions of complex phenomena is a key goal in science and understanding the world. They encourage younger researchers to seek these descriptions as a way to capture and make sense of complexity.

Keywords

💡Applied Mathematics

Applied Mathematics refers to the use of mathematical methods to solve practical problems in science, engineering, and other disciplines. In the video, the speaker, an applied mathematician, emphasizes its utility as a 'toolkit' for understanding the world around us, rather than focusing solely on calculations and theoretical proofs. The concept is central to the video's theme, as it underpins the speaker's approach to various problems, from football statistics to social interactions.

💡Statistical Thinking

Statistical Thinking involves the application of statistical methods and theories to analyze data and draw inferences. The speaker discusses this concept as one of the four stages of his thinking process. It is exemplified in the script through historical references to Ronald Fisher, who contributed significantly to statistical methods, and through modern applications like analyzing football player performance and TED Talk claims, demonstrating its relevance to both historical and contemporary contexts.

💡Effect Size

Effect Size is a measure of the magnitude of a phenomenon or an outcome in statistical analysis. It is mentioned in the video to illustrate the limitation of statistical significance, where a result may be statistically significant but have a small effect size, meaning it does not have a large practical impact. The speaker uses the example of 'grit' and its impact on success to clarify this concept, noting that grit only explains four percent of the variance, a small effect size in the context of predicting success.

💡Chaos Theory

Chaos Theory is a branch of mathematics that deals with systems that are highly sensitive to initial conditions, meaning small changes can lead to significantly different outcomes. The speaker introduces this concept by discussing the work of Edward Lorenz and Margaret Hamilton, highlighting the 'butterfly effect' through a simple doubling experiment. The concept is integral to understanding the limits of predictability and control in various systems, including weather prediction and football match outcomes.

💡Cellular Automata

Cellular Automata are mathematical models used to simulate complex systems based on simple rules and local interactions. The speaker refers to a project by a master's student, Michael Hansen, who created a cellular automata model that generates complex patterns from simple rules. This concept is used to illustrate the emergence of complexity from simplicity, a key idea in understanding complex systems without needing to control every variable.

💡Ecological Interactions

Ecological Interactions refer to the relationships and influences between different organisms and their environment. The speaker uses the Lotka-Volterra equations, which model the predator-prey dynamics between foxes and rabbits, to explain how ecological interactions can be mathematically represented. This concept is crucial for understanding the balance and changes in ecosystems, as well as the broader theme of interactive thinking.

💡Correlation vs. Causation

Correlation vs. Causation is a fundamental statistical concept that distinguishes between two variables being associated with each other (correlation) and one variable causing changes in the other (causation). The speaker warns against confusing these two, using the example of a player's activity when their team goes a goal down in a football match. The concept is essential for understanding the limitations of statistical analysis and the need for careful interpretation of data.

💡Ronald Fisher

Ronald Fisher is a key figure in the development of modern statistical methods, known for his work in experimental design and statistical inference. The speaker discusses Fisher's contributions to statistics, including his innovative approach to experimental design and his development of various statistical tests. Fisher's work is foundational to the speaker's discussion of statistical thinking and its applications and limitations.

💡Interactive Thinking

Interactive Thinking is an approach that involves building models to understand the interactions within a system. The speaker describes this as a progression from statistical thinking, where one uses mathematical models to explore how different elements within a system influence each other. Examples from the script include modeling the behavior of football players and fish, illustrating how this approach can lead to a deeper understanding of complex systems.

💡Complexity

Complexity, in the context of the video, refers to the intricate patterns and behaviors that emerge from simple rules or interactions. The speaker discusses the concept of complexity in relation to cellular automata and the work of Kolmogorov, emphasizing that a pattern's complexity is tied to the shortest description that can reproduce it. This concept is central to the video's exploration of how simple interactions can lead to complex outcomes, which is a key theme in understanding various systems in the world.

Highlights

The speaker, an applied mathematician, emphasizes the secondary role of calculations in understanding the world through mathematical tools.

The speaker's interest in mathematics stems from a desire to understand the world around them, rather than a passion for calculations.

The talk is structured around four different ways of thinking: statistical, interactive, chaotic, and complex.

Ronald Fisher's work in experimental design and combinatorics is highlighted, showing the power of using mathematics to solve real-world problems.

The story of Fisher's tea party experiment demonstrates the application of combinatorics in a light-hearted scenario.

The speaker discusses the limits of statistics, using the example of 'grit' and its minimal explanatory power in predicting success.

The importance of context in statistics is underscored, noting that correlation does not imply causation.

Alfred J. Lotka's introduction of interactive thinking through his work on ecological interactions and differential equations is presented.

The concept of chaos theory is introduced through the work of Margaret Hamilton and Edward Lorenz, emphasizing the unpredictability in systems.

The 'butterfly effect' in chaos theory is illustrated through a simple mathematical process showing how small differences can lead to vastly different outcomes.

The speaker uses football as an analogy to explain the balance between control (order) and randomness (chaos) in various aspects of life.

The importance of understanding the difference between statistical significance and practical significance is discussed.

