A related problem: replacing the majority with the norm.
Most Americans are Christians. Given a random American, he/she is more likely to be Christian than anything else. It may be a safe bet to say Merry Christmas (especially since few people are offended by hearing Merry Christmas even if they’re not Christian.) So far, that’s just reacting rationally to the fact that Christians are a majority.
But it starts to get unsettling when the majority is regarded as the norm—when people refer to the United States as “a Christian nation,” for instance, with a normative rather than a statistical implication. There’s a difference in thinking “Most Americans are Christian, but some are not,” and thinking “Americans are Christian. (Except for a few aberrations.)” The latter has the connotation that non-Christians are less American.
You can apply this to all kinds of majority/minority things. “Most people are straight, but some are gay” as opposed to “People are straight. Except for some aberrations.” “Most mathematicians are men, but some are women” as opposed to “Mathematicians are men. Except for some aberrations.” “Many cultures share a similar standard of beauty, but there are some differences” as opposed to “There is one standard of beauty. Except for aberrations.”
People are known to have a bias of rounding up high probabilities (treating 90% as practically certain) and rounding down low probabilities (treating 10% as practically impossible.) It’s possible that this has an effect on the way we think about minority populations—we mentally approximate a population that’s 95% A and 5% B as “basically” 100% A, and we don’t always distinguish in our intuitions between a 5% and a 0.05% population.
Moral of the story: it may be rational to assume that a given person in a group is a member of the majority, but remember to correct for your tendency to slip over the edge from “majority” to “normal” or “standard”.
In absolute terms, 5% of the United States’ population is about 15 million. That’s 1.5 times the population of Belgium, Portugal, etc., and only 65 out of 224 countries have a population higher than that.
I don’t detect a difference between the two universes being described by
“most people are straight and some are gay” and
“people are straight, except for some aberrations”
except that all things being equal, I would suspect that who uttered the second phrase was more likely to disapprove of homosexuality than who uttered the first phrase. But is reality being described any less accurately by one of these two phrases? How would we go about discovering which phrase was more accurate?
But is reality being described any less accurately by one of these two phrases?
When comparing two predictions, the better prediction is the one that leads to less surprise.* That means that oftentimes you may treat false positives as much less important than false negatives, or vice versa. To me, the difference between the two is which error they favor- the first is likely to overestimate the chance someone is gay, whereas the second is likely to underestimate it. Given that the damage done by a wrong guess is asymmetric, which error you favor should likewise be asymmetric.
*I say this instead of “is right more often” because when it’s wrong in a spectacular way that should be counted multiple times. If you say “well, 5/6ths of people haven’t been sexually assaulted, so I can make rape jokes and be ok 5/6ths of the time!” then you are cruelly underestimating the damage done by making a rape joke to a rape survivor. When you count it in terms of surprise, you get the better result of “always assume someone could be a rape survivor.”
Agreed with what SarahC says, but will add to it that your suspicions about the speaker (“that who uttered the second phrase was more likely to disapprove of homosexuality than who uttered the first phrase”) are not irrelevant.
That is, if the speaker doesn’t disapprove of homosexuality, then the second phrase is conveying misleading information about the speaker, who is real. In this case, yes, reality is being described less accurately by the second phrase.
By the same token, if the speaker does disapprove, then reality is being described less accurately (along this axis) by the first phrase.
Also, it’s worth asking why you conclude what you do about the speaker. It seems likely to me that it’s not an idiosyncrasy of yours, but rather that you are responding to connotations of the word “aberration” which are communicated by the second phrase… specifically connotations involving, not only the statistical likelihoods, but the perceived social value of people who are/aren’t straight.
One could therefore determine which phrase was more accurate in a particular society by looking at how people’s value to that society varies with their orientation.
Something similar might be true about perceived moral value, but talking about moral value as part of reality is more problematic.
Also, it’s worth asking why you conclude what you do about the speaker. It seems likely to me that it’s not an idiosyncrasy of yours, but rather that you are responding to connotations of the word “aberration” which are communicated by the second phrase… specifically connotations involving, not only the statistical likelihoods, but the perceived social value of people who are/aren’t straight.
I think this is only due to the fact that we’re both aware of political battles over homosexuality. I don’t read any disapproval of six-fingered people into SarahC’s comment below.
Are you suggesting that people with different political opinions should use different language to describe the same reality, or merely that they do?
3-8% of Americans are gay (more like 5% in the UK.) That’s a true statement. Guessing that an arbitrary person is straight is perfectly kosher, from a Bayesian perspective.
