In mathematical statistics

Mathematical statistics

Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis...

, economics

Economics

Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

, and econophysics

Econophysics

Econophysics is an interdisciplinary research field, applying theories and methods originally developed by physicists in order to solve problems in economics, usually those including uncertainty or stochastic processes and nonlinear dynamics...

, the King effect refers to the phenomenon where the top one or two members of a ranked set show up as outlier

Outlier

In statistics, an outlier is an observation that is numerically distant from the rest of the data. Grubbs defined an outlier as: An outlying observation, or outlier, is one that appears to deviate markedly from other members of the sample in which it occurs....

s. These top one or two members are unexpectedly large because they do not conform to a statistical distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

 or rank-distribution

Rank-size distribution

Rank-size distribution or the rank-size rule describes the remarkable regularity in many phenomena including the distribution of city sizes around the world, sizes of businesses, particle sizes , lengths of rivers, frequencies of word usage, wealth among individuals, etc...

 which the remainder of the set obeys well.

Distributions typically followed include the power-law distribution, that of a stretched exponential, or a parabolic fractal

Parabolic fractal distribution

In probability and statistics, the parabolic fractal distribution is a type of discrete probability distribution in which the logarithm of the frequency or size of entities in a population is a quadratic polynomial of the logarithm of the rank...

.

The King effect has been observed in the distribution of: French

France

The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...

 city sizes, where the point representing Paris

Paris

Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...

 is the "King", failing to conform to the stretched exponential; country populations, where only the points representing China

China

Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...

 and India

India

India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...

 fail to fit a stretched exponential; and the popularity of musicians, where Cliff Richard

Cliff Richard

Sir Cliff Richard, OBE is a British pop singer, musician, performer, actor, and philanthropist who has sold over an estimated 250 million records worldwide....

 and Elvis Presley

Elvis Presley

Elvis Aaron Presley was one of the most popular American singers of the 20th century. A cultural icon, he is widely known by the single name Elvis. He is often referred to as the "King of Rock and Roll" or simply "the King"....

are the outliers not fitting on a stretched exponential. In the latter study, the author notes that the name "King effect" is "strangely appropriate for Elvis' data.