Notation in probability and statistics

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Probability theory and statistics have some commonly used conventions, in addition to standard mathematical notation and mathematical symbols.

Probability theory[edit]


  • Greek letters (e.g. θ, β) are commonly used to denote unknown parameters (population parameters).
  • A tilde (~) denotes "has the probability distribution of".
  • The hat symbol ^ denotes an estimator of the true parameter.
  • An estimate of a parameter is often denoted by placing a caret or "hat" over the corresponding symbol, e.g. \hat{\theta}, pronounced "theta hat".
  • The arithmetic mean of a series of values x1, x2, ..., xn is often denoted by placing an "overbar" over the symbol, e.g. \bar{x}, pronounced "x bar".
  • Some commonly used symbols for sample statistics are given below:
  • Some commonly used symbols for population parameters are given below:
    • the population mean μ,
    • the population variance σ2,
    • the population standard deviation σ,
    • the population correlation ρ,
    • the population cumulants κr.

Critical values[edit]

The α-level upper critical value of a probability distribution is the value exceeded with probability α, that is, the value xα such that F(xα) = 1 − α where F is the cumulative distribution function. There are standard notations for the upper critical values of some commonly used distributions in statistics:

Linear algebra[edit]

  • Matrices are usually denoted by boldface capital letters, e.g. A.
  • Column vectors are usually denoted by boldface lower case letters, e.g. x.
  • The transpose operator is denoted by either a superscript T (e.g. AT) or a prime symbol (e.g. A′).
  • A row vector is written as the transpose of a column vector, e.g. xT or x′.


Common abbreviations include:

See also[edit]


  • Halperin, Max; Hartley, H. O.; Hoel, P. G. (1965), "Recommended Standards for Statistical Symbols and Notation. COPSS Committee on Symbols and Notation", The American Statistician 19 (3): 12–14, doi:10.2307/2681417, JSTOR 2681417 

External links[edit]