Here’s the thing. Most of us would say that being left-handed is a minority trait, while being six-fingered is an anomaly or aberration. About 15% of people are left-handed; about 0.2% of people are six-fingered. Take that as a benchmark. Then being gay is more like being left-handed than it is like being six-fingered.
And, at 20-30%, women scientists should definitely belong in the “left-handed” rather than “six-fingered” category.
If you start thinking of a sizable minority as though it’s as rare and strange as a very small minority, then you’re making a mistake.
You’re literally saying it’s a fuzzy measure of magnitude, like the difference between “big” and “huge”? That makes the stakes seem pretty low. Why quibble over them?
Yeah, it’s a fuzzy measure of magnitude. I was trying to quantify why a stereotype can be wrong (in addition to just bothering some people) and I think that what makes stereotypes actually incorrect is the human tendency to approximate “most” by “all.”
In Kahneman and Tversky’s prospect theory, there is evidence that people do not react to the differences between small probabilities. It’s conceivable that sometimes people treat a small minority as though it were a tiny minority, virtually non-existent. (On the other hand, there’s some evidence that people overestimate small probabilities, which makes this argument weaker. So I take it back.)
The other way stereotyping can be a mistake has to do with the conjunction fallacy. Perhaps most X’s are A, most X’s are B, and most X’s are C. It does not follow that most X’s are A and B and C. Something that is A and B and C is a “most representative” element, but most X’s are not “representative.”
This is the old platitude that “there is no typical student at our school.” It would be truer to say that the most typical students are usually rare. But people will assume that a student from that school is like that rare “typical student.” This is a form of stereotyping which is actually inaccurate. (As opposed to “offensive but accurate,” which is what many people claim stereotypes to be.)
A third type of inaccurate stereotyping is mistaking P(B|A) for P(A|B). Most criminals are men, but most men are not criminals.
These are all good points. Given that 20-30% of scientists are women, it’s misleading to say “scientists are normally men” without quantifying “normally.” And though most users of this website are americans, and men, and hetero, and college-educated, possibly it is not normal for them to be all at once (I could have picked better examples). But I don’t like the idea of people scoring less-parochial-than-thou points off of each other through trivial mistakes along these lines. Maybe that doesn’t happen.
A third type of inaccurate stereotyping is mistaking P(B|A) for P(A|B). Most criminals are men, but most men are not criminals.
The usual emphasis on this website is the close relationship between these probabilities, and its important consequences.
A related problem: replacing the majority with the norm.
Most Americans are Christians. Given a random American, he/she is more likely to be Christian than anything else. It may be a safe bet to say Merry Christmas (especially since few people are offended by hearing Merry Christmas even if they’re not Christian.) So far, that’s just reacting rationally to the fact that Christians are a majority.
But it starts to get unsettling when the majority is regarded as the norm—when people refer to the United States as “a Christian nation,” for instance, with a normative rather than a statistical implication. There’s a difference in thinking “Most Americans are Christian, but some are not,” and thinking “Americans are Christian. (Except for a few aberrations.)” The latter has the connotation that non-Christians are less American.
You can apply this to all kinds of majority/minority things. “Most people are straight, but some are gay” as opposed to “People are straight. Except for some aberrations.” “Most mathematicians are men, but some are women” as opposed to “Mathematicians are men. Except for some aberrations.” “Many cultures share a similar standard of beauty, but there are some differences” as opposed to “There is one standard of beauty. Except for aberrations.”
People are known to have a bias of rounding up high probabilities (treating 90% as practically certain) and rounding down low probabilities (treating 10% as practically impossible.) It’s possible that this has an effect on the way we think about minority populations—we mentally approximate a population that’s 95% A and 5% B as “basically” 100% A, and we don’t always distinguish in our intuitions between a 5% and a 0.05% population.
Moral of the story: it may be rational to assume that a given person in a group is a member of the majority, but remember to correct for your tendency to slip over the edge from “majority” to “normal” or “standard”.
Scope insensitivity could also be a factor here.
In absolute terms, 5% of the United States’ population is about 15 million. That’s 1.5 times the population of Belgium, Portugal, etc., and only 65 out of 224 countries have a population higher than that.
I don’t detect a difference between the two universes being described by
“most people are straight and some are gay” and
“people are straight, except for some aberrations”
except that all things being equal, I would suspect that who uttered the second phrase was more likely to disapprove of homosexuality than who uttered the first phrase. But is reality being described any less accurately by one of these two phrases? How would we go about discovering which phrase was more accurate?
