Statistical Syllogisms and Misleading with Numbers 2 (Mean, Median, and Distribution)

Introduction

If I know that most students like pizza and I select a student at random, I can reasonably infer that the student likes pizza. When you make this kind of inference you are making an inference from what you know about a group to what you can know about an individual or subset of the group. In fancy-talk we call this kind of argument a statistical syllogism. We can also think of statistical syllogisms as ‘reverse generalizations.’ Generalizations make conclusions about groups based on what we know about a subset of the group while statistical syllogisms make inferences about individuals or subsets based on what we know about the group.

Because statistical syllogisms rely on generalizations, evaluating syllogisms requires we know something about averages (mean), medians, and distributions. In this lesson we’ll learn how to systematically evaluate statistical syllogisms.

Part 1: The Basics of Evaluating Statistical Syllogisms

Consider the following dialogue:

Person A: Evan likes pizza.
Person B: How do you know?
Person A: Well, cuz he’s a student.

Person A implies the following argument:

(P1). Most students like pizza.
(P2). Evan is a student.
(C). Evan likes pizza.

More generally statistical syllogisms take the following form:

Statistical Syllogism: Standard Form

(P1). n proportion of population X are/have/like/do/etc… Y.
(P2). a is an X.
(C).  a has n probability of being/having/liking/doing/etc Y.

Things to notice: The conclusion is probabilistic; that is, since it’s an inductive argument it isn’t guaranteed to be true. It’s probability of being true will depend on:

1. The proportion of the target population that have the trait in question.

In this example it will be the proportion of students that like pizza. The higher the proportion, the stronger the Relevance (RRAR) of P1 and P2 to to the conclusion. [Note: The relevance of P1 and P2 are both affected because statistical syllogisms are dependent/linked premises].

  • If “most”=95% then the conclusion is very likely to be true.
  • If “most”=55% then the conclusion is only slightly likely to be true.

2. The homogeneity of the target population with respect to the trait in question.

More generally, we need to know if there are important sub-groups within a target population that better predict the likelihood of a conclusion being true. If there are important subgroups relevant to the trait in question, then P1 will have weak Relevance (RRAR) to the conclusion. If the target population is highly homogenous (i.e., there aren’t relevant subgroups) then P1 will be strongly relevant.

E.g., Suppose 20% of the student population is lactose intolerant and Evan is part of this sub-population. Knowing that Evan is a lactose intolerant student better predicts his attitude toward pizza than merely knowing he’s a student.

E.g., The average age of an American is around 38. It would be a mistake to infer that an American chosen at random is 38 years old. Since ‘age of Americans’ is an extremely heterogeneous category, knowing the average age is weakly relevant to any conclusion about an individual American.

Here we should stop and note that if P1 is an average, special problems will arise with respect to homogeneity and in turn relevance. We’ll look at those in detail in Part 2 below.

3. Acceptability of P1: Assess the Generalization

Statistical syllogisms rely on generalizations for their first premise. Therefore, any time we encounter a statistical syllogism, we should apply our method of evaluating generalizations to it. This might mean doing a bit of background research to see where the number came from: Was the sample representative and large enough? Were there potential measurement errors? Our evaluation will determine the acceptability (RRAR) of the first premise.

4. Acceptability of P2: Is a and X?

Perhaps this criterion is too obvious to state. However, sometimes people do make genuine error in regards to whether an individual or subgroup is indeed a member of a larger group. For example, someone might make a generalization about refugees and infer something about a subgroup of immigrants. This is a category mistake as there are important traits that differentiate these two groups. Immigrants are a distinct category from refugees as they are from illegal immigrants. Despite these differences, these groups often get lumped together leading to bad inferences about subgroups.

Part 2: Averages and Distribution in Statistical Syllogisms

The average American has one testicle. From this fact I shouldn’t infer that an American selected at random will have one testicle. Similarly, since the average American has 1.7 children, I shouldn’t expect to find an individual family with 1.7 children. Averages are arithmetical fictions and as such should be treated with caution when they are the first premise of a statistical syllogism.

Importantly, generalizations that are averages tell us nothing about distribution.

Example:

Hey, I just read that the average price of a home in the US increased by 10% last year. Your brother’s house must also have increased by 10% since last year.

Statistical syllogism in standard form:

(P1) In 2016, the average price of a home in the US increased by 10% year over year.
(P2) Your brother owns a home.
(C) The price of your brother’s home must have increased by 10% year over year.

If real estate regional and local markets across the US moved at widely different rates, then knowing the average rate for the US has little predictive value. Put another way, if the average comes from a wide distribution or a heteronomous data set, it’s of little predictive value.

Consider two cases, both where the average is 10%.

Case A: 
Region 1=10%
Region 2=10%
Region 3=10%
Region 4=10%
Region 5=10%
                  50/5=10

Case B:
Region 1=46%
Region 2=1%
Region 3=10%
Region 4= –3%
Region 5= –4%
                   50/5=10

Both data sets have the same average of 10%. If the data set looks like Case A (narrow distribution/homogenous) then P1 is strongly relevant to a conclusion about a particular subset or individual. If the data set looks like Case B (wide distribution/heterogeneous) then P1 is not relevant to a conclusion about a particular subset or individual.