The speaker's personal journey in mathematics and science, including working on a wide range of different areas, is shared.

The application of mathematical models in understanding social behaviors, such as applause in a group, is demonstrated.

The limitations of using mathematics to explain everything are discussed, using Lotka's unsuccessful attempt to create a 'Grand Theory of Everything' as an example.

The final way of thinking, complexity, is introduced with a cellular automata model showcasing how simple rules can create complex patterns.

The concept of complexity is related to the shortest description that can produce a pattern, as defined by Kolmogorov.

Transcripts

[Music]

thank you

thank you very much for the lovely

introduction and and being allowed to

come here for the third time it's a real

privilege to come and and talk to you

here I worked earlier in Oxford

University I think I left here about

2005 so it's very nice for to walk

around and all these old memories come

back so it's lovely to be here so what

am I going to talk about today well for

me

and this this was a quote that Daryl

took and used actually in advertising

this talk and I hadn't thought so much

about it but I wanted to to lift up

again because for me as an applied

mathematician the calculations are

secondary now I'm quite good at maths

I'm quite good at calculating things and

doing the manipulations required and I

suppose I teach it so I have to be

reasonably good at it but that's never

been the thing that motivates me first

I'm not one of these people who likes to

sit and do long calculations or well

maybe I like it a little bit but not a

great deal not as much as many of my

pure mathematics colleagues the reason I

got interested in mathematics and this

really came from an early age the reason

I got interested was I wanted to

understand the world around me and I

felt that mathematics was the toolkit

which I could use to get that

understanding

and there that's what we're going to

look at today that's the story I'm going

to tell I'm going to try and give you an

insight

into how my own thinking Works in four

different stages looking at statistical

interactive chaotic and complex thinking

and I'm going to illustrate it with

stories some of them are going to come

from football because Daryl has told me

that uh I'm very popular in my football

talks I thought I'd throw in a little

bit of football for you

some of it is going to come from other

parts of science so I'm an active

scientist working on a large range of

different areas

and some of it also comes from my

personal life so how I use mathematics

to think about the types of social

problems that I encounter every day how

how I interact with people when things

go wrong how I can find a solution for

them so I'm going to take all of those

three different branches Science

Football and our personal lives and use

them to illustrate these four different

ways of thinking

so the first way of thinking

we have is statistical

and when I did this I also went back in

time and really really Applied

Mathematics is just a little bit more

than 100 years old really the the things

that we use today so I went back in time

and I started with various

historical figures who built these

things

and this is Ronald Fisher this is him in

1912 when he was a student at Cambridge

University

and Fisher

was an incredibly arrogant young man he

believed that he was smarter than

everybody around him and at school he

was smarter than everybody around him

and he went to Cambridge University

which was this is going to sound

controversial but at the time if you

wanted to study mathematics Cambridge

University was the best place to study

mathematics in the world and he went

there and he found that he was pretty

much smarter than all the other students

and he also thought that he was smarter

than pretty much all of the professors

and so you can imagine Fisher he was

sitting in his room a few weeks before

the tripos exams which of course he aced

he was sitting in the room not studying

for the exams but he was trying to work

out how the mathematics he was using was

coupled to reality he felt that when the

the people who taught in mathematics the

professors who taught him when they used

it they didn't see the coupling between

what they were proving and the results

they had and how that actually could be

used and that's what he wanted to find

he was sitting looking for that solution

and it didn't go well for him right he

wrote an article nobody read it nobody

was interested in his ideas and he ended

up pretty much in the wilderness he

wanted to fight in World War One in

1914. he couldn't because he was too

short-sighted and he ended up buying a

farm and trying to run this Farm which

he was absolutely terrible at the

terrible at he just couldn't manage to

he wasn't very good at working hard he

was good at having theoretical ideas but

not getting anything anything done but

he was rescued and he was rescued by a

guy called Sir John Russell and Sir John

Russell

he ran rothemsted experimental station

and he actually said he was looking for

an oddball mathematician to look at all

their experimental results and so he

recruited Fisher and here is Fisher

pictured on the left-hand side this is

how he could be typically be seen at

rothenstead he would be sort of puffing

smoke and explaining ideas to people

and in this picture he's at a tea party

now Sir John Russell

um he instigated the tea party because

at rothamstead in 1919

um they started to have women working

there and he felt that if they had lady

employees he didn't really know what to

do with them but he knew one thing they

needed to drink tea and so he made he

made tea for these uh so they started

having a tea party for these um uh for

the woman ostentation essentially but he

went on oh well everybody had the tea in

the end and they became a very central

part of what was done at rothenstead

now at one of these tea parties

um though there was a a Dr Muriel

Bristol who was one of the

experimentalists and one of the people

who did the studies on crops and Fisher

was about to serve her tea and she said

stop

I need to have my milk first and Fisher

as usual in his arrogant way he just he

said nonsense this can't be true it

doesn't matter if you have your milk

first in your tea you can have your milk

afterwards you know it all mixes up

together there's no difference if you

have your milk first or if you have your

milk afterwards

and so and he wasn't he wouldn't just

stop there he wasn't happy until he'd

done an experiment and tested if Dr

Muriel Bristol could tell the difference

between uh if she had her milk first or

her tea first in her tea

and so he set up an experiment to do the

test or he got his some of his

colleagues they suggested various

methods how they might do that

and

I'm now going to ask you actually so

we want to test if Dr Muriel Bristol can

tell the difference between if the milk

goes into her tea first or if the tea is

put in before the milk

and I'm going to allow you to consider

three different ways or two different

ways plus another alternative for doing

this test so the first is to offer her a

pairwise challenge we offer her a milk

cup and a non-milk cup

and then we maybe randomize these in

different ways and we have four tests

for her

the other one is we present her with a

tray with milk and non-milk tea on the

tray and we ask her to identify the ones

that will milk first and the ones that

were not milk first okay so hands up who

thinks that the pairwise test is the

best way to do this

we've got a few people at the back hands

up who thinks that the milk first tea

first tray is the best

you can do better than this okay now

we've got a few more hands up so I'm

taking the rest of you uh going for

option C and you're also thinking that

there's no difference between these two

methods that they're both the same

okay so I'm not sure if that's okay

hands up if you do think there's no

difference between the two methods

I think we've got a little bit of a

majority there for for option b okay so

let's have a look at this and this is

the working that um

that fisher did

so if you've got a pairwise challenge

option A

when you're setting up the experiment

and this is key to how Fischer was

thinking when you're setting up the

experiment

you have 16 different ways you can

organize the cups so the black one the

black circle there is is T first the

white circle is milk first and you have

four different pairs and there's 16

different ways of arranging those pairs

so that's two to the power of 4 is 16

and the probability and this is the the

key here the probability that Muriel

Bristol gets this right is 1 in 16. she

has to in order to get them all right

well if she if she can't tell the

difference the probability she gets them

all right is 1 in 16.