When comparing two predictions, the better prediction is the one that leads to less surprise.* That means that oftentimes you may treat false positives as much less important than false negatives, or vice versa. To me, the difference between the two is which error they favor- the first is likely to overestimate the chance someone is gay, whereas the second is likely to underestimate it. Given that the damage done by a wrong guess is asymmetric, which error you favor should likewise be asymmetric.
*I say this instead of “is right more often” because when it’s wrong in a spectacular way that should be counted multiple times. If you say “well, 5/6ths of people haven’t been sexually assaulted, so I can make rape jokes and be ok 5/6ths of the time!” then you are cruelly underestimating the damage done by making a rape joke to a rape survivor. When you count it in terms of surprise, you get the better result of “always assume someone could be a rape survivor.”
Agreed with what SarahC says, but will add to it that your suspicions about the speaker (“that who uttered the second phrase was more likely to disapprove of homosexuality than who uttered the first phrase”) are not irrelevant.
That is, if the speaker doesn’t disapprove of homosexuality, then the second phrase is conveying misleading information about the speaker, who is real. In this case, yes, reality is being described less accurately by the second phrase.
By the same token, if the speaker does disapprove, then reality is being described less accurately (along this axis) by the first phrase.
Also, it’s worth asking why you conclude what you do about the speaker. It seems likely to me that it’s not an idiosyncrasy of yours, but rather that you are responding to connotations of the word “aberration” which are communicated by the second phrase… specifically connotations involving, not only the statistical likelihoods, but the perceived social value of people who are/aren’t straight.
One could therefore determine which phrase was more accurate in a particular society by looking at how people’s value to that society varies with their orientation.
Something similar might be true about perceived moral value, but talking about moral value as part of reality is more problematic.
I think this is only due to the fact that we’re both aware of political battles over homosexuality. I don’t read any disapproval of six-fingered people into SarahC’s comment below.
Are you suggesting that people with different political opinions should use different language to describe the same reality, or merely that they do?
Merely that they do.
3-8% of Americans are gay (more like 5% in the UK.) That’s a true statement. Guessing that an arbitrary person is straight is perfectly kosher, from a Bayesian perspective.
Here’s the thing. Most of us would say that being left-handed is a minority trait, while being six-fingered is an anomaly or aberration. About 15% of people are left-handed; about 0.2% of people are six-fingered. Take that as a benchmark. Then being gay is more like being left-handed than it is like being six-fingered.
And, at 20-30%, women scientists should definitely belong in the “left-handed” rather than “six-fingered” category.
If you start thinking of a sizable minority as though it’s as rare and strange as a very small minority, then you’re making a mistake.
Isn’t 5% just 3-8% that forgot to state its error margins?
You’re literally saying it’s a fuzzy measure of magnitude, like the difference between “big” and “huge”? That makes the stakes seem pretty low. Why quibble over them?
How do you feel about six-fingered scientists?
Yeah, it’s a fuzzy measure of magnitude. I was trying to quantify why a stereotype can be wrong (in addition to just bothering some people) and I think that what makes stereotypes actually incorrect is the human tendency to approximate “most” by “all.”
In Kahneman and Tversky’s prospect theory, there is evidence that people do not react to the differences between small probabilities. It’s conceivable that sometimes people treat a small minority as though it were a tiny minority, virtually non-existent. (On the other hand, there’s some evidence that people overestimate small probabilities, which makes this argument weaker. So I take it back.)
The other way stereotyping can be a mistake has to do with the conjunction fallacy. Perhaps most X’s are A, most X’s are B, and most X’s are C. It does not follow that most X’s are A and B and C. Something that is A and B and C is a “most representative” element, but most X’s are not “representative.”
This is the old platitude that “there is no typical student at our school.” It would be truer to say that the most typical students are usually rare. But people will assume that a student from that school is like that rare “typical student.” This is a form of stereotyping which is actually inaccurate. (As opposed to “offensive but accurate,” which is what many people claim stereotypes to be.)
A third type of inaccurate stereotyping is mistaking P(B|A) for P(A|B). Most criminals are men, but most men are not criminals.
These are all good points. Given that 20-30% of scientists are women, it’s misleading to say “scientists are normally men” without quantifying “normally.” And though most users of this website are americans, and men, and hetero, and college-educated, possibly it is not normal for them to be all at once (I could have picked better examples). But I don’t like the idea of people scoring less-parochial-than-thou points off of each other through trivial mistakes along these lines. Maybe that doesn’t happen.
The usual emphasis on this website is the close relationship between these probabilities, and its important consequences.