Questions about Quiz #4? https://bgsu.instructure.com/courses/1137459/quizzes/1526334/statistics
Lesson 4.0: Generalizations
Generalizations - definition
Formalizing & assessing arguments for generalizations
Review selected homework questions
Lesson 4.1: Statistical syllogisms
Statistical syllogism - definition
Formalizing & assessing statistical syllogisms
Review selected homework questions
Applies to both: Importance of operationalizing the sampling frame
[Takeaway points for today @ bottom of post]
Lesson 4.0: Generalizations
General claims are probabilistic: -Most BGSU students like candy -70% of Americans will hand out Halloween candy this year -Only a small proportion of spiders are dangerous (to humans)
Think of generalizations as two-premise arguments: P1) S is a representative sample of group P. P2) [Some proportion] of S is Y (Y=the trait we're interested in). C) [Some proportion] 2 of P is Y.
Example: P1) The students in this class are a representative sample of BGSU undergraduates. P2) 20% of the students in this class have seen those creepy clowns everyone's talking about. C) Therefore, 20% of BGSU undergraduates have seen those creepy clowns.
1001 Definitions - make sure to familiarize yourself with these prior to the exam
Sampling frame - defines the trait and the total population (group) you want to study
Target population (group) - The total population you want to study
Sample - The subset of individuals (from the total population) you plan to study
Sample size - The number of subjects being studied, i.e., your chosen subset of the total population
Hasty generalization - You commit this fallacy when you rely on a sample that is too small to support a general conclusion (cannot represent the diversity of the total population)
Representative sample - A sample such that the relevant traits or variables in the sample are represented proportionally in the total population
Biased sample - Does not have the same distribution of variables represented in the same proportion as in the total population
Random sample - To get a random sample you must use a selection method to ensure that every member of the total population has an equal chance of being included in the sample
This is random, but it's not a random sample:
Margin of error - Refers to the probability of a sample being biased. As sample size goes down (under a certain limit) the margin of error goes up
Stratified random sampling - First, identify relevant population clusters; second, make sure that they are represented in the sample in the same proportion that they exist in the population
Anecdotal evidence - Evidence collected in a casual or informal manner, relies heavily or entirely on personal testimony. Usually too small sample size & a biased sample
Operationalize - A fancy word for defining your terms clearly
Takeaway points Roughly speaking:
1) Sufficiently large sample size, representative sample, low margin of error, stratified random sampling, terms operationalized --> good generalizations
2) Small sample size, non-representative (i.e. biased) sample, high margin of error, hasty generalization, anecdotal evidence, terms not operationalized --> bad generalizations
THIS WEEK: ASSESS PREMISES FOR ADEQUATE SAMPLE SIZE & REPRESENTATIVENESS
Homework 4.0, Part A
A.Suppose you are asked to conduct a study but only given vague terms.Operationalize the sampling frame to eliminate vagueness in the target population and in the trait we're interested in. You are free to operationalize any way you like so long as you eliminate vagueness.
3. Small dogs are aggressive.
Target population: ? Trait: ?
For those interested, here's a great article (note: focus is on dog bite-related fatalities, and not dog bites simpliciter):
Patronek, Gary J., et al. "Co-occurrence of potentially preventable factors in 256 dog bite–related fatalities in the United States (2000–2009)." Journal of the American Veterinary Medical Association 243.12 (2013): 1726-1736. Search Google Scholar to find PDF.
Homework 4.0, Part B
B. Put the argument into standard form for generalizations. Evaluate each premise according to criteria we covered in class:
(P1) Check for sample size, representativeness. (P2) Check for measurement errors. (Probably not relevant for this set of exercises. We'll look at this in more detail next class). (C) Is the proportion in (P2) the same as the one in the conclusion.
3. I never found calculus to be useful, therefore it isn't useful.
P1)? P2)? C)? P1, Sample size - sufficient, insufficient, or not sure? P1, Representativeness - representative, not representative, or not sure?