now if we do hit option b

the one that you liked and this is going

to prove to be the better choice

there are eight places to put out the

first cup seven places for the second

cup these are the milk cups

um six places for the third cup and five

places for the fourth cup and you've put

them out at random and then you fill up

with the these would be the the milk

ones then you fill up with the non-milt

ones and then you can also think of the

ordering of the cups there's four times

three times two times one ways to do it

and using combinatorics you can find out

well there's 70 ways of placing the cups

and if you don't believe the math you

can sit and write them all out and I got

I got a little yeah I told you I don't

like calculating so I got a little bit

bored before I wrote all of the

different ways you can arrange the cups

out but you can arrange them in these

different ways and so if Muriel Bristol

can't tell the difference the

probability of getting all four right is

now only one in in seventy so the the

this test is the better way to do it and

this I think is a perfect example of

using a nice piece of mathematics in the

form of combinatorics and that's what

Fisher did he used different parts of

combinatorics to solve a problem in

experimental design

and he went on to write a book

which became is still a sort of handbook

used today of how you design different

experiments this is a slightly this is

his Latin Square design which is a

different design than the randomized

design we've just talked about but he

could then from there start to spread

his statistical ideas

and there's just an incredible power I

think in being able to think about the

correct way to do an experiment or the

correct way to analyze data

now I like to so I've as Daryl said I I

have worked on football and I've worked

in mathematics and football and this has

taken me on some amazing Journeys and

it's very nice to for example I was I

had a um thing with Gary Neville so I

like to show off now about my sort of

football and contacts I think Gary

Neville is the most famous uh footballer

I've I've met on my journey so I just

wanted to sort of throw that in there

um and what's what's invariably happens

when I talk to footballers or former

footballers or coaches about about

mathematics and football

is that they have this thing where they

say oh well you know numbers can tell us

something numbers can tell us a few

things but you can't measure a player's

attitude you can't measure and so Gary

said that when we did this thing

together he said oh you can't measure if

a team goes a goal down

you can't measure the player who really

gets everyone going and really rallies

the team and gets them going again

and I was sitting there thinking yeah

actually that's exactly what you can

measure with Statistics and so a few

days after

um Gary Gary said this I sent him an

analysis where we analyzed exactly that

we look to see what happens when a team

um so when a team can see the goal so

this is Trent Alexander Arnold in a game

in the 21-22 season and the line the

central line here the central dotted

line

which is highlighted

is when they conceded a second goal

against United they lost the game 2-1 in

the end and just after half time they

conceded a second goal

and the squiggly line that's going up

and down that's Trent Alexander Arnold's

performance on the ball

and

as time goes on you can see that he's

getting better and better he's he's

actually producing lots of good passes

for his teammates

and then the goal goes in dotted line

and then you see his performance drops

back down again and so in this

particular case if we do the Gary so we

ended up calling this the Gary Neville

statistic so if we do the Gary Neville

statistic on this his performance goes

down after they concede a goal compared

to his performance in the 15 minutes

before they concede a goal so it's a

measurable statistic and in this way you

can for example we looked at the top

Strikers in 2122 and it was quite

interesting because you might call

um Jamie vardy a player who's got a lot

of attitude or character or something

like that and then it turns out that's

what we got when we measured it when

when

um when Jamie vardy's team went down

they went down a little bit more often

than some of the other teams but he got

better 29 of the times worse 13 of the

times and his performance was the same

about five of the times and you can see

yeah there's a ranking of the top five

players for That season in these

different situations

and it's very nice I've put in

parenthesis here because we used

precisely one of Ronald Fisher's tests

Fisher's exact test in order to test if

these players were statistically better

when they're when their team went down

or not

and so there's all types of ways in

which we can use statistics another

um example

and now we're kind of moving over to the

edges of the limits of statistics

and I think that this is a very

important point because

while while I think Gary is wrong in

that you can't measure attitude at all

he's right in another way because you

can't measure everything you can measure

certain aspects of how a player gets

better it's one piece of information you

have but you can't measure everything

and this is from a study uh from a TED

Talk and during the writing of the book

I watched the 25 most popular TED Talks

because I was very interested how they

use statistics in order to assess the

validity of claims in TED Talks and this

was a talk by Angela Duckworth and she

said that Grit and grit is the idea of

determination how determined you are to

succeed the grit is the strongest

predictor of success and it comes from a

study that she conducted

um together with some colleagues and

what they did is they looked at Ivy

League undergraduates I think at Yale

they looked at U.S military cadets and

they looked at people who were competing

in a spelling bee and before they

started doing these activities they

asked them questions about how if you

start something do you always see it

through it those types of questions a

series of 12 questions about if they

were determined gritty types of people

and they found that grit this the

answers that people gave in those

