The Value of Stereotyping

George Rebane

Stereotyping is the use of a template of characteristics that are thought to belong to members of a particular class more frequently than members of the general population of which the class is a subset.  The template of characteristics is also known as the stereotypical characteristics like, say, a plastic pocket protector full of writing instruments more often seen in the shirt pockets of male engineers than in the pockets of other men.

Stereotyping has gotten a bad rap in our society, and its use is roundly criticized in the public forum.  However, stereotyping, so as to assign or exclude someone from membership in a given class, is built in to almost every critter that lives on this planet.  Why?  Simply because it is a low cost survival technique that has evolved in all species to simplify quick decisions about the famous ‘Three F’ functions – feed, fight, or mount - important to everyone.

But if you understand the underpinnings of stereotyping, it can be a valuable tool in making all kinds of decisions, and also serve as a mirror for better understanding yourself.  I was motivated to write this piece when recently reading Daniel Kahneman’s monumental Thinking, Fast and Slow which dances around the subject without getting into the nitty-gritty of it because of the little math involved.  Kahneman, recent Nobelist and co-father of behavioral economics with the late Amos Tversky, is also a giant in the field of psychology.  In that field stereotyping comes up under the forbidding label of representativeness (q.v. – which is short for quod vide or ‘which see’, and I’ve concluded that its modern version is simply ‘google it’.)

Now consider seeing a good looking woman, finely coiffed, beautifully dressed, and dripping with expensive jewelry, getting out of a chauffeured limousine in front of a fancy restaurant.  You wonder if she’s a member of the currently notorious ‘1%’.  More particularly, what are the chances (friendly word for ‘probability’) that she is a member of that exclusive class, given she has some of the stereotypical characteristics of that class that you just observed?  If you had been facing the other way and your friend told you that a woman just got out a car behind you; then without seeing her, what would you answer if he mused whether the woman belonged to the '1%'?

Your reasonable answer is to quote the base rate (psychological term for prior probability) for such women being randomly observed in public.  And for our purposes, we’ll assume that the base rate for the '1%' is, surprisingly, 1% or 0.01.  So you’d tell your friend that the chances are one in a hundred that she’s a member of the 1%.  Within reason, you could do nothing else.

The important conclusion here in the broader sense is that with no additional information or with evidence of unknown reliability, we should stick to the base rate chance when deciding on class membership.  But now we look at how that base rate should be modified when more evidence comes in.

Suppose in our scenario you had information (perhaps even based on your own subjective experience) that such a scene (the stereotype) would be at least a hundred times more prevalent with someone belonging to the '1%' than the same scene with someone NOT belonging to the '1%'.  In a slightly more formal view, you assess that it was at least a hundred times more likely (probable) that someone looking as fancy as that lady would actually belong to the '1%' (the class in question), than the chance that someone would look like that who did NOT belong to the 1% - i.e. actually belonged to the '99%', the complement of the base rate.  The techies will recognize the number 100 here as the likelihood ratio calculated by the probability of seeing the stereotype given the member belongs to the class, divided by the probability of seeing the stereotype given the member NOT belonging to the class (Pearl, 1988).

So, how do you now update your belief that the lady getting out of the limo actually belongs to the '1%'?  It turns out that we can calculate that chance or probability from the famous theorem by the equally famous Reverend Thomas Bayes (q.v.).  And now for the simple and elegant formula for updating your prior belief or knowledge (of class membership) given new evidence (here stereotypical characteristics) – the envelope please.  The left hand term below is read ‘the probability that class (membership) is TRUE, given that the stereotypical characteristics are (or evidence is) observed is TRUE’.


Where PF is the Prevalence Factor or likelihood ratio (here equal to 100) described above.  Note that (1 – BaseRate) is the probability of NOT belonging to the class in question.  Plugging in the values into our trusty calculator gives us


Applying our knowledge of stereotypical characteristics at even a modest level, such evidence being only a hundred times more prevalent for the 1% than the 99%, updates our belief from the base rate of 1% to 50% that the lady may indeed be a member of the filthy rich.  Next, it’s instructive to look at a more reasonable value of PF such as 1,000.  Plugging those numbers into the formula in our calculator, we get Pr(belongsto1%|stereotype) = 10/10.99 = 0.91 or 91%.  This now says that there is less than one chance in ten  -actually 9% - that some not so rich lady is masquerading as a member of the really rich, and is a more acceptable probability that I would put my money on.

Let’s take a more detailed look at the prevalence factor PF.  But first, everyone clearly sees that the value of the base rate can only be a number between zero (0%) and one (100%).  Equally obviously, and as we have seen, the more reliably the stereotype is displayed by members of a class, the higher the value of PF should be in our formula.  If the prevalence of the stereotype gets really large (say, 10,000), then we would expect our updated belief to approach one or 100% or certainty when we observe the stereotype.  And sonovagun, that’s exactly what happens.

