Statistics of Illumination

Beth Chance

Roxy Peck

Cal Poly, San Luis Obispo

STATISTICS SAY… Increasingly daily life involves

statistical information– interpretations of graphical and numerical

summaries – comparisons of groups– poll results from random samples– conclusions from randomized experiments– predictions of future outcomes

Most people use statistics as a drunkard uses a

lamppost- more for support than

for illumination.

Predicting Variable Behavior

Predicting Variable Behavior

(a) Height of students in this class

(b) Students’ preference for coca-cola vs. pepsi-cola

(c) Number of siblings of individuals

(d) Amount paid for last haircut

(e) Gender breakdown

(f) Students’ guesses of my age

Matching Variables to Graphs

Matching Variables to Graphs

Think about context! Anticipate patterns and variations

– variable intuition– graph-sense

STATISTICS SAY… Students’ heights would show more

variability than guesses of my age KDC Pursues High-Return, Low-Risk

Strategy

What is Variability?

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What is Variability?

Class F Class G Class H Class I Class J

range 6 8 8 8 8

IQR 2.75 3 0 8 5

Std. Dev.

1.769 2.041 1.180 4.000 2.657

Describing Variability The “bumpiness” of a histogram does

not determine the variability of the observations

The number of distinct values the variable takes does not determine the variability of the observations

STATISTICS SAY… 5236 drivers age 65 and over were

involved in fatal accidents, compared to only 2900 drivers aged 16 and 17, so young people are safer drivers...

65% of motorcycle fatalities occurred in states with mandatory helmet laws...

Counts Versus Ratios Simple counts are often not a good

basis for comparison of two or more groups.

Group size isn’t always obvious—two groups of 25 U.S. states may have very different sizes even though both include the same number of states.

Deciding on a sensible basis for comparison requires thought!

STATISTICS SAY… 85% of software developers predicted

that Microsoft's integration of Internet functions into Windows would help their company

Some Simple Questions Question 1

Lost ticket

Yes: 6

No: 9

Lost $20

Yes: 8

No: 6

Some Simple Questions

People are more likely to say “yes” when they have lost a $20 bill

People tend to answer “not surprising” to both expressions

People are more likely to choose program A with the “save” version and program B with the “die” version

Some Simple Questions

Be careful when wording survey questions – ask to see the phrasing!

Bill Gates: It would help me EMENSELY to have a survey showing that 90% of developers believe putting the browser into the operating system is a good idea…– Browser vs. “browser technologies”

STATISTICS SAY …

Researchers in Philadelphia investigated whether pamphlets containing information for cancer patients are written at a level that the cancer patients can comprehend– Median reading levels are equal

Readability of Cancer Pamphlets

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Readability of Cancer Pamphlets

Graphs can illuminateLook at the data!

Think about the question

STATISTICS SAY…

American men were randomly selected for the 1970 draft

Draft numbers (1-366) were assigned to birthdates

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Draft Lottery Calculate the median draft number for

each month– 31 days: 16th value– 30 days: average 15th and 16th values– 29 days: 15th value

Draft Lottery

month median

January 211.0

February 210.0

March 256.0

April 225.0

May 226.0

June 207.5

month median

July 188.0

August 145.0

September 168

October 201

November 131.5

December 100

Draft Lottery

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Draft Lottery

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Draft Lottery

Statistics matter Summaries can illuminate Randomization can be difficult

STATISTICS SAY…

The average time between eruptions of the Old Faithful Geyser is 71 minutes– August, 1985

Geyser Eruptions

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Geyser Eruptions

Looks can be deceiving! Use the graph that summarizes

without losing important details

STATISTICS SAY… The average major league baseball

salary in the United States is about $1.5 million

Rowers’ Weights

2000 Men’s Olympic Rowing Team

Rowers’ Weights

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Rowers’ Weights

Mean Median

Full Data Set 197.29 207.5

Without Coxswain 200.11 210.00

Without Coxswain or 210.57 210.00 lightweight rowers

With heaviest at 320 215.33 210.00

Resistance....

Rowers’ Weights

Know what your numerical summary is measuring

Investigate causes for unusual observations

Baseball: median salary ~ $500,000

STATISTICS SAY…

People live longer in countries with more televisions

Televisions and Life Expectancy

Buy another television? Association is not causation

STATISTICS SAY… Overall survival rates:

– A: 80% B: 90%

Fair condition:– A: 98.3% B: 96.7%

Poor condition:– A: 52.5% B: 30.0%

Hospital Recovery Rates “Simpson’s Paradox”

– Hospital A gets most of the poor condition cases

– Patients in poor condition are less likely to survive

– Thus: hospital A has the lower survival rate despite being the better choice for either condition

Beware of lurking variables

Hospital Recovery Rates (cont.)

Fair

% survive

Hospital A Hospital B0%

100%

Hospital Recovery Rates (cont.)

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% survive

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Hospital Recovery Rates (cont.)

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STATISTICS SAY… Taking an aspirin each day reduces

the risk of heart attack for men, but less so for women

How Experiments Take Variability Into Account

Direct control

Blocking

Randomization

Randomization

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Blocking Scheme B

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Results from 100 Trials

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Completely Randomized First Blocking Scheme

Second Blocking Scheme

Controlling for Variability Blocking reduces variability in the

estimated mean difference Homogeneous blocks are desirable

Randomization evens out the effects of extraneous variables

STATISTICS SAY… A log was selected at random…

Sampling Logs Does choosing times at random result

in a random sample of logs?

_______________________________

Estimating Mean String Length

Does the sampling procedure produce a simple random sample?

How is this related to the log problem??

Can you suggest a better sampling method?

Selecting a Sample

Random Sampling eliminates human selection bias so the sample will be fair and unbiased/representative of the population.

While increasing the sample size improves precision, this does not decrease bias.

STATISTICS SAY… 45% +/- 1% of people surveyed claim

to prefer watching soccer to baseball

Reese’s Pieces

Reese’s Pieces Take sample of 25 candies Sort by color Calculate the proportion of orange

candies in your sample Construct a dotplot of the distribution

of sample proportions

Reese’s Pieces Did everyone obtain the same sample

result? Is there a pattern to the sample results? Is it possible to make predictions about

the population based on only one sample?

Can you be “confident” of your prediction?

Thank You bchance@calpoly.edu rpeck@calpoly.edu