Generalizations are averages that presuppose homogeneity within the target population. To the degree that the target population isn’t homogeneous, to that degree conclusions of statistical syllogisms become weaker. To put it a in a familiar way: If an average is a consequence of a wide distribution/heterogeneous then it is only weakly Relevant (RRAR) to the conclusion.

Practice:

(a) Put the statistical syllogism into standard from and (b) Identify what relevant sub-groups we’d need to know about to strengthen the inference to the conclusion.

Example:

1. 25% of Americans don’t complete high school, therefore there’s about a 25% chance that Stephanie didn’t complete high school.

(a) Standard form:

(P1) 25% of Americans don’t complete high school.
(P2) Stephanie is an American.
(C)  Therefore, there’s a 25% chance that she didn’t complete high school.

(b) Relevant sub-groups:

(Lack of) Educational achievement in the US isn’t evenly distributed. To strengthen the inference I’d want to know

(a) her socio-economic class,
(b) the state she grew up in,
(c) her parents’ level of education,
(d) any other relevant sub-groups predictive of level of education… .

 

2. About 68% of American adults aged 20 and older are obese. Therefore, 68% of my classmates are obese.

3. 15% of cars are subject to a recall within their first 3 years on the road, therefore there’s a 15% chance my car will be recalled.

4. 32% of BGSU students graduate on time (for a 4 year degree), therefore there’s a 32% chance of you graduating within on time. Link (Links to an external site.)

5. The US unemployment rate is 4.9%, therefore a Black man in Detroit is 4.9% likely to be unemployed.

Part 3: How to Fool and Be Fooled Using Distribution, Averages, and Absolute Values

True fact: You can make data support almost any argument you want depending on how you present it. This is possible due to the nature of averages and differences in distribution.

Average/Mean vs Median and Distribution

Which ten-person island would you rather live on? (Answer first, then scroll down)

  • Country A: GDP is $1 010 000 and average income is $101 000
  • Country B: GDP is $600 000 and average income is $60 000.

 

 

 

 

 

 

To answer the question, we need more information. First we need to know the distribution. Is it wide or narrow?

Country A distribution: {1 000 000, 2 000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000}
Country B distribution: {60 000, 60 000, 60 000, 60 000, 60 000, 60 000, 60 000, 60 000, 60 000, 60 000}

Now which one would you rather live in?

As mentioned in the previous section, when statistics are presented as a mean/average be wary. Two sets of numbers can have similar averages but very different distributions. Often we don’t have access to the raw data but there is another number that can tell us something about the distribution: The median is the value in the middle of the distribution. Since the median tells us about distribution and is therefore much more valuable for evaluating data in light of an argument:

The median income of Country A is 1000. The median income of Country B is 60 000.

Looking at the median rather than average income suggests that Country B is the better choice. But wait! There’s more!

We need to know a few more things:
(a) What is the occupation of the person making all the money in Country A? What is my occupation? If my occupation is the same as the guy making all the money in Country A, it might be better for me to go there

(b) What is the cost of living? That is, I must apply the Relativity (RRAR) criterion. Suppose the cost of living on Country A is only 60% of the median income while on Country B is 90% of the median income. This information might also affect my choice.

In summary, averages tell us little unless the data set or target population is homogenous and so we should always look for median values. However, median values alone might not tell us everything we need to know depending on our interests. Finally, we should always ask of any conclusion, Relative to what?

Here’s an example of how distribution matters:

Practice:

You aren’t very good at stats but next semester you have to take your stats requirement. You want to take stats with teacher that gives you the best shot at a good grade. Professor Plum’s class GPA is 75% and Professor Apple’s’s is 70%. You decide to take stats with Professor Apple. Using the concept of distribution and median, explain why you might make this decision?

Even if Professor Plum’s median class score is lower than Professor Apple’s class, you still might choose to take Professor Plum’s class. Why might someone do this? (assume both have open enrollment).

 

HOMEWORK

A. Statistical Syllogisms: (a) Put the statistical syllogism into standard from 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 in the Theatre program scored well on the math component of the SAT.

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.

3. The earth’s average surface temperature has risen about .80 degrees C since 1906. It follows that winter in my town this year will be .80 degrees warmer than in 1906.

4. The average annual cost of health care in the US is about $9, 000/person. It follows that most Americans will each consume about that much on health care this year.

B. Critical Thinking in the Real World 

1. Question 4 (of Part A) refers to the average annual cost of health care per person. Relative to other countries spends almost triple the OECD (an organization of 34 economically developed market economies) average yet gets only moderate health care outcomes and has the highest rates of preventable diseases. Every OECD country has some form of universal health care while the US has only started moving in that direction with the Affordable Care Act. Hypothesize why universal health care might make the average cost of health care cheaper and the average outcomes better. Hint: Think about who buys health care insurance in the US, who doesn’t, and why. Note: There’s no one right answer here. There are many factors contributing to the above mentioned effects.

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