Statistical risk is a quantification of a situation's

risk In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environ ...

using

statistical methods Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

. These methods can be used to estimate a

probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...

for the outcome of a specific variable, or at least one or more key

parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

s of that distribution, and from that estimated distribution a risk function can be used to obtain a single non-negative number representing a particular conception of the risk of the situation. Statistical risk is taken account of in a variety of contexts including

finance Finance refers to monetary resources and to the study and Academic discipline, discipline of money, currency, assets and Liability (financial accounting), liabilities. As a subject of study, is a field of Business administration, Business Admin ...

and

economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ...

, and there are many risk functions that can be used depending on the context. One measure of the statistical risk of a

continuous variable In mathematics and statistics, a quantitative variable (mathematics), variable may be continuous or discrete. If it can take on two real number, real values and all the values between them, the variable is continuous in that Interval (mathemati ...

, such as the return on an investment, is simply the estimated

variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...

of the variable, or equivalently the square root of the variance, called the

standard deviation In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...

. Another measure in finance, one which views upside risk as unimportant compared to downside risk, is the downside beta. In the context of a binary variable, a simple statistical measure of risk is simply the

probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...

that a variable will take on the lower of two values. There is a sense in which one risk A can be said to be unambiguously greater than another risk B (that is, greater for any reasonable risk function): namely, if A is a mean-preserving spread of B. This means that the

probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...

of A can be formed, roughly speaking, by "spreading out" that of B. However, this is only a partial ordering: most pairs of risks cannot be unambiguously ranked in this way, and different risk functions applied to the estimated distributions of two such unordered risky variables will give different answers as to which is riskier. In the context of

statistical estimation Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value ...

itself, the risk involved in estimating a particular parameter is a measure of the degree to which the estimate is likely to be inaccurate.

See also