Ecological Fallacy - Mean and Median

Mean and Median

A third example of ecological fallacy is when the average of a population is assumed to have an interpretation in term of likelihood at the individual level.

For instance, if the average score of group A is larger than zero, it does not mean that a random individual of group A is more likely to have a positive score. Similarly, if a particular group of people is measured to have a lower average IQ than the general population, it is an error to conclude that a randomly selected member of the group is more likely to have a lower IQ than the average general population. Mathematically, this comes from the fact that a distribution can have a positive mean but a negative median. This property is linked to the skewness of the distribution

Consider the following numerical example:

• Group A: 80% of people got 40 points and 20% of them got 95 points. The average score is 51 points.
• Group B: 50% of people got 45 points and 50% got 55 points. The average score is 50 points.
• If we pick two people at random from A and B, there are 4 possible outcomes:
  • A - 40, B - 45 (B wins, 40% probability)
  • A - 40, B - 55 (B wins, 40% probability)
  • A - 95, B - 45 (A wins, 10% probability)
  • A - 95, B - 55 (A wins, 10% probability)
• Although Group A has a higher average score, 80% of the time a random individual of A will score lower than a random individual of B.