Representativeness is the collective term used to describe the following range of fallacies people make when judging probabilities.
The problem of base-rate neglect
Insensitivity to prior probability of outcomes
Insensitivity to sample size
Misperception of chance and randomness
We consider these in turn:
The problem of base-rate neglect
Consider the following problem
A particular heart disease has a prevalence of 1/1000 people. A test to detect this disease has a false positive rate of 5%. Assume that the test diagnoses correctly every person who has the disease. What is the chance that a randomly selected person found to have a positive result actually has the disease?
This question was put to 60 students and staff at Harvard Medical School.
Almost half gave the response 95%.
The average answer was 56%.
Click here for the correct answer (and full explanation) that was given by just 11 participants.
Insensitivity to prior probability of outcomes
Suppose you are given the following description of a person:
'He is an extremely athletic looking young man who drives a fast car and has an attractive blond girlfriend.'
Now answer the following question:
Is the person most likely to be a premiership professional footballer or a nurse?
If you answered professional footballer then you were sucked into this particular fallacy. You made the mistake of ignoring the base-rate frequencies of the different professions simply because the description of the person better matched the stereotypical image. In fact there are only 400 premiership professional footballers in the UK compared with many thousands of male nurses, so in the absence of any other information it is far more likely that the person is a nurse.
The hypothesis that people evaluate probabilities by representativeness in this way (thereby ignoring the prior probabilities) was tested by [Kahneman and Tversky 1973]. Subjects were shown brief personality descriptions of several individuals, allegedly sampled at random from a group of 100 professionals - all engineers or lawyers. The subjects were asked to assess, for each description, the probability that it belonged to an engineer rather than a lawyer. There were two experimental conditions:
1. Subjects were told the group consisted of 70 engineers and 30 lawyers
2. Subjects were told the group consisted of 30 engineers and 70 lawyers
The probability that a particular description belongs to an engineer rather than a lawyer should be higher in 1 than in 2. However, in violation of Bayes rule, the subjects produced essentially the same probability judgements. Subjects were apparently evaluating the likelihood of a description being an engineer rather than a lawyer by the degree to which it was representative of the two stereotypes; they were paying little or no regard to the probabilities of the categories.
When subjects were given no personality sketch, but were simply asked for the probability that an unknown individual was an engineer the subjects correctly gave the responses 0.7 and 0.3 in 1 and 2 respectively. However, when presented with a totally uninformative description the subjects gave the probability to be 0.5 in both experiments 1 and 2.
Kahneman & Tversky concluded that when no specific evidence is given, prior probabilities are used properly; when worthless evidence is given, prior probabilities are ignored.
Insensitivity to sample size
Consider the problem of two hospitals of different sizes in the same town. In the large hospital, 45 babies are born each day, whereas only 15 are born in the smaller hospital. 50% of all babies are boys, but on some days the percentage will be higher and on other days, it will be lower.
Which hospital would you expect to record more days per year, when over 60% of the babies born were boys?
Click here for the correct answer and explanation.
Misperception of chance and randomness
This is an error of "local" randomness known as "the gambler's fallacy" or a belief in the "law of small numbers". People believe that when flipping a coin for example, after several "heads" the next flip will surely be "tails". The sequence H-T-H-T-T-H is considered more likely than H-H-H-T-T-T, for example. It seems to be more random, or it is more representative of the expected sequence generated by such a random process.
In other words, the fallacy is that the characteristics of a process will not only be represented globally in an entire sequence, but also locally in each of it's parts, and it is a fallacy with which even experienced researchers frequently expose themselves. For example, by expecting 10 samples to provide statistically significant results in an experiment or trial, in the same way as if there were 10,000 samples.