questions were some of the biggest

predictors of success and that's what

she said in the in the talk and that

sounds very impressive a bit like me

trying to persuade Gary Gary Neville

that we can we can measure

um attitude in football players

but

if you look a little bit more closely at

this the actual study as opposed to the

yes we shouldn't measure we shouldn't

measure success on how many times people

have watched the YouTube video because

this YouTube video has been watched 25

million times

and it's it's not Angela Duckworth who

wrote the headline on it but

her paper

reveals quite clearly and as she doesn't

try to hide this in any way at all the

grit just explains four percent of the

variance between people

now four percent of the variance how

much is that it isn't

it doesn't mean that only four percent

of people are explained by this I'm

going to try and show you what four

percent of the variants look like four

percent of the variance looks like this

so

if you measure grit on the scale one to

five on the bottom here and you look at

grade point averages for example now

this isn't real data this is data I've

just made up in order to illustrate what

four percent of the variance looks like

four percent of the variants would have

some type of relationship a bit like

that

and you can see if you squint carefully

I can't see it from this angle but I

think maybe you can squint and see this

there is a kind of increasing Trend

there between grit and grade point

average but you also see that some

people who yeah and there's there's lots

of people who are very gritty and are

successful and there's people who aren't

gritty and unsuccessful there's also

lots of people who are very gritty and

don't get a high grade and there's lots

of people who aren't very gritty and do

get a high grade and so when you're

interpreting this you shouldn't confuse

so I often say that you shouldn't

confuse the forest for the tree you're a

tree right every person in this in this

room is a tree so if we tested all of

your grittiness and your success in life

we'd find some kind of relationship like

this but it wouldn't mean that you

necessarily as an individual

um had this relationship between grit so

if you're not a gritty person if you

never see through any projects you start

you don't need to worry at all you're

going to be absolutely fine there's lots

and lots of ways in which your life can

succeed

and this what I've tried to do here and

I'm not sure if I've got the art quite

right here but the the arc I want to

describe here is I I want to start by

saying statistics is very powerful I can

show off to Gary Neville about it but

then at some point statistics doesn't

actually give you all of the answers

right and that's very well Illustrated

if we actually go back to Ronald Fisher

this rather arrogant young undergraduate

student because Ronald Fisher also has

another scientist story

he was from a very early age and this

picture was taken in 1912

very interested in Eugenics and he had

this idea that we needed to breed people

to be more like him like to be more

clever and smart and good at maths and

so on and he campaigned all the way up

to the war I think after the war he kept

more quiet about this

but all the way up to the war he

campaigned for for example sterilizing

people who were considered feeble-minded

and this is this is of course horrible

and I mean it's a horrible thing to

think about but not only is it like

morally repugnant it's also

scientifically wrong they couldn't find

any kind of Gene for single uh for

feeble-mindedness there is no

correlation between or there's a very

weak correlation or no correlation at

all between feeble-mindedness in one

generation in mothers and in their

daughters so this was a relationship

that was scientifically dubious that he

continued to press forward and the

reason he was successful is because or a

reason he was successful in pushing us

forward is is thing forward was that he

would use statistics in order to sort of

attack his opponents he would call them

all stupid they didn't understand

statistics so he would actually use

statistics to undermine other people's

arguments in a really counterproductive

way he took a fake Theory and then used

statistics to defend it and he didn't

just do this once after the war when

he'd given up on on Eugenics or at least

stopped talking about it

he then did the same thing on smoking so

as we saw in the first picture he was a

very keen smoker

and for him there was just no

possibility that smoking could cause

cancer

and so he spent a lot of time

investigating very narrow areas of

science doing his statistics on that and

trying to convince people that smoking

didn't cause cancer and I don't know

the effect that this research had but

certainly you've got one of the leading

statisticians in the world who's

defending this position for a long long

time

and that

illustrates a lot of why statistics is a

limited approach so I've written down a

few a few sort of bullet points here so

statistical thinking doesn't provide all

the answers

um

one problem is and I I didn't really get

into this but I mean what a dick he is

right

I mean why do you need to test if she

you know can tell the difference between

the milk first everyone was very happy

they were all just enjoying their tea

party and suddenly he's doing a

statistical day I mean you know he's an

idiot so so that's one thing and and I

really think you know we joke about that

but we see it at work all the time you

know that's always being told that we

should be there should be statistical

tests and metrics made about us and

things like that and we don't need as

much quantification as we have

then there's the effect size thing and

we sometimes talk about statistical

significance you can have statistical

significance but still have a very small

effect size and that's the confusing the

tree in the forest many non-gritty

individuals are successful in life

context is always important so just

because a player is more active when

their team goes a goal down does not

imply the team plays better just because

Ronaldo demands that everyone gives him

the ball when they go a goal down or

Jamie vardy demands that he gets the