6. 40% of bees in Nebraska die every winter, therefore 40% of bees in the US die every winter.
7. 90% of people in LA, San Fransisco, Manhattan, San Diego, Austin, and Seattle support gay marriage, therefore 90% of Americans support it.
More examples
1. My cats love Fancy Feast. Therefore, your cats will love Fancy Feast too.
2. A randomized study of 2,000 adult Americans found that the average person has three friends.
3. Students in my class are really good at recognizing invalid inferences. BGSU undergraduates must be good at recognizing invalid inferences. Measurement errors <-- More on this subject in upcoming lectures, so not our focus today
Sometimes the way we collect my data affects whether we are actually measuring what we think we are measuring.
Sometimes two properties are closely correlated, and we confuse one (specifically, the causal influence of one) for the other.
Often with human trials, participants will drop out over the course of the study.... This affects the size of the sample, and it may affect whether we can trust the results.
Lesson 4.1: Statistical syllogisms
Syllogism = def. (for our purposes) a two-premise argument
Don't be intimidated by terminology. It's a fancy word for something really simple!
Example of a deductive syllogism:
P1) All humans are mortal P2) Socrates is human C) Therefore, Socrates is mortal
This comic will only be funny to those who've taken Phil 1010 (or read some Plato on their own):
Socrates "apologizes" We're talking about inductive syllogisms, aka, statistical syllogisms. Statistical syllogisms are also two-premise arguments... but you'll need to fill in the implicit premise. Filling in the implicit premise is really simple.
You're given an argument like this, which relies on a generalization: Most clowns are creepy, so the clown at your niece's birthday party is probably going to be creepy.
Filling in the implicit premise, you wind up with a two-premise argument that looks like this:
P1) Most clowns are creepy P2) There's going to be a clown at my niece's birthday party C) The clown at my niece's party will probably be creepy
Statistical syllogisms formalized - general form
P1) Generalization P2) Instance C) Conclusion regarding the instance (based on the generalization)
Example 2:
P1) 99% of cats like boxes P2) I have a box & a cat C) 99% chance my cat's going to like the box
Assessing statistical syllogisms:
Probability of conclusion being true depends on two things
1) The proportion of the target population that has the trait in question (as indicated by the generalization) More likely to be true, in decreasing order: 99%...75%... 66%... 51%* Notice that all these percentages fall under the umbrella term "the majority." 2) Homogeneity of the target population with respect to the trait in question. In other words: How similar the individuals in a certain group are, with respect to a certain trait. Related: Are there important subgroups within my target population that affect the likelihood of the conclusion's being true? The less homogeneous the target population is, the less likely the conclusion is to be true.
Homework 4.1, Part A
(a) Put the statistical syllogism into standard form and (b) Identify what relevant sub-groups we'd need to know about to strengthen the inference to the conclusion.
1. 25% of BGSU students scored well on the math component of the SAT, therefore there's a 25% chance that my classmate scored well on the math component of the SAT.
a) P1) Generalization P2) Instance C) Conclusion regarding an instance (based on generalization)
b) More information needed? Relevant sub-groups?
2. 51.1% of Americans voted for Obama last election, therefore there's a 51.1% chance that the person sitting next to me on my flight to Texas voted for Obama.
a)
b)
Are you still with me?? Here's a cat video for Halloween:
(Cats who hate their costumes)
Homework 4.1, Part B
How to manipulate your audience: Averages and percentages vs. absolute numbers
Depending on whether we express a value as a percentage or as an absolute value we can tell contradictory stories using the exact same data!
Basic math review
1) Calculating percentages ("percentage of X")
Convert percentages to decimals, then multiply.
Example: What is 1% of 20?
1% = 1/100= 0.01
0.01 x 20 = 0.2
2) Calculating percentages ("X out of Y = Z%")
Divide one number by the other, then multiply x 100
Example: 300 is what percentage of 100,000?
