Lecture Notes

Lecture Notes by Christopher Lay

Los Angeles Pierce College

Department of History, Philosophy, and Sociology

Mathew Van Cleave's 2016 Introduction to Logic and Critical Thinking

https://open.umn.edu/opentextbooks/BookDetail.aspx?bookId=457

Chapter 3 "Evaluating Inductive Arguments and Probabilistic and Statistical Fallacies," § 1 "Inductive Arguments and Statistical Generalizations"

Recall: "an inductive argument is an argument whose conclusion is supposed to follow from its premises with a high level of probability, rather than with certainty."

"This means that although it is possible that the conclusion doesn’t follow from its premises, it is unlikely that this is the case."

Statistical Generalizations

"Statistical generalizations are generalizations arrived at by empirical observations of certain regularities."

"Statistical generalizations can be either universal or partial."

Universal and Partial Generalizations

"Universal generalizations assert that all members (i.e., 100%) of a certain class have a certain feature."

"[P]artial generalizations assert that most or some percentage of members of a class have a certain feature."

Evaluating Generalizations in Arguments

"[D]eductive arguments typically use universal statistical generalizations whereas inductive arguments typically use partial statistical generalizations."

"[B]eing able to evaluate when a statistical generalization is good or bad is crucial for being able to evaluate arguments."

"[I]n evaluating arguments we want to know about the strength of the inference from the premises to the conclusion, but we also want to know whether the premise is true!

"[I]n evaluating statistical generalizations [we seek to determine] whether the premise in our argument is true (or at least well-supported by the evidence)."

The Truth of Good Statistical Generalizations

"We can assess whether or not a statistical generalization is true by considering whether the statistical generalization meets certain conditions."

"There are two conditions that any statistical generalization must meet in order for the generalization to be deemed 'good.'"

"Adequate sample size: the sample size must be large enough to support the generalization."

"Non-biased sample: the sample must not be biased."

Bias, according to the New Oxford American Dictionary, is "prejudice in favor of or against one thing, person, or group compared with another, usually in a way considered to be unfair."

Representative Sample

"The 'adequate sample size' condition and the 'non-biased sample' condition are ways of making sure that a sample is representative."

"A sample is simply a portion of a population."

"A population is the totality of members of some specified set of objects or events."

"A good statistical generalization is one in which the sample is representative of the population."

"When a sample is representative, the characteristics of the sample match the characteristics of the population at large."

Getting it Wrong

"It is perhaps easiest to illustrate these two conditions by considering what is wrong with statistical generalizations that fail to meet one or more of these conditions."

Getting it Wrong: Inadequate Sample Size

"First, consider a case in which the sample size is too small (and thus the adequate sample size condition is not met)."

The Fallacy of Hasty Generalization

"One commits the fallacy of hasty generalization when one infers a statistical generalization (either universal or partial) about a population from too few instances of that population."

"Hasty generalization fallacies are very common in everyday discourse, as when a person gives just one example of a phenomenon occurring and implicitly treats that one case as sufficient evidence for a generalization."

Getting it Wrong: Biased Sample

Recall that bias, according to the New Oxford American Dictionary, is "prejudice in favor of or against one thing, person, or group compared with another, usually in a way considered to be unfair."

Consider a generalization about the kinds of food eaten in Los Angeles based off of the food eaten on this campus.

Bias in Polling

Let's come up with a generalization about all Pierce College students that is biased because we only polled people in this room.

If our generalization about grade inflation at Pierce College only includes students in the classroom, it is likely to be biased insofar as it does not include other Pierce College students or the professors.

Questionnaire Bias

"[B]ias can creep into a statistical generalization through a biased way of asking a question."

With biased ways of asking a question, polls can fail to be properly representative "because the responses would be skewed by the biased phrasing of the options."

"Which do you favor?"

"a. Preserving a citizen’s constitutional right to bear arms"

"b. Leaving honest citizens defenseless against armed criminals"

Random Sampling

"Random sampling is a common sampling method that attempts to avoid any kinds of sampling bias by making selection of individuals for the sample a matter of random chance (i.e., anyone in the population is as likely as anyone else to be chosen for the sample)."

"The basic justification behind the method of random sampling is that if the sample is truly random (i.e., anyone in the population is as likely as anyone else to be chosen for the sample), then the sample will be representative."

"The trick for any random sampling technique is to find a way of selecting individuals for the sample that doesn’t create any kind of bias."

Randomness Example

"A common method used to select individuals for a random sample (for example, by Gallup polls) is to call people on either their landline or cell phones."

"Since most voting Americans have either a landline or a cell phone, this is a good way of ensuring that every American has an equal chance of being included in the sample."

"Next, a random number generating computer program selects numbers to dial."

"In this way, organizations like Gallup are able to get something close to a random sample and are able to represent the whole U.S. population with a sample size as small as 1000 (with a margin of error of +/- 4)."

Exercise 22

"What kinds of problems, if any, do the following statistical generalizations have? If there is a problem with the generalization, specify which of the two conditions (adequate sample size, non-biased sample) are not met. Some generalizations may have multiple problems. If so, specify all of the problems you see with the generalization."

1. "Bob, from Silverton, CO drives a 4x4 pickup truck, so most people from Silverton, CO drive 4x4 pickup trucks."

2. "Tom counts and categorizes birds that land in the tree in his backyard every morning from 5:00-5:20 am. He counts mostly morning doves and generalizes, "'most birds that land in my tree in the morning are morning doves.'"

3. Tom counts and categorizes birds that land in the tree in his backyard every morning from 5:00-6:00 am. He counts mostly morning doves and generalizes, “most birds that land in my tree during the 24-hour day are morning doves.”