Math is Music - Stats is Literature, 2004

7/30/2019 Math is Music - Stats is Literature, 2004

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September 18, 2004 AWL Workshop -- HACC 1

Math is MusicMath is MusicStats is LiteratureStats is Literature

Dick De Veaux - Williams CollegeThanks also to Paul Velleman,

Cornell University

Or why are there no six year old novelists?Or why are there no six year old novelists?


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ProdigiesProdigies Math, music, chess

Gauss, Pascal Mozart, Schubert, Mendelssohn

Bobby Fischer

Why these three areas? Each creates its own world with its

own set of rules

There is no experience required Once you know the rules, you are freeto create anything


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Prodigies in LiteratureProdigies in Literature

Mary Wollstonecraft Shelley Age 19

Created Frankenstein, an imaginarycreature

Others? Why?

Literature is about the world, not aboutrules. It deals with lifes experience andthe wisdom we develop over time.


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StatisticsStatistics What doWhat dostudents find hard?students find hard?

Understood the material in class, butfound it hard to do the homework

Should be more like a math course,with everything laid out beforehand

More problems in class should be like

the HW and tests


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What is easy?What is easy?

The math part Give them the formula, they can get the

answer with some training

The hard part Putting it all together Real world

Experience

Methods


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Whats Hard?Whats Hard? ----ExampleExample


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TT--CodeCode


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Whats Hard?Whats Hard?Five Unnatural ActsFive Unnatural Acts

Think Critically

Be Skeptical

Focus not on what we know, buton what we dont know

Think first about Variation

Think clearly about Conditioningand Rare events


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Statistics is UnnaturalStatistics is Unnaturaland Subversiveand Subversive

We ask students to Question the data

Examine the assumptions

Reject the null hypothesis

Have they done this in mathclass?

Convincing them to be subversivemay be easier than you think


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ThinkThink CriticallyCritically Challenge the datas credentials.

Look for bias. Know what we want to know.

Whats the QUESTION?

Look for Lurking variables. Check Assumptions and Conditions.

Critical thinking requires creativity. Youmust think about things that are not infront of you and imagine ways in whichthings might have gone wrong.


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Be SkepticalBe Skeptical Be cautious about making claims

based on data. Trust every analysis, but plot the

residuals. Skeptical statisticians expect the

unexpected, so we go looking for it.

SHOW that the analysis isappropriate


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Ancient HistoryAncient History

The vote in the 2000 Presidential election forBuchanan and the vote for Nader, (the two

principal alternatives to Bush and Gore), has

a correlation of 0.65 over the counties ofFlorida.

Ask: Is the relationship linear?

Is the data set homogeneous or are there subgroups?

Are there any outliers?


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Plot the DataPlot the Data

750

1500

2250

3000

2500 5000 7500

NADER

B

U

C

H

A

N

A

N

Without Palm Beach county and itsbutterfly ballot, the correlation is 0.91.


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Hypothesis TestingHypothesis Testing

Skepticism formalized

The null hypothesis is a skepticalclaim about the data

Its unnatural to show theopposite


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Critical Thinking andCritical Thinking and

SkepticismSkepticism

Critical thinking is open-endedquestioning of the datas credentials. We wonder whether the data are competent to tell

us what we want to know.

Skepticism questions whether whatthe data appear to be telling us is the

whole truth.


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Focus on What We DontFocus on What We Dont

KnowKnow

In most science and mathcourses, we focus on what we

know Statisticians are a bit perverse


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Confidence IntervalsConfidence Intervals

We dont say The mean is 31.2.

We dont say The mean is probably 31.2

We dont say The mean is close to 31.2.

All we can manage is The mean is close to 31.2. Probably

(and, in fact, Im willing to admit I may bewrong and to spend the effort to give you a

whole interval of plausible values and then to

spend extra effort to estimate how likely it is

that even that interval is wrong.)


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All Models are WrongAll Models are WrongGeorge Box:

All models are wrong but some areuseful

Statisticians, like artists, have the badhabit of falling in love with their models

But, statisticians love models--because theyare wrong.

What do we focus on?residuals!

what the model fails to account for


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VariationVariation

Students find it easier to thinkabout values rather than variation,

butStatistics is about Variation


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ExampleExample

A town has two hospitals Large hospital about 100 babies a day

Smaller hospitals about 15 babies a day

Over the course of the year, whichhospital (if either) would probably have

more days in which more than 60% of

the babies born are male?


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The Standard DeviationThe Standard Deviation

is the Statisticians Ruleris the Statisticians Ruler

Most of the inference seen in theintroductory course compares a

statistic to its standard deviationto see whether it is big.

This idea carries into advancedmethods as well.


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Thinking aboutThinking about

Conditional EventsConditional Events

This is just plain hard.

It is easy to show that we dont

naturally think clearly aboutconditional probabilities.

But we must for rational decision

making.


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LindaLinda((TverskyTversky && KahnemanKahneman))

Linda is 31 years old, single,outspoken, and very bright. Shemajored in philosophy. As a student,she was deeply concerned withissues of discrimination and social

justice, and she participated inantinuclear demonstrations.


