# Pareto distribution

by Admin Updated: August 3, 2018 |

## The 80:20 rule is a special case of a Pareto distribution which is a type of power law

The 80:20 rule - the idea that 80% of something is attributable to only 20% of something else - is generally accepted as a fundamental truth, especially in business. While it’s a useful principle, it may not always be exactly true so it’s worth knowing a little more about it.

Vilfredo Pareto (1848-1923) was an Italian scientist, economist and mathematician who helped add more mathematical formality to the field of economics and contributed, among other things, mathematical analyses of the unequal distribution of resources. The Pareto principle, a.k.a. the *80:20 rule* was named after him.

### Power law

A power law states that some quantity is proportional to the power of some other quantity. A simple example is the area of a square being proportional to the square of the length of one of its sides:

Area = x^{2}

where x is the length of a side.

### Pareto Type I distribution

There are several types of distribution attributed to Pareto. Type I is the simplest and takes the form of the generalized power law:

y = x^{α}

with the following constraints:

- The result (y) must be a probability, i.e. a real number between 0 and 1 (zero and 100%).
- The base (x) must be a ratio, i.e. greater than or equal to 1 (1:1, 2:1, 3.7:1 etc).

which means the power (α) must be a real number less than zero.

### Why this is useful

Equations allow you to make predictions.

Pareto studied inequalities in land ownership in Italy and compared the number of land owners with the amount of land they owned (a measure of wealth distribution).

Necessarily, 100% of people owned 100% of the land (x=1:1) and the percentage decreased thereafter as the imbalance increased: 45% of people owned at least twice as much as everyone else (x=2:1), 28% of people owned at least 3× (x=3:1) etc.

The data formed a curve:

He noticed two things:

- 20% of the population owned at least 4× more than everyone else, equal to 80% of the total, i.e. 4:1 is 4/(4+1) = 4/5 = 0.8.
- This looked like a power function and, if it was, he should be able to back-calculate a value of α such that 4
^{α}= 0.2.

For y = x^{α}, α = Log_{x}(y) = Log_{n}(y) ÷ Log_{n}(x) where n can be any base because the equation is a ratio.

The 80:20 rule applies in the special case when α ≈ -1.161.

and when this is true it is possible to predict, for example, that approximately 7% of the population are at more than 9× (90%).

### Strategies for business

In the distribution equation, the exponent (α) is not always -1.161 but it is often close. For this reason the “80:20 rule” has been adopted as a practical reminder of the unequal distribution of resources.

It is safe to say, for example, that a minority of customers will generate the majority of income but it is not safe to say 20% generate 80% without carrying out an original analysis.

If you find that your data is explained by α ≈ -0.732, you’re looking at a 90:10 rule and 80% is probably attributable to about 36% of the sample.

Nevertheless, the 80:20 rule is used to indicate that it may prove wise to invest in a more profitable subset of the customer base.

### Caveats

- Pareto distributions are recursive. This means that if an 80:20 rule applies to the whole data, it also applies to a subset of it. In other words, if you get rid of your least profitable 80% of customers, you can expect to find that an 80:20 rule applies to the ones that are left.
- Statistical models can be misleading. Sometimes, the least profitable customers perform a catalytic role, e.g. they may not buy much themselves directly but they are your biggest fans and bring in tons of new business.
- Sometimes a customer who is seemingly less important in terms of profitability can be devastatingly outspoken if treated unfairly.

### More info

*Statistics How To* has a good qualitative review with some interesting examples at pareto-principle-the-8020-rule.

More details (with equations and graphs) are in a PDF document from subversion.american.edu (PDF, 177kB).

*Applied Probability and Statistics* (updated 2017) has an in-depth review at the-pareto-distribution.