300/100,000 = 0.003
0.003 x 100 = 0.3%
1. Suppose you're the CEO of company that did 10 billion dollars in annual sales last year. Last year the economy grew by 5% but your sales only grew by 3% compared to the previous year. You have a shareholder meeting coming up and these numbers don't look good. Your sales haven't kept pace with the economy. Thinking about absolute numbers vs. percentages, how can you present the sales numbers to the shareholder meeting in a positive way? Hint: Translate your annual sales into an absolute number.
How are you going to pitch your company's sales data to the shareholders?
4. Here are some statistics on Syrian refugees and US immigration. Obama has set a goal to accept 10,000 Syrian refugees currently residing in Jordanian refugee camps. There are about 10 million Syrian refugees and 500,000 Syrian refugees in Jordan are awaiting resettlement to a new country.
(a) Suppose you write a pro-Syrian immigration blog. Thinking about absolute numbers vs percentages, how could present the numbers in a way that makes it seem like the US isn't doing very much to help refugees? Present them in a headline that conveys this point of view.
Your answer:
(b) If you were writing an anti-immigration blog, how would you present the numbers? Present them in a headline that conveys this point of view.
Your answer:
Key points from today's class (not the cat video):
Evaluating generalizations
Sample size: definition and importance
Hasty generalizations
Representativeness: definition and importance
Biased samples
Evaluating statistical syllogisms
Form of the (implied) argument
Reliance on generalizations (averages)
Importance of relevant sub-groups
How to manipulate audiences: Absolute vs. average values
QUIZ DUE WEDNESDAY (10/19) P.M. Check Canvas this afternoon!
Plan for today: 1) Review:Clarify vagueness (incl.vague normative terms), semantic ambiguity, syntactic ambiguity, group ambiguity 2) BRIEF review, if students desire: fallacies of equivocation, composition, and division 3) Homework 3.5, Part A - questions and examples in bold (If time, Part C) 4) BRIEF review, if students desire: weasel words, euphemism/dysphemism 5) Review: Misleading comparisons 6) Homework 3.6, esp. Parts B and D - questions and examples in bold.
Takeaway points (emphasized today): -The difference between vagueness and ambiguity -Vague normative terms (normative vs. descriptive) -Three types of ambiguity (semantic,syntactic, group*) *subspecies of semantic ambiguity -Ami's favorite: Fallacy of equivocation -Ways to mislead by comparison: differentiate between 1) omitted/ambiguous comparison class, 2) apples-to-oranges, and 3) puffery. -"Critical thinking in the real world:" Misleading comparisons in context, 3.6 Part D - Muslims & terrorism.
Lesson 3.5
Vagueness: A statement is vague when it contains a term (or terms) with no specific meaning for the intended audience. Sometimes context will clarify the meaning of a vague term.
Examples:
"We don't tolerate that kind of thing around here."
"Learn to drink the cup of life as it comes."
And we all have that one friend on social media...
"Some people don't know what's best for them. Live and learn."
Rough test for vagueness vs. ambiguity: A statement is vague if you have no idea WTF it's saying. If you can narrow the number of potential interpretations down to 2-3, the statement is probably ambiguous rather thanvague.
Normative vagueness
Normative = def. Establishing, relating to, or deriving from a standard or norm, especially of behavior.
Example:
"You cut in front of someone in line" is a descriptive claim - a claim about the way the world is. "You shouldn't cut in front of people in line" is a normative claim - a claim about the way the world ought to be.
Careful of these vague, normative terms:
"Dangerous," "harmful," "risky," "hazardous..."
These terms need further specification because people often interpret them terms in light of a background ideology.
For instance, it is "dangerous" to have premarital sex according to some religious groups because it puts your immortal soul at risk (of going to hell).
By contrast, premarital sex is only dangerous according to a secular ethos when people don't take precautions to prevent STIs.
Ambiguity: When something has two (or more) possible meanings* *NOT so many meanings you can't narrow it down to a few possibilities. That's vagueness.
Semantic ambiguity: When the meaning of a statement is unclear because it contains a word or phrase with more than one potential meaning.