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Order these in order of LikelihoodOrder these in order of Likelihood

a) Linda is a teacher in an elementary school

b) Linda works in a bookstore and takes yogaclasses.

c) Linda is active in the feminist movement.

d) Linda is a psychiatric social worker

e) Linda is a member of the League of WomenVoters.

f) Linda is a bank teller.

g) Linda is an insurance salesperson.

h) Linda is a bank teller who is active in the feministmovement.


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Pick a number at RandomPick a number at Random


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Random?Random?

0 1 2 3 4


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Random IIRandom II

10

20

30

Count

1 2 3 4 5 6 7 8 9 10 12


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Is Statistical ThinkingIs Statistical Thinking

Unnatural?Unnatural?

We havent evolved to be Statisticians. Our students who think Statistics is an

unnatural subject are right. This isnt howhumans think naturally.

But it is how humans think rationally. And itis how scientists think. This is the way wemust think if we are to make progress inunderstanding how the world works and,

for that matter, how we ourselves work.


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How can we help?How can we help?

Give them an outline for putting

the real world into a framework Whats the problem?

The Ws

The model The method

What are the mechanics?

What have we learned?


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ThinkThink ShowShow ---- TellTell

THINK: What techniques apply?

SHOW: Mechanics how to do it.

TELL: Explain what you learned.


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The Three Rules of DataThe Three Rules of Data

AnalysisAnalysis

I. Make a Pictureit will help you think about the data

II. Make a Picture

it may show unexpected features

III. Make a Pictureit will help you tell others what

youve found.

These are made easier with technology!


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Know the Datas WsKnow the Datas Ws Who is the data about?

Whats a row?

What is measured? What are the columns?

And in what units?

When was it measured?

Where was it measured?

Ho(W) was it measured?

Why was it measured?


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The WsThe Ws

14718833912040.5383.36.02USALance Armstrong2004






3611444881428.7156.09.31FranceLucien Petit-Breton1908

339344881428.5156.22.30FranceLucien Petit-Breton1907

148246371324.5185.47.26FranceRene Pottier1906

246029751127.3112.18.09FranceLouisTrousselier1905

23882388624.396.05.00FranceHenri Cornet1904

21602428625.394.33.00FranceMaurice Garin1903

FinishStartDis (km)StagesSpeedTimeCountryWinnerYear


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The ModelThe Model Statistics is about models

A model is a simplification ofreality.

We know its not perfect

Two quotations from George Box


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Common ModelsCommon Models Probability models

Regression model


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CommonCommon ModelsModels Simulation


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Pay Dirt ModelsPay Dirt Models

Sampling distribution models

By now students know that models areidealized

Theyve seen probability models and

simulations: CLT follows naturally Null hypothesis models

Wrong (we hope) but useful


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ModelsModels

Require assumptionsBecause they are idealized, they are only really

true under idealized assumptions

Are described by parametersParameters refer to models of populations, not

to the populations themselves


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Assumptions andAssumptions and

ConditionsConditions

Some assumptions we must justassume. (Pretend)

Many can be checked forplausibility with appropriateconditions Often the conditions are graphical

(Remember the 3 rules)

Few are really true


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Conditions to CheckConditions to Check Summary statistics

Quantitative data condition.

T-test Assumption is that data are Normal

Rule of thumb? 30? 50? 100?

Nearly normal condition--make a picture

Variable -- TCODE

Mean 54.41

Std Dev 957.50

Std Err Mean 3.11

upper 95% Mea 60.51lower 95% Mean 48.31


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Show, with TechnologyShow, with Technology

Calculation is for calculators andstatistics packages. Let them do it, so students can think about

statistical thinking.

Show generic output rather than aparticular package.

Let them do it so we can play Statistics


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Play StatsPlay Stats

M H lM H l


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More HelpMore Help

Reality ChecksReality Checks The answer is wrong if it makes no sense --

even if you pushed the buttons you meant topush or gave the command you intended

Check that the results are plausible

Remember the units!


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Draw ConclusionsDraw Conclusions

Plot the data, but then say what you

see. Give guidance for how to see

Reject the null hypothesis, but then

provide a CI to assess effect size. Emphasize interplay between tests and CI

Think about costs and consequences.

Dont be satisfied with I rejected Ho


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What Can Go Wrong?What Can Go Wrong?

Acknowledge common

misapplications and misinterpretationsof statistics.

(Hope to) Minimize them in Tellingwhat was found.


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StepStep--ByBy--StepStep

Encourage students to bring all ofthese ideas together when they

solve a statistical problem.

Illustrate how, step-by-step


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Take Home MessagesTake Home Messages

Stats is about the real world: Technology frees the student to think

about the world

Give the student a structure for a chaotic

world Root the course in examples taken from

the students lives to make the connection

apparent

Help them with unnatural thinking


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Thank you !!Thank you !!

Documents

Math is Music - Stats is Literature, 2004