How is Statistics Different
from Mathematics, and
Why Should Teachers Care?
Allan Rossman and Beth Chance
Cal Poly - SLO
Context matters
120
120
170
170
220
220
weight (lbs)

Weights of members of 2000 U.S.
Men’s Olympic Rowing team
2
Context matters
Gesell (aptitude) score
vs. age (in months) of
first speaking
125
125
115
115
105
105
Gesell score

95
95
85
85
75
75
65
65
55
55
10
10
20
20
30
30
40
40
age of first speaking
3
Context matters
Without outliers
120
110
Gesell score

100
90
80
10
15
20
age of first speaking
4
Context matters
0.3
0.25
patients
pamphlets
proportion
0.2
0.15
0.1
0.05
above 12
12
11
10
9
8
7
6
5
4
3
under 3
0
level

Are the cancer pamphlets written at appropriate levels
to be read and understood by the cancer patients?
5
Measurement matters








Unemployment
Intelligence
Highway safety
Authoritarian personality
Memory ability
Ambidextrous-ness
Teaching effectiveness
Pace of life
6
Measurement matters

Is a geographic region’s “pace of life” associated
with its heart disease rate?



Average walking speed of pedestrians over a distance of
60 feet during business hours on a clear summer day along
a main downtown street
Average time a sample of bank clerks take to make change
for two $20 bills or to give $20 bills for change
Average ratio of total syllables to time of response when
asking a sample of postal clerks to explain the difference
between regular, certified, and insured mail
7
Measurement matters
heart
30
20
10
10
20
30
walk
30
heart
heart
30
20
10
20
10
15
25
bank
35
10
20
30
talk
8
Data collection design matters

“Ladies, do you give more emotional support
to your husband or boyfriend than you
receive in return?”



Study A: 96% of a sample of 4500 said “yes”
Study B: 44% of a sample of 767 said “yes”
Which study do you have more confidence in?
9
Data collection design matters

“Ladies, do you give more emotional support
to your husband or boyfriend than you
receive in return?”


Study A: Sociologist Shere Hite distributed over
100,000 questionnaires through women’s groups,
got 4500 responses
Study B: ABC News - Washington Post conducted
interviews with a random sample of 767 women
10
Data collection design matters

Study A:



Study B:



Group 1: 42 successes in 61 trials (.689)
Group 2: 30 successes in 62 trials (.484) P-value = .011
Group 1: 806 successes in 908 trials (.888)
Group 2: 614 successes in 667 trials (.920) P-value = .015
Study C:


Group 1: 3274 successes in 3775 trials (.867)
Group 2: 6438 successes in 7225 trials (.891) P-value = .000
11
Data collection design matters

Study A: Social experiment that randomly
assigned three- and four-year-old children to
2 years of preschool instruction or control
group


Strong evidence of causal benefit of preschool
Study B: Observational study of court
records, comparing violent crime rates
among those abused as children and control
group

Strong evidence of association, but no causal link
12
Data collection design matters

Study C: On-time flight arrivals in one month
for Alaska Airlines and America West


America West had higher on-time arrival rate
Airport-by-airport analysis reveals that Alaska had
higher on-time arrival rate for every airport



America West had most flights to Phoenix, with very
high on-time arrival rate
Alaska had most flights to Seattle and SF, with lower ontime arrival rates
Lurking, confounding variable
13
Data analysis requires substantial judgment

Outliers



Should outlier(s) be removed?
Should I apply a more resistant method?
Technical conditions

Are they satisfied?



Never perfectly, but close enough?
Is the technique robust enough to proceed anyway?
Transformations


Should I apply one at all?
How do I choose which one to use?
14
Inductive vs. deductive reasoning

Mathematics





Deductive reasoning
Logical thinking
Problem solving
Probability
Statistics



Inductive reasoning, conditional reasoning
Draw conclusions from data
Make inferences from data
15
Uncertainty abounds!


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“Statistics is never having to say you’re
certain.”
“You never know. You really never know.
Really.”
Correct



“We have strong evidence that ….”
“The data strongly suggest that …”
Incorrect

“The data prove that …”
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Uncertainty abounds!


Rarely is there a definitive conclusion
Often there is not even a definitive approach
to a problem
17
Terminology crucial?


Also true in mathematics, but …
Everyday language has technical meaning


Bias, sample, statistic, accuracy, precision,
confound, correlation, confident, significant,
normal, random
Analogous to studying foreign language
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Communication crucial

Explanations in layperson terms essential


Statistics is a consulting enterprise
Must constantly interact with clients whose
technical skills vary greatly



Must elicit from them what problem is
Must communicate to them results and conclusions
Most AP Students will be consumers not
producers of statistics
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Much newer discipline

Think about when these ideas/tools were first
developed



Much mathematics that we teach is millenia old


Geometry, logic, proof, trigonometry, function, calculus
Boxplot, stemplot, randomized comparative experiment,
least squares regression, t-test
All is at least many centuries old
Some statistics that we teach is 100 years old

Much is a few decades old
20
Summary: How is Statistics Different from
Mathematics?

Context matters




Question of interest matters
Measurement method matters
Data collection design matters
Substantial judgment involved



Outliers, resistance
Technical conditions, robustness
Transformations
21
How is Statistics Different from
Mathematics? (Summary)


Inductive vs. deductive reasoning
Uncertainty abounds



Terminology crucial


Everyday phrases adopt technical meanings
Communication crucial


Few definitive conclusions
Few definitive approaches
Explanation in layperson terms essential
Much newer discipline
22
Why should teachers care?

Different preparation needed



Real data, meaningful contexts
Technology
Understand different kinds of concepts, skills


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Often weren’t taught in teacher preparation
Development of students’ communication skills
Successful teaching strategies in other classes
may not work as well here
23
Why should teachers care?

Different for students

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Research shows difficulty of reasoning with
uncertainty
Many students very uncomfortable with
uncertainty, lack of definitive conclusions, need for
detailed explanations, individual interpretation
Challenge of promoting healthy skepticism without
extremes of cynicism or naïve acceptance
Many mathematically strong students will be
frustrated

But many less stellar math students will be empowered
24
How is Statistics Different from
Mathematics? (Final Analysis)

It’s more fun!!
25