Examples:
"There was not a single man at the party."
"The geese are by the bank."
Syntactic ambiguity: A statement is syntactically ambiguous when a reader or listener can reasonably interpret one sentence as having more than one possible structure.
Examples:
"The fish is ready to eat."
"The assailant threatened the man with a knife."
Group ambiguity ***NEW*** - not in lecture notes Some of your homework examples exhibit a typeof (semantic) ambiguity called group ambiguity. Group ambiguity: Referring to a group of individuals without clarifying whether one means the
group as a whole or the members in it taken separately.
Examples:
"Cows release more greenhouse gases than cars."
"Teachers earn more than principals." Note that semantic ambiguity, syntactic ambiguity, and group ambiguity are NOT MUTUALLY EXCLUSIVE categories. That is, statements may contain more than one type of ambiguity. What we're most concerned with is whether you are able to identify and describe the problem with a claim. This is more important than you remembering the right terminology.
Other definitions:
Fallacy of equivocation: When a key term in an argument isn't used with a consistent meaning throughout the premises and/or conclusion. In other words, a term might be used differently between premises or between the premises and the conclusion. **Test hint: This is Ami's favorite fallacy.**
Examples:
From class: "Since, as scientists tell us, energy neither comes into being nor goes out of being, there can be no energy crisis."
"Noisy children are a real headache. Two aspirin will make a headache go away. Therefore, two aspirin will make noisy children go away."
Fallacy of composition: Inferring that since the parts have a property, then the whole must have that property.
Example:
"Your brain is made of molecules. Molecules do not have consciousness. Therefore, your brain cannot be the source of consciousness."
Fallacy of division: Inferring that what's true of a whole must also be true of its parts
Example:
Homework 3.5 Note that I've omitted examples we discussed in lecture
A. Vagueness or Ambiguity? (a) If a premise is vague explain why and/or two or more possible interpretations of the term. (b) If a statement is ambiguous identify whether it is semantic, syntactic ambiguity and give at least two possible interpretations of the statement. (HW: Do 1-12, Optional 13-18)
2. A recent study shows that 1 in 3 women have been sexually harassed at work. Link (Links to an external site.)
6. The government is not to be trusted.
7. Coconut water helps to detoxify the toxins.
8. Headline: Lung Cancer in Women Mushrooms.
9. The rich control the political system.
10.You should avoid rollercoasters because they're dangerous.
11. Drinking frequently may be hazardous to your health.
12. The fish is ready to eat.
14. Teachers earn more than principals.
15. Headline: Reagan Wins on Budget, but More Lies Ahead.
16. Headline: Squad helps dog bite victim.
17. I don't like it when you smoke.
18. I'm not a big banana pancake fan.
B. Fallacies of Equivocation, Composition, and Division. Identify which fallacy is being committed and briefly explain your answer. (HW: Do odd numbers. Do even numbers for optional extra practice.)
A few examples (cover in class if questions)
3. The average salary US salary is around 50,000/year, therefore most people are doing well.
5. A ton of feathers is lighter than a ton of lead.
8. While it's true that studying critical thinking allows you to argue better, is it really a good idea to encourage people to argue?
C. Thinking About Cases in Real Life (cover in class if time)
1. Suppose it's true that no police officers are intentionally discriminatory toward visible minorities. (a) Does it follow that the law enforcement institution cannot possibly be discriminatory? In more general terms: If nobody in an institution intentionally discriminates does it follow that the institution cannot be discriminatory? (b) If I argue that an institution cannot be discriminatory because none of its members are discriminatory, do I commit the fallacy of composition?
ARE YOU STILL PAYING ATTENTION??? HERE'S A CAT VIDEO:
Lesson 3.6
Weasel words: Allow for intentional vagueness, mislead an audience. Common weasel words: "up to x percent/x number", "some", "as many as", "reportedly", "virtually", "many", "seems", "perhaps", "may" "one of"...
Euphemism: Substitutes mild and indirect ways of speaking for ways that might seem blunt, harsh, or impolite for social context.