ball that doesn't mean that the team

actually does better as a result of that

so it's very important to think about

the context of these types of things and

then as we all know correlation and

causation they're not the same thing

but that leads us to the to the next

step we need to think about ways to

tease out causation we want to be able

to tease out our understanding of the

world one thing causes another

and it brings us very nicely on to

interactive thinking and that's the next

step on from statistical thinking

and I have another hero I can reassure

you that this hero is not going to turn

out to be a raving racist who bullies

all of his co-workers so um there are a

few mathematicians who haven't done that

in their lives just a few of them but

they're around

and this is Alfred J Locker

um and he was originally a chemist

and he started his uh he's he was

originally from Poland but he did his

undergraduate degree in Birmingham

and

I I really like his story because

he started working in this chemistry lab

and he was kind of disappointed with

what he saw when he was doing his

experiments I mean it's a long time

since I did chemistry at school but it

can be a little bit disappointed you

know you get the acid and The Alkali you

mix them together and there's some salt

and water it's not always the most

exciting thing you've ever seen and so

he'd see these stable stable reactions

just come to equilibrium

but in the evening he was reading all of

these books like Charles Darwin's book

and he was thinking about biology and

just all of the exciting patterns we see

all the motion and movement of animals

or the firing of our brains everything

that happens in society and he was

thinking why can't chemistry produce

anything like that I mean we know that

chemistry must be the underlying

building block of it but it's not

something we see we can't make that

relationship together

and the way he solved the problem was he

basically cheated and he he did the

following thing so if

we've all done this in school

we've balanced equations right through a

balanced reactions and if you've got uh

two two h two well you've got four

hydrogen atoms two oxygen atoms and they

react to get to make two water molecules

so that's a a standard chemical reaction

and the important Point here is that

this is balanced so there's four

hydrogens and two oxygens on the left

and there's four hydrogens and two

oxygens on the right

but what loter said is well I'll just

forget about that balance thing even

though I can't find a chemical reaction

that is unbalanced I'll just think in my

head I'll do a thought experiment and

this is where it's lovely with

mathematics I'll do a thought experiment

where I ignore the fact that I can't

balance my reactions and so he take he

wrote down these equations he said that

imagine an r that becomes two r's and

imagine an R plus an F which becomes two

F's and you can see that these aren't

balanced there's one r on the left hand

side of the first one two r's on the

right yeah there's you can see that

they're just not balanced and I've

written down here below the way you can

think about these things are rabbits and

foxes so

and and they're not it's not a realistic

model of rabbits and foxes you've got to

think of the idea of one little rabbit

hopping around and suddenly there's two

little rabbits hopping around we know

it's a little bit more complicated than

that but we'll start with that idea and

then a fox comes and eats a rabbit and

then it makes another Fox that's that's

what the model says it gives a a rough

idea of how ecological interactions work

and he took that and he wrote down

differential equations

I wanted to put in a few of these

equations here to give you a feeling for

how they work

the the equations on the left here one

of them I I'm not going to get you to

understand every detail of the equation

but what I want to give you a feeling

for is on the left is the rate of change

of the rabbits and the foxes so Dr by DT

is a rate of change of the rabbits DF by

DT is the rate of change of the foxes

and on the right are the things which

cause that change so I mentioned here

that we want to get causation into our

equations so on the right of the things

which cause that change and rabbits

increase when there's lots of other

rabbits they have lots of little bunny

rabbits and then they are eaten by the

foxes so the more F they are if we look

at this brf term that's the rate at

which the

um the rabbits are eaten by the foxes

and then when we come down that we have

the opposite relationship for for The

Foxes the foxes grow when there's more

rabbits and then they eventually die off

um of old age there's nothing which

hunts the foxes in in this in this

scenario

now again I'm not going to solve all of

these equations but I did want to

mention a little bit about how you can

think about them and understand them

and on the right here actually learned

this from Philip Maney when I was here

in in Oxford about these types of method

but he had a very lovely method

a professor of mathematical biology here

for solving these equations without

solving them

so you can split up the plane of foxes

and rabbits and you can identify a point

on that plane and look to see do the

rabbits increase or do the fox's

increase so in the bottom left-hand

Corner the foxes go down because there's

not enough rabbits to eat but the

rabbits go up because there's not enough

foxes eating them and that goes on until

there's a sufficient number of rabbits

and then the foxes start increasing so

in the bottom right hand corner this

Arrow points up and if that Arrow points

up well then the foxes increase

and when the foxes increase they start

to eat the rabbits and the rabbits go

down and you start to get this circling

round and round of foxes and rabbits and

you could do basically show this without

explicitly solving the equations that

there's going to be Cycles round and

round of these foxes and rabbits we're

going to get this interesting

interaction and if we look at it over

time

um we have these periodic oscillations

of the foxes and the rabbits

now this whole way of thinking

which latke introduced turned out to be

useful in all sorts of situations now

