Standard score

Under standardization or z - transformation is understood in mathematical statistics, a transformation of random variables, so that the resulting random variable has a zero mean and variance one. The standard deviation is the square root of the variance and is therefore also equal to one.

Standardization is necessary, for example to different distributed random variables can be compared with each other. In addition, standardized random variables for some statistical methods such as factor analysis, is necessary.

Is a random variable with mean and variance ( and therefore standard deviation) we obtain the corresponding standardized random variable by:

Shall apply to those obtained in random variable:

In many statistical programs such as SPSS and Statistica the possibility of standardization of the measurement results is already installed. Strictly speaking, we should speak of a Studentisierung here but since the exact distribution of the underlying random variables is unknown and has to be used instead of the expected value of the arithmetic mean and variance instead of the sample variance. Often, however, the terms of Studentisierens and standardizing be used interchangeably.

The concept of z- transforming, or the use of letters is common in so far as, in particular in the central limit theorem ( that starts with a Z) standardized random variables are used.

Swell

  • Bortz, Schuster statistics for human and social sciences, 7th edition, 2010, Springer
  • Falk et al., Foundations of statistical Analyses and applications with SAS, 2002, Birkhäuser
  • Random variable
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