Dysphemism: The opposite of a euphemism. A derogatory or unpleasant term is used instead of a pleasant or neutral one.
E.g.: "Loony bin," "kick the bucket," "feminazi"
Questions on Part A (weasel words) or Part C (euphemism/dysphemism) of Homework 3.6???
Misleading comparisons: Any time someone claims that something is good, bad, better, worse, cheaper, nicer, etc you should immediately ask "compared to what?"
Three ways to mislead by comparison:
1) Omitted/ambiguous comparison class -
40% less fat than what? (Note: a lot of ads specify what the comparison is... in fine print.)
"Religion may have been wrong about a few things, but science has been wrong about many more things!"
3) Puffery (legal term) - An exaggerated statement based on opinion, not fact. Examples:
The "world's cutest frog:"
Homework 3.6 Part B. Misleading comparisons 1. Identify the kind of misleading comparison (Omitting/ambiguous comparison class, apples to oranges, or puffery). 2. Explain why it's a misleading comparison.
Example:
Student use of methamphetamines has gone up 50%.
Answer: 1. Omitting comparison class. 2. It's misleading because we don't know what the appropriate comparison is. Is it a unit of time or since a point in time? I.e., in the last 10 years, the last 10 months? Since midterms? Is it compared to another drug? E.g., alcohol, the pot?
1. Healthcare costs are rising.
2. Critical thinking is a very important class.
3. The average man's salary is more than the average woman's, therefore women are underpaid.
4. Barrie Bonds was a better batter than Willie Mays. (Hint: One player used steroids)
5. Chipotle makes the best burritos in the world.
6. (From an April report on the retail sector) Retail sales have fallen 20% since December. The retail sector is in trouble.
7. Today's pole-vaulters are much better than those of 30 years ago because they vault much higher. (Hint: Think about changes in technology).
8. What's better for you, coffee or tea?
9. New Hippy Organic Nature Spiritual Woman Flower skin cream is now 25% more effective!!!!11!!!
Part D. Critical Thinking in The "Real" World
In some parts of American (and Western) culture and media, it's believed that Muslims are likely to be terrorists. This belief is often cited as justification for more intense security screening and proposed immigration bans. Notice that "likely" implies a comparison to other groups. I.e., X is more likely if they have some property MORE THAN some other group(s). So, the belief is that Muslims are more likely than other groups to be terrorists. This is still vague. Here's part of the problem: What's the correct comparison class?
2. What's the correct period of measurement? The past matters, but how much? If more Americans have died since 9-11 from Right wing extremists do we weight that more heavily than terrorist acts from 15 years ago? 25 years ago? 50 years ago? If we go back far enough, the British will say the American colonists were the worst terrorists. In other words, how do we select our time range to pick out our groups. Interestingly in much of the 20th century, many terrorist attacks were perpetrated by the Left.
3. On one hand, since we live in America we're concerned about the likelihood of someone being a terrorist in our immediate environment and so comparing groups within America might be the correct comparison class. On the other hand, if our claim is that a group (in the most general sense) is more likely than other groups to engage in terrorism then we need to look at all terrorist acts around the world. When we do this, the data shows that Muslims are in fact not very likely to commit acts of terrorism especially when you take into account their relative population in the world. For example of the around 800 acts of terrorism in Europe from 2011-2014 less than 1% where perpetrated by Muslims. Most were by Left and Right wing nationalist and extremist groups. If we measure likelihood of committing an act of terrorism relative to a group's population (and include the last 30 years) then the Irish and the Basque are probably at or near the top of the list.
4. So, back to the original question--and we can make it even more general--if someone wants to say that a member of a particular group is likely to engage in terrorist acts, what is the best way to evaluate that statement since it's inherently comparative? "Likely" implies more than baseline. But what's the baseline? i.e., what are the other groups we're comparing to? What is the period of history? What do we count: number of terrorists involved or number of civilians killed? Prepare to discuss your answer in recitation.
