Pareto Distribution

What is Pareto Distribution?

The Pareto Distribution was named after Italian economist and sociologist Vilfredo Pareto. It is sometimes referred to as the Pareto Principle or the 80-20 Rule. The Pareto Distribution is used in describing social, scientific, and geophysical phenomena in society. Pareto created a mathematical formula in the early 20^th century that described the inequalities in wealth distribution that existed in his native country of Italy.

Pareto observed that 80% of the country’s wealth was concentrated in the hands of only 20% of the population. The theory is now applied in many disciplines such as incomes, productivity, populations, and other variables. The Pareto distribution serves to show that the level of inputs and outputs is not always equal.

History of Pareto Distribution

The Pareto Distribution principle was first employed in Italy in the early 20^th century to describe the distribution of wealth among the population. In 1906, Vilfredo Pareto introduced the concept of the Pareto Distribution when he observed that 20% of the pea pods were responsible for 80% of the peas planted in his garden.

He related this phenomenon to the nature of wealth distribution in Italy, and he found that 80% of the country’s wealth was owned by about 20% of its population. In terms of land ownership, the Italian observed that 80% of the land was owned by a handful of wealthy citizens, who comprised about 20% of the population.

The definition of the Pareto Distribution was later expanded in the 1940s by Dr. Joseph M. Juran, a prominent product quality guru. Juran applied the Pareto principle to quality control for business production to show that 20% of the production process defects are responsible for 80% of the problems in most products.

According to Juran, focusing on the 20% causes of defects allowed organizations to implement more effective quality control measures and make better use of their resources. Juran’s additions to the Pareto distribution concept were contained in his 1951 book titled “Quality Control Handbook.”

Pareto Distribution Formula

The formula for calculating the Pareto Distribution is as follows:

F(x) = 1 – (k/x)^α

Where:

• x – Random variable
• k – Lower bound on data
• α – Shape parameter

On a chart, the Pareto distribution is represented by a slowly declining tail, as shown below:

The chart is defined by the variables α and x. It provides two main applications. One of the applications is to model the distribution of wealth among individuals in a country. The chart shows the extent to which a large portion of wealth in any country is owned by a small percentage of the people living in that country.

The second application is to model the distribution of city populations, where a large percentage of the population is concentrated in the urban centers and a lower amount in the rural areas. The population in urban centers continues to increase while the rural population continues to decline as younger members of the population migrate to urban centers.

Practical Applications of the Pareto Distribution

1. Business Management

One of the applications of the Pareto concept is in business management. A business may observe that 20% of the effort dedicated to a specific business activity generates 80% of the business results. A business can use this ratio to identify the most important segments that it can focus on and thereby increase its efficiency.

For example, if marketing contributed to increased business results, the business can allocate more time and resources to marketing activities to increase the company’s revenues and profits.

2. Company Revenues

The 80-20 Pareto rule may also apply in evaluating the source of the company revenues. For example, when the company observes that 80% of reported annual revenues come from 20% of its current customers, it can focus its attention on increasing the customer satisfaction of influential customers.

From this observation, the company can also deduce that 80% of customer complaints come from 20% of customers who form the bulk of its transactions. Also, focusing on solving the complaints of 20% of its customers can increase the overall customer satisfaction of the company. The company should focus on retaining 20% of its influential customers and on acquiring new customers.

3. Employee Evaluation

A company can also use the 80-20 rule to evaluate the performance of its employees. The company may observe that 80% of its overall output is the direct result of about 20% of its employees. Using the ratio, the company can focus on rewarding the 20% most productive employees as a way of motivating them and encouraging the lower cluster of employees to work harder. The productivity ratio could also show the company that 80% of human resource problems are caused by 20% of the company’s employees.

Limitations of the Pareto Distribution

While the 80-20 Pareto distribution rule applies to many disciplines, it does not necessarily mean that the input and output must be equal to 100%. For example, 20% of the company’s customers could contribute 70% of the company’s revenues. The ratio brings a total of 90%. It shows that the Pareto concept is merely an observation that suggests that the company should focus on certain inputs more than others.

Additional Resources

CFI is the official provider of the Financial Modeling and Valuation Analyst (FMVA)® certification program, designed to transform anyone into a world-class financial analyst.

To keep learning and developing your knowledge of financial analysis, we highly recommend the additional CFI resources below:

• Externality
• Key Performance Indicators (KPI)
• Normative Economics
• Sustainability
• See all economics resources