the one thing we should we try not to

mention but can't be unmentioned in any

mathematical modeling thing we try not

to mention this but but it's also used

in pandemic modeling and you've all

heard about these epidemic curves and

our values and so on

um but that's the same thing that a

susceptible plus an infective becomes

too infectives and that's an example of

one of lotka's unbalanced equations

which allows us to describe how an

epidemic reaction epidemic will spread

through a group of people

now

I'm not going to go into as I said I

don't want to talk too much about

epidemics but this example I really love

so I'm going to talk about this example

this is a study we did I did together

with some colleagues and this is a very

cruel experience it's not it's not it's

not that cruel but we have we had a

group of undergraduate students we got a

third a third year undergraduate student

to give a seminar to first-year

undergraduate students and we told the

first year undergraduate students you

know remember to give a round of

applause after the seminar just to show

your appreciation and then what we were

really interested in was how people

applaud it what are the cues that make

people applaud and we could see that

people start applauding when other

people around them start applauding and

you basically have an epidemic of

Applause and that's what the first green

curve shows that's the number of people

clapping that's the spread of the

Applause virus going through the group

but then

and this doesn't happen in real diseases

you also have a social recovery

so when people stop clapping they look

around and they hear the other people

have stopped clapping and it was

actually a little bit like when Daryl

left the room just now there was a sort

of start signal there that there might

be something going on that we might be

about to start and you all started to go

down in volume and suddenly everyone was

quiet

and that's the type of social effect

we're very aware of all kinds of small

social details and these spread through

us in a group and the the conclusion

that what I love about the recovered

thing is because we have that social

recovery so

and I try to always remember this that

if at the end of a talk you've given or

a presentation if the Clapping goes on

for a long time that's not because you

gave a good talk it's just because your

audience aren't particularly coordinated

so they couldn't they could they

couldn't manage to to stop together

and I think that's what that's what I

would encourage you to think about and I

said that I also wondering about the

personal aspects of this

it's nice just to sit sometimes and

think about the social reactions that

you have in your life and how they work

and I've written down a few of them I

haven't told you what they are yet so

I'll tell you what what they are the top

the top one

that I was imagining this is a person p

and then it's plus an O this is a sofa

that the person has outside the house

and so we have P plus o goes to P plus o

if you're just one person and you've got

a sofa outside the house you're going to

still be one person and you can't get

the sofa into the house so what you have

to do and this is the bottom equation is

you have to get a friend and so this

bottom one is 2p 2 people plus a sofa

that's outside the house is still two

people you've still got your people

afterwards but you've moved your sofa

into the living room and you can write

those social interactions for every type

of activity the one on the right here

the ones on the right here I was

thinking about smiling so if you're a

smiley person why and you meet a

non-smiley person whose ex then if you

smile then hopefully they become a

smiley person too but that's not always

the case sometimes you know you don't

always start smiling because somebody

else is smiling they might just be an

idiot who's just smiling for no reason

at all what happens most often actually

in human social interactions is they as

you believe the following Equator the

one the equation at the bottom this is

the most common equation I think that

describes human behavior and that's that

a non-smiling person plus two smiley

people will become three Smiley people

because then they're convinced that

actually must be something to smile at

and I use that a lot in my thinking if

if I'm thinking about how to in the book

I take an example of if I'm trying to

get a group if a group of friends are

trying to get going with some kind of

healthy activity maybe they spend a lot

of time sitting in the pub together

don't really go out and do any exercise

together it's not enough for one of them

to become a why to to try and get them

going that you have to have two of them

and they have to have a really sustained

effort and over time then you get this

Tipping Point effect where everybody

starts to move over and starts it starts

to engage in the healthy activity so

those are the types of things you have

to think about what type of chemical

reaction what type of social reaction am

I involved in

and that's been a lot of to be honest

this has taken up a lot of my adult life

is studying these types of things and to

give you a little bit of a flavor of the

sorts of things we do

this is just to give a sort of overall

representation but

um when we modeled fish for example we

would create models which described

their social interactions described how

one fish turned left if another fish

turned left if another fish turned right

and so on then we were building the top

there it's a mathematical model we've

built a fish movement and so on so we'd

show that these simple rules of

interaction would produce their

Collective Behavior then we'd study also

the the movement of individual fish

that's the colored idea at the the

bottom then we'd actually frighten all

of the fish and we'd look at how they

made an escape wave we we'd measure that

escape wave and then we'd use models to

understand that escape wave and it's a

very powerful way of thinking throughout

science that you can build up these

models of interaction you compare them

to reality and build a better and better

understanding of fish Behavior and we do

a lot of similar things in football so

this is an example of an attacking run