On the other hand, if we maintain that the stereotype occurs just as frequently in the general population as with the class in question – here, proportionately just as many ladies blow their savings on the fancy rig from the 99% as do from the 1% - then the likelihood ratio (defined above) or Prevalence Factor is unity or one.  Plugging one into our formula gives BaseRate as the answer.  In other words, your belief is unchanged after observing the stereotypical characteristics – you have discounted the evidence completely and your updated belief that the lady belonged to the '1%' is the same as it was before considering the displayed stereotype.  Again this is the reasonable conclusion for people who discount stereotypical evidence.

Finally, if you believe that the stereotypical characteristics are less frequently or seldom seen in that class, then your value for PF becomes less than one, and from the formula your updated belief goes below or smaller than the base rate.  Consider if you saw a lady getting out of a beat-up family minivan wearing a print dress, cheap coat, and a run in her pantyhose.  You would most likely say that the chances that such a stereotype was connected to the '1%' class was, say, one in a hundred or PF = 0.01.  Putting that into our formula then yields


In other words, the chances of a lady looking like that belonging to the '1%' is one out of ten thousand, or essentially nil.

One last thing.  As the base rate gets larger, even approaching 100%, we see from the formula that the benefits of observing a stereotype, or any additional supporting evidence for that matter, contributes less to strengthening our belief that the member belongs to the class in question.  A moment's thought confirms that to agree with our intuition.  As a simple exercise, insert either 0.99 or 1 in for BaseRate and see how much any additional supporting evidence strengthens your belief.  And now try it with contradicting evidence represented, as above, with PF equaling less than one.

Hopefully, you now understand that Bayesian analysis is powerful stuff indeed, and when correctly used can expose our unfounded biases and also reinforce the reasonable ones in a very explicit way.  The careful reader will see that the formula can also be used to assess the worth of competing sets of evidence before effort is made and costs committed to obtain such evidence - in other words, answering questions like 'What difference would that evidence make?'

Addendum (aka Lucky Strike Extra):  For those who have made it this far, let’s consider the case of terrorism.  Suppose you’re at a large computer exhibition held in a huge conference hall.  The place is packed with exhibit stalls and thousands of people, a busy scene with visitors carrying bags of vendor supplied literature and promotional cshotskies.  You see a man placing his attaché case under a display table and quickly walking away, disappearing into the busy throng.

Seconds later it suddenly dawns on you that the man definitely looked like a swarthy mid-easterner, no doubt a Muslim.  A moment’s reflection recalls hundreds of attendees putting down their bags and cases while talking to vendors and seeing demonstrations, nothing special there.  But then the stereotypical characteristics of that mid-eastern looking man come flooding back as you look at the case there under the display table.

What should you do?  Nothing?  Get the hell out of there as fast as possible?  Or shout out an alarm and try to clear the area before the ‘bomb’ explodes, knowing it could just be an innocent attaché case with its owner heading for a bathroom in a hurry?

Comments

[Russ Steele]

I choked on representativeness when I arrived at that chapter. I had to keep asking my self, what the hell is he talking about. I like your explanation better and will just make a mental transfer. Using my systems one, I am voting for a bathroom run, having myself left a back pack or two at the booth when making a pit stop.

[Douglas Keachie]

Go steal the case, and throw it into the largest possible open empty area as quickly as possible. Out a window onto a roof below would be good. You have a bit of time, because he will not want it to go off until after he is well away from the area. If it doesn't explode, then you should disappear too. BTW, you already identified it as a "case of terrorism."

[Douglas Keachie]

Or give it to your favorite hyper serious flyboy....

[Gregory]

The flyboy would like to thank Keachie for responding to the charge that he's a malicious liar with more fabrications and empty rhetoric.

[Ben Emery]

I didn't read through your statistical analysis of probabilities with respect to the reality of stereotyping but found it funny that you would use a 1% example who possibly became part of that economic class by exploiting the natural resources of the lineage nation of that middle easterner with the US military who then felt the need for retaliation. I am just playing along with your absurd stereotyping scenarios.

[Ben Emery]

George,
This is Ben Emery. Once again another name has popped up when I posted. I am not Iliad but in this case I guess I am. I will go through typepad and try to correct what is wrong.

[Ben Emery]

I have no idea where Iliad came from but I think it is fixed at this point.

[George Rebane]

BenE - It looks like you're back with your own name.

The '1%' scenario was just made up to explicate the Bayesian approach to updating knowledge in the context of what we call stereotyping. The point made is that stereotyping is and always has been extremely valuable for humans and other critters. When we vilify it without reason, we wind up behaving poorly and make bad public policy.

That having been said, the Bayes rule for its application is both correct and useful as claimed.

[Douglas Keachie]

Douglas Keachie would like to thank Gregory Goodknight for responding to the charge that he's a malicious liar with more fabrications and empty rhetoric.

Are you starting to get the message, Greg?