Pareto Principle - Mathematical Notes

Mathematical Notes

The idea has rule of thumb application in many places, but it is commonly misused. For example, it is a misuse to state a solution to a problem "fits the 80–20 rule" just because it fits 80% of the cases; it must be implied that this solution requires only 20% of the resources needed to solve all cases. Additionally, it is a misuse of the 80–20 rule to interpret data with a small number of categories or observations.

This is a special case of the wider phenomenon of Pareto distributions. If the Pareto index α, which is one of the parameters characterizing a Pareto distribution, is chosen as α = log₄5 ≈ 1.16, then one has 80% of effects coming from 20% of causes. It follows that one also has 80% of that top 80% of effects coming from 20% of that top 20% of causes, and so on. Eighty percent of 80% is 64%; 20% of 20% is 4%, so this implies a "64-4" law; and similarly implies a "51.2-0.8" law. Thus, the 80-20 rule would imply that 64% of wealth is held by 4% of the people; however, one cannot reverse this and say that 4% of the wealth is held by the poorer 64% of the people.

The term 80–20 is only a shorthand for the general principle at work. In individual cases, the distribution could just as well be, say, 80–10 or 80–30. There is no need for the two numbers to add up to the number 100, as they are measures of different things, e.g., 'number of customers' vs 'amount spent'). However, each case in which they do not add up to 100%, is equivalent to one in which they do; for example, as noted above, the "64-4 law" (in which the two numbers do not add up to 100%) is equivalent to the "80–20 law" (in which they do add up to 100%). Thus, specifying two percentages independently does not lead to a broader class of distributions than what one gets by specifying the larger one and letting the smaller one be its complement relative to 100%. Thus, there is only one degree of freedom in the choice of that parameter.

Adding up to 100 leads to a nice symmetry. For example, if 80% of effects come from the top 20% of sources, then the remaining 20% of effects come from the lower 80% of sources. This is called the "joint ratio", and can be used to measure the degree of imbalance: a joint ratio of 96:4 is very imbalanced, 80:20 is significantly imbalanced (Gini index: 60%), 70:30 is moderately imbalanced (Gini index: 40%), and 55:45 is just slightly imbalanced.

The Pareto principle is an illustration of a "power law" relationship, which also occurs in phenomena such as brush fires and earthquakes. Because it is self-similar over a wide range of magnitudes, it produces outcomes completely different from Gaussian distribution phenomena. This fact explains the frequent breakdowns of sophisticated financial instruments, which are modeled on the assumption that a Gaussian relationship is appropriate to, for example, stock movement sizes.