by Marcus rashford and the model that we

build for these types of situations

this red area here shows the territory

that he controls

and this is a physics-based model where

we say how far how fast can he run where

can he get to and we can actually

describe what area he he occupies and

also the value of that area so How

likely is he is getting a pass at that

particular Point going to lead to a goal

and that allows us to actually Scout

players based on their runs and it even

allows us to scout runs where they don't

get the ball so in this example

where

we're interested in Luke Shaw here and

he's doing a run here on the left

and he doesn't get the pass he'd love to

have this pass but he doesn't get it but

we can still measure the value that that

pass created so you can look at these

counterfactual situations for

um for for football players

and this is a very powerful method the

interactive way of thinking allows us to

build up our understanding of systems it

doesn't have the same kind of

I I suppose the statistics has a sort of

more of a grounding feeling to it this

we use our imagination much more we try

to use our imagination to increase our

understanding and then build

mathematical models to test that

understanding

now I wanted to go back to lotka because

um

there are also limits to this way of

thinking and of course I wouldn't have

four if we'd if we'd solved it all now

so there's there's limits to this and

there's limits were limits that lotker

himself hit he wrote a book called

elements of physical biology and

he's one of these these mathematicians

who and this happens a lot to us

is we sort of just get carried away and

we believe that we can just explain

everything with mathematics that there's

nothing that we can't explain and so he

built models he built models of

Consciousness he built models of of our

whole society and he believed that all

of them could be understood using his

reaction Dynamics and it really

yeah he didn't I mean and this was I

suppose it was a very Valiant effort

this is in 1922 he finished his his

Magnus Opus so he didn't even have a

computer or anything to simulate these

types of models on but he never really

succeeded in pinning down one essential

way in which you should approach all

sorts of problems he he ended up kind of

split between lots and lots of different

small things and that I can personally I

can relate to that very well because

that tends to be how I work with lots of

problems there's lots of different

methods and you're doing lots and lots

of small different things in order to

get your solution the day of day of an

applied mathematician isn't it's not

like these theoretical physicists you

know they have like this beautiful

Theory of Everything and they can come

here and just say oh it's all this and

wow but no it's not like that it's more

that you're sort of tinkering around

with small different problems in lots

and lots of different ways so lotska

never found his Grand Theory of

Everything using interactive thinking

and one of the reason

one of the reasons he never found is

Grand theory was because he didn't know

about chaos

which is the third way of thinking

now to introduce chaos

I'm going to go to another mathematical

hero

this is Margaret Hamilton and

she was also like the other two we've

met prodigious at school

very talented undergraduate student she

wanted to go on and do a PhD in pure

mathematics

but her husband also wanted to do a PhD

and this is now in the 1960s and she

ended up moving to Boston and she also

had to get a job she had a daughter to

support a husband to support and so she

had to get a job to support them but the

job that she got was programming this

machine the lgp 30 and she fell

immediately in love with this Computing

machine because

she hated making mistakes she hated

errors whenever she calculated anything

she calculated it perfectly and now she

found that she could actually program

this first computer to do the same

calculations and she got access to this

computer because she was working in the

lab of a person called Edward Lorenz who

was a professor of meteorology but also

with a mathematical background there are

a lot of mathematicians in this talk so

um and he

he wanted to predict the weather he

wanted to predict the future weather

based on temperature pressure and so on

in different areas could he predict the

weather into the future and she started

writing a computer code to do this and

this involved writing and doing Punch

Cards at the time and she'd run her

computer code

and they did this one thing is that they

simulated they simulated the weather one

day and the next day they decided to

check their results by simulating making

the exact same simulation on the

computer to check that everything worked

but they found on the second day they

got a different result than on the first

day

and Margaret was distraught because she

didn't like making mistakes she didn't

want to think there was a mistake in her

code but she started going through the

code and there was no errors in the code

and what they found was that the output

of the simulation was in six decimal

places

well the input they put into it was in

three decimal places

so there was an error in the input in

the fourth decimal place and this meant

that the weather simulation made

completely different predictions in the

future going like a few 10 days into the

future in the simulated World it made

completely different predictions in the

future and I didn't mention that this

was a system of 14 differential

equations that she solved we've moved on

from lockter and Voltaren too so she

solved these 14 differential equations

and they make just this small error in

the value you put in makes a massive

difference and that is the first

indication of the butterfly of chaos

which many of you will be familiar with

and Lorenz went on he worked with

um I say Lorenz went on Margaret

Hamilton we're going to find also went

on to do some very impressive things but

Lorenz went on with the help of Ellen

Fetter who replaced Margaret Hamilton as

his programmer to produce what we now

know as the we often think of this

picture I think or I think of it as

being the butterfly of chaos and what it

