Flaw Questions: Percent vs Amount

[Emmeline Ndongue]

Percent vs Amount:
As seen in the examples of Manhattan LR, the argument goes "The idea that comic book collectors are especially likely to attend science fiction movies on opening weekend is wrong. A largely scale study found that comic book collectors make up only 15% of opening weekend audiences for science fiction movies."

The book offered an explanation: if, say, only 5% of the population collects comic books, then the fact that these collectors make up 15% of opening weekend audiences at sci-fi films suggests that they are three times more likely to attend than others.

I'm having a trouble understanding this explanation mathematically. What does three times more likely mean? Is it referring to the probability of people attending the opening weekend? Is it that the higher the percentage a certain group group take up the total people attending opening weekend means it's more likely that they will attend ? How do get "3 times" more likely given that the study is true?

[ohthatpatrick]

The idea here is that if a given trait appears in 5% of the overall population, then that trait should appear in 5% of any other population, assuming that the trait we're talking about has no causal bearing on the other population.

More lucidly:
Right handed people are 85% of the overall population.
We would expect 85% of prisoners to be right handed, unless we think that right-handed people are more/less likely to be convicted of crimes than are lefties.

This sort of argument appears in the news a lot (I'm making up specific stats, but just trying to provide some common examples):

HOW DO WE ARGUE THAT BLACK PEOPLE ARE UNFAIRLY TREATED BY OUR CRIMINAL JUSTICE SYSTEM?
We say that black people are only 15% of the overall population but they are 35% of the prison population.

HOW DO WE ARGUE THAT ILLEGAL IMMIGRANTS ARE NOT ESPECIALLY LIKELY TO BE CRIMINAL?
We say that illegal immigrants are 2% of the overall population but are only committing 1% of crimes.

When we see that a certain trait is showing up disproportionately, it makes us scratch our heads and usually leads us to posit some causal backstory.

If we saw that 40% of biologists were female, but only 10% of the Nobel prize winners for biology were female, we would wonder, "Is there some bias within the Nobel prize awarding process?"

So if comic book collectors are 5% of the overall population, but 15% of the people who saw this sci-fi movie on opening weekend, then a disproportionately high percentage of comic book collectors are seeing the movie.

15% is "5% times 3", so we can say comic book collectors are 3 times more likely to see the movie than others.

The baseline for likelihood is that "your percent of the bigger population would match your percent of the specific population".

If comic book collectors were equally likely to see the movie as anyone else (and comic book collectors are 5% of the bigger population), then 5% of the moviegoers would be comic book collectors.

Hope this helps.

[VendelaG465]

So if comic book collectors are 5% of the overall population, but 15% of the people who saw this sci-fi movie on opening weekend, then a disproportionately high percentage of comic book collectors are seeing the movie." so if i was to state 4% or 5% that would be correct ? but anything higher than that would raise suspicious b/c we've established 5% as a base # for the overall/entire population correct? it'd be like saying i have 5 apples in a bag, but pulled out 10 apples from that same bag. I'm just trying to make sure I'm on the same page.

[ohthatpatrick]

The apple analogy doesn't really work because you're just using raw numbers, and what we were talking about was specifically about percentages.

If you told me that a bag full of fruit contained 20% apples, and then you had a random claw grab some fruit out of the bag, we would expect 20% of the fruit it grabbed to be apples.

If it were under 20%, we'd think, "This claw has a bias against grabbing apples."

If it were over 20%, we'd think, "This claw has a bias in favor of grabbing apples."

If it were 20% apples, we'd think, "This claw is good at grabbing a random, representative assortment of the fruit in the bag"