An empirical statistical law or (in popular terminology) a law of statistics represents a type of behaviour that has been found across a number of datasets and, indeed, across a range of types of data sets. Many of these observances have been formulated and proved as statistical or probabilistic theorems and the term "law" has been carried over to these theorems. There are other statistical and probabilistic theorems that also have "law" as a part of their names that have not obviously derived from empirical observations. However, both types of "law" may be considrered instances of a scientific law

Scientific law

A scientific law is a statement that explains what something does in science just like Newton's law of universal gravitation. A scientific law must always apply under the same conditions, and implies a causal relationship between its elements. The law must be confirmed and broadly agreed upon...

 in the field of statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

.

For example, both Zipf's law  and Heaps' law

Heaps' law

In linguistics, Heaps' law is an empirical law which describes the portion of a vocabulary which is represented by an instance document consisting of words chosen from the vocabulary. This can be formulated as V_R = Kn^\beta...

 have been described as "empirical statistical laws" in the field of linguistics.

Examples of empirically inspired statistical laws that have a firm theoretical basis include:

• Statistical regularity
  Statistical regularity
  Statistical regularity is a notion in statistics and probability theory that random events exhibit regularity when repeated enough times or that enough sufficiently similar random events exhibit regularity...
  
  
• Law of large numbers
  Law of large numbers
  In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times...
  
  
• Law of truly large numbers
  Law of Truly Large Numbers
  The law of truly large numbers, attributed to Persi Diaconis and Frederick Mosteller, states that with a sample size large enough, any outrageous thing is likely to happen. Because we never find it notable when likely events occur, we highlight unlikely events and notice them more...
  
  
• Central limit theorem
  Central limit theorem
  In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...
  
  
• Regression towards the mean

Examples of "laws" with a weaker foundation include:

• Safety in numbers
  Safety in numbers
  Safety in numbers is the hypothesis that, by being part of a large physical group or mass, an individual is proportionally less likely to be the victim of a mishap, accident, attack, or other bad event...
  
  
• Benford's law
  Benford's law
  Benford's law, also called the first-digit law, states that in lists of numbers from many real-life sources of data, the leading digit is distributed in a specific, non-uniform way...
  
  

Examples of "laws" which are more general observations than having a theoretical background:

• Rank-size 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...
  
  

Examples of supposed "laws" which are incorrect include:

• Law of averages
  Law of averages
  The law of averages is a lay term used to express a belief that outcomes of a random event will "even out" within a small sample.As invoked in everyday life, the "law" usually reflects bad statistics or wishful thinking rather than any mathematical principle...