# How is Statistics Different from Mathematics, and Why Should Teachers Care? Allan Rossman and Beth Chance Cal Poly - SLO.

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How is Statistics Different from Mathematics, and Why Should Teachers Care? Allan Rossman and Beth Chance Cal Poly - SLO

2 Context matters (lbs) Weights of members of 2000 U.S. Men’s Olympic Rowing team

3 Context matters Gesell (aptitude) score vs. age (in months) of first speaking

4 Context matters Without outliers

5 Context matters Are the cancer pamphlets written at appropriate levels to be read and understood by the cancer patients?

6 Measurement matters Unemployment Intelligence Highway safety Authoritarian personality Memory ability Ambidextrous-ness Teaching effectiveness Pace of life

7 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

8 Measurement matters

9 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?

10 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

11 Data collection design matters Study A:  Group 1: 42 successes in 61 trials (.689)  Group 2: 30 successes in 62 trials (.484) P-value =.011 Study B:  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

12 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

13 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 on- time arrival rates Lurking, confounding variable

14 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?

15 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

16 Uncertainty abounds! “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 …”

17 Uncertainty abounds! Rarely is there a definitive conclusion Often there is not even a definitive approach to a problem

18 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

19 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

20 Much newer discipline Think about when these ideas/tools were first developed  Geometry, logic, proof, trigonometry, function, calculus  Boxplot, stemplot, randomized comparative experiment, least squares regression, t-test Much mathematics that we teach is millenia old  All is at least many centuries old Some statistics that we teach is 100 years old  Much is a few decades old

21 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

22 How is Statistics Different from Mathematics? (Summary) Inductive vs. deductive reasoning Uncertainty abounds  Few definitive conclusions  Few definitive approaches Terminology crucial  Everyday phrases adopt technical meanings Communication crucial  Explanation in layperson terms essential Much newer discipline

23 Why should teachers care? Different preparation needed  Real data, meaningful contexts  Technology  Understand different kinds of concepts, skills Often weren’t taught in teacher preparation  Development of students’ communication skills  Successful teaching strategies in other classes may not work as well here

24 Why should teachers care? Different for students  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

25 How is Statistics Different from Mathematics? (Final Analysis) It’s more fun!!

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