Statistical regularity
Encyclopedia
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. It is an umbrella term
that covers the law of large numbers
, all central limit theorem
s and ergodic theorems.
If one throws a die once, it is difficult to predict the outcome, but if we repeat this experiment many times, we will see that the number of times each result occurs divided by the number of throws will eventually stabilize towards a specific value.
Repeating a series of trials will produce similar, but not identical, results for each series: the average, the standard deviation and other distributional characteristics will be around the same for each series of trials.
The notion is used in games of chance, demographic statistics
, quality control
of a manufacturing process, and in many other parts of our lives.
Observations of this phenomenon provided the initial motivation for the concept of what is now known as frequency probability
.
This phenomenon should not be confused with the Gambler's fallacy
, it only concerns regularity in the (possibly very) long run. Gambler's fallacy does not apply to statistical regularity because the latter considers the whole rather than individual cases.
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....
and probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
that random events exhibit regularity when repeated enough times or that enough sufficiently similar random events exhibit regularity. It is an umbrella term
Umbrella term
An umbrella term is a word that provides a superset or grouping of concepts that all fall under a single common category. Umbrella term is also called a hypernym. For example, cryptology is an umbrella term that encompasses cryptography and cryptanalysis, among other fields...
that covers the 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...
, all 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...
s and ergodic theorems.
If one throws a die once, it is difficult to predict the outcome, but if we repeat this experiment many times, we will see that the number of times each result occurs divided by the number of throws will eventually stabilize towards a specific value.
Repeating a series of trials will produce similar, but not identical, results for each series: the average, the standard deviation and other distributional characteristics will be around the same for each series of trials.
The notion is used in games of chance, demographic statistics
Demographic statistics
Among the kinds of data that national leaders need are the demographic statistics of their population. Records of births, deaths, marriages, immigration and emigration and a regular census of population provide information that is key to making sound decisions about national policy.A useful summary...
, quality control
Quality control
Quality control, or QC for short, is a process by which entities review the quality of all factors involved in production. This approach places an emphasis on three aspects:...
of a manufacturing process, and in many other parts of our lives.
Observations of this phenomenon provided the initial motivation for the concept of what is now known as frequency probability
Frequency probability
Frequency probability is the interpretation of probability that defines an event's probability as the limit of its relative frequency in a large number of trials. The development of the frequentist account was motivated by the problems and paradoxes of the previously dominant viewpoint, the...
.
This phenomenon should not be confused with the Gambler's fallacy
Gambler's fallacy
The Gambler's fallacy, also known as the Monte Carlo fallacy , and also referred to as the fallacy of the maturity of chances, is the belief that if deviations from expected behaviour are observed in repeated independent trials of some random process, future deviations in the opposite direction are...
, it only concerns regularity in the (possibly very) long run. Gambler's fallacy does not apply to statistical regularity because the latter considers the whole rather than individual cases.