Unit 5. Thinking Statistically

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1 Unit 5. Thinking Statistically Supplementary text for this unit: Darrell Huff, How to Lie with Statistics. Most important chapters this week: 1-4. Finish the book next week. Most important chapters: 8-10.

2 Starting to Think Statistically Philosophy and Logic Unit 5 Sections 5.1, 5.2

3 Inductive v. Deductive A deductively valid argument is something like a proof. An argument is deductively valid if and only if it is impossible for all the premises to be true and the conclusion false. The premises, if true, would prove the conclusion. But there are plenty of arguments which do not even try to do this!

4 Example My wife does not have male pattern baldness. My mother does not have male pattern baldness. My grandmother does not have male pattern baldness. None of my wife s or mother s female friends have male pattern baldness No women have male pattern baldness.

5 Inductive arguments A strong inductive argument =df. an argument which is not deductively valid, yet whose premises provide good reason for believing the conclusion. In these the premises make the conclusion very probable or highly likely. So it is unlikely that the conclusion is false if all the premises are true, but it is still possible. This is why these aren t quite proofs.

6 A contrast Deduction is prototypically a proof. Induction is prototypically statistical. The paradigm use: inductive generalization That is, to argue from some sample of observed cases to a generalization (of the form All S are P).

7 My wife does not have male pattern baldness. My mother does not have male pattern baldness. My grandmother does not have male pattern baldness. None of my wife s or mother s female friends have male pattern baldness No women have male pattern baldness.

8 My wife does not have male pattern baldness. My mother does not have male pattern baldness. My grandmother does not have male pattern baldness. None of my wife s or mother s female friends have male pattern baldness No women have male pattern baldness. Sample

9 My wife does not have male pattern baldness. My mother does not have male pattern baldness. My grandmother does not have male pattern Sample baldness. None of my wife s or mother s female friends have male pattern baldness No women have male pattern baldness. Population

10 Two terms the population =df. the entire class of individuals about which the generalization is made Sometimes called the reference class Its size may be unknown, or have no upper limit. the sample =df. the class of individuals actually observed. This is always a finite number of individuals.

11 The problem of induction Take any generalization you believe. Ask: How do you know that it is true? Often the evidence rests on observations. There can only be a finite number of them. Yet the conclusion is unrestricted. It applies to cases that are not observed. Therefore these inferences cannot be justified deductively!

12 Problem of induction, II Inductive generalizations proceed from samples to populations. How is this possible? How can it be done rationally? The answers lie in statistical methods.

13 Two goals for unit 5 Learn some of the principles of inductive generalization Learn how to recognize some of the common ways that it goes wrong. By the sheerest of coincidences, these correspond to the two kinds of test item on tests 5!

14 The Five Steps of the Path (How to do a statistical study correctly:) A. Choose a sample B. Measure each member of the sample C. Compute a statistic D. Generalize to the population E. Infer relations between groups

15 Sample Measure Compute Generalize Infer A sample is just the finite group of individuals who are actually observed or measured. To measure is to assign numbers to properties in accord with some rule. A statistic is a number summarizing a bunch of such measurements. (an average, a percent, a rate, a fraction)

16 Sample Measure Compute Generalize Infer The population is a much larger group of individuals, not all of whom are actually observed, but to which the statistic is generalized. The goal: that the statistic say something meaningful about the population as a whole.

17 The fundamental principle A river cannot, we are told, rise above its source. It is equally true that the result of a sampling study is no better than the sample it is based on. (Huff, p 18) Or: A statistic is no better than the collection of measurements of individuals on which it is based, and from which it is computed.

18 The little figures that are not there : a. b. c. d. how a sample is selected how an attribute is measured how the statistic is computed how it is generalized These hide many fallacies!

19 A. Choose a Sample Inductive generalization works best if the sample is random and it is sufficiently large to be representative. Example: a barrel filled with red and white beans. What percent are red? (see Huff p 13, and chapter 1)

20 Which is the best sample? taking a small handful off the top taking several handfuls off the top mixing the barrel thoroughly, taking several handfuls from different places mix the barrel thoroughly, dump it onto a conveyor, and use dice to decide which of every twelve handfuls that pass by to take out

21 Random vs. biased a sample is random =df. every member of the population has an equal chance of being sampled. (Huff, p 21) a sample is biased if some members of the population have a greater chance of being sampled than others. So biased non-random.

22 1. Biased Sample A non-random sample. That is: A procedure in which some members of the population have an greater chance of being a member of the sample than others. You find this if you find some variable which makes some individuals more likely to be selected than others.

23 Biased sample Example: wealthy Yale graduates (pp );

28 Biased sample Two More Examples: 1936 Literary Digest poll (p 20); companies which do not engage in price gouging, and so are more likely to return the poll (p. 125).

31 How to do survey research correctly list the entire population (a sampling frame) randomly select a proportion track down every individual selected deal with the non-respondents

32 The problem with biased samples Those who were not sampled may differ in significant ways from those who were. With a biased sample there is absolutely no way to tell how big a difference this might be.

33 Why you should never trust election polls It is impossible to know the population in advance. Likely voters 1996 predictions: Clinton had a 21% lead. (Actual margin: 7%) 1998 predictions: Republicans will gain 8-15 seats. (Actual results: they lose 5.)

34 2 Inadequate Sample Size Def: A statistic based on just a few cases. Particularly blatant: N = 1. The report is based on 1 case One doesn't know how many cases in toto were sampled.

35 2 Inadequate Sample Size Examples: toothpaste statistics based on a sample of 12 (p. 38); a coin toss (p 39); A polio experiment with two cases (40); hidden camera interviews on TV. Husky polls.

39 The problem with using a small sample Because the sample is small, if you do it all over again, mere chance is likely to give you a different result. Results from repeating a study using small samples will likely vary a lot. This variation is from chance alone. So studies with small samples have a high margin of error or high probable error (as will be defined later).

40 The problem with using a small sample (cont) Bigger sample = more reliable = lower margin of error Repeated studies with big samples are likely to vary less from one another. A poll of 500: 5% A poll of 1000: 3% So a bigger sample has a lower margin of error.

Study Design. Study design. Patrick Breheny. January 23. Patrick Breheny Introduction to Biostatistics (171:161) 1/34

Study Design. Study design. Patrick Breheny. January 23. Patrick Breheny Introduction to Biostatistics (171:161) 1/34 Study design Patrick Breheny January 23 Patrick Breheny Introduction to Biostatistics (171:161) 1/34 in the ideal world In an ideal world, We have a list of everyone in the population of interest We randomly