illustrates is if you do start with two

points very close to each other we've

moved now down from 14 Dimensions to

three dimensions again if you start with

two points very close together and they

start to diverge

they'll move around on the same

attractor on this shape that we have

here but they'll never come close or

they might come close to each other for

a short amount of time but they'll then

live their own life and so when we move

from two Dimensions up to three we have

this chaotic movement where things never

come back to the same place again

I think I think I think we're going to

do my experiment okay so I think I think

we're going to do the experiment and

then I'll I'll boot out something else

because you've listened to me patiently

for 50 minutes so you have to get to do

the experiment okay so here we're going

to we're going to do this I want you to

work in pairs

I want you one of you should think of it

so it's a look at the person next to you

and you might be a new friend that

you've got today

um or it might be somebody that you came

with

and then I think I want one of you to

think of a number between 1 and 99

then you tell that number to the other

person

and the other person follows the follow

the the following rules so if a number

is less than 50 double it and this is

the new number I chose 42 because you

can never have a math talk without 42 in

it so 42 times 2 is 84 and so that's all

you do you just double the number now if

the number is greater than 50 take it

away from 100 and then double it to get

the new number so if I have 84 then I

have 100 minus 84 is 16 times 2 is 32.

now say the new number to your partner

and they repeat step one and two so

we'll do this for um do this with either

with the person you came with or

somebody who's nearby to you we'll do

this for about two minutes and then

we'll see where we get to

I think you've done it very nicely done

I can see the I can see I hear the

murmur of numbers everywhere very very

lovely

um

I'm I'm not going to get you all to come

up here and present your results I just

wanted to give you get get you to get a

feeling of this type of process

um you're not generating purely chaotic

numbers when you do this if you'd

started with 20 for example you would

have found yourself cycling around quite

quickly but if you started with a number

that's not divisible by five you would

have probably been on quite a long

trajectory through different numbers

and the point I want to make about this

process is the following is that close

together numbers very quickly diverge so

if one group over there had started with

13 and another group over here had

started with 14

by the end of just this short period

where you got to say the numbers to each

other you would have been on very

different numbers so you have 13 26 52

96 8 16 32 64. 14 28 28 is not so far

from 26 56 52 they're still together

88.96 are starting to get away from each

other but the big jump is now one of

them sort of goes over the threshold and

one of them doesn't so you've got 8 and

24.

1648 and then you've got 32 and 96 and

64 and 8. So within a few steps these

numbers have diverged quite far from

each other I don't know if any any of

you took decimal numbers

um you didn't think of that but if you

do take decimal numbers then you get

true chaos from this thing for almost

any real number you choose you will get

if you take plus 0.1 in this case only

so this is 14.1 compared to 14.2 you

start to they're together for a few

steps but after about seven eight nine

ten they go apart they come a little bit

together again for a while but then they

diverge and you've got very different

paths for those two numbers

and we often illustrate this

um using something called a cobweb

diagram so the idea here is you take the

number from One Step and the previous

number might be around 20 for example it

will jump up to be around 40 then it

will go to 80 then it will crash down to

around 20 again and then it will start

to move around everywhere on this

and one of the reasons I wanted you to

do this experiment is what was being

what I could hear from your perspective

was a Mumble of uniform distributed

random numbers you were essentially

going through a lot of integers and

everywhere in the room there was a

different point in this distribution you

basically had this uniform distribution

of numbers that were sort of kind of

coming up to me and I think it's really

lovely to think of that that you're all

doing the same process you're all doing

exactly the same thing yet you kind of

have this hum this distribution this

background of very different numbers

and

that is the butterfly of chaos and and

for me it illustrates

there's an important Point here

I I think chaos is wonderful

Margaret Hamilton she hated chaos right

and she she left Lorenzo's lab and she'd

learned a valuable lesson from working

on these weather simulations and it was

that she doubled down and made even

fewer errors and she wanted to work in

the most extreme

conditions possible where you couldn't

make errors and so she got a job for

NASA

and she became the head of the software

engineering which created the software

that sent that was on the Apollo moon

mission and so she was she created the

software that the astronauts used to

tell them how to to do navigational

decisions to control the thrusters to um

uh to update to know where the position

of the ship was

and she was in the control room when

they when they made the actual landing

on the moon and so I see this as a

situation where you sort of have to

choose right in if you're if you're

going to control something because of

chaos if there's something you really

care about or there's something that's

really important then you have to treat

it like Margaret Hamilton does you had

to treat it like the moon landing

there's no error there's no room for any

type of error

but you can't have control over

everything so I often think about this

in football because

there's always going to be butterflies

in other situations so here this isn't

the uniform distribution as you

generated but it's the poisson

distribution

there is lots of other situations

football being one of them where we just