# Statistical Variable

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An observable characteristic of a statistical element is called variable. The actual values assumed by statistical variables are called observations, measurements or data. The set of possible values a variable can take is called sample space. Variables are denoted by script capitals $X,Y,\ldots$ , whereas corresponding realizations are written in lower case: $x_{1},x_{2},\ldots$ , the indices reflect the statistical elements sampled.

Variable Observations
$X$ $x_{1},x_{2},x_{3},\ldots$ $Y$ $y_{1},y_{2},y_{3},\ldots$ It is useful to differentiate between variables used for identification and target variables. Identification variables In assigning a set of fixed values the elements of the population are specified. For example, restricting a statistical investigation to female persons involves setting the identification variable “sex” to “female”. Target variables These are the characteristics of interest, the phenomena that are being explored by means of statistical techniques. E.g., the age of persons belonging to a particular population. Example: Objective of the statistical investigation is to explore Berlin’s socioeconomic structure as of December 21, 1995. The identification variables are chosen to be:

• legal: citizen
• spatial: permanent address in Berlin
• temporal: 31 December 1995

Statistical element: a registered citizen of Berlin on 31 December 1995 Population: all citizens of Berlin on 31 December 1995 Possible target variables:

Symbol Variable Sample space
$X$ Age (rounded to years) $\left\{0,1,2,\ldots \right\}$ $S$ Sex {female, male}
$T$ Marital status {single, married, divorced}
$Y$ Monthly income $\left[0,\infty \right)$ 