Studentization

Under Studentisierung or Studentisieren is understood in mathematical statistics, a transformation of the realizations of a random variable, so that the resulting values ​​have the arithmetic mean zero and the sample variance one. Since the sample standard deviation of the square root of the sample variance, corresponds to this is thus also equal to one.

Studentisieren is necessary for example to different distributed random variables can be compared with each other.

The name comes from the pseudonym " Student" Sealy Gosset of the statistician William ( 1876-1937 ).

Are the realizations of a random variable with arithmetic mean, we obtain the associated studentized values ​​in that subtracting the arithmetic mean and dividing by the sample standard deviation:

Then for these values ​​thus obtained:

  • Arithmetic mean:
  • Sample variance:

In many statistical programs such as SPSS and Statistica the possibility of Studentisierens of the measurement results is already installed. Often in this case the concept of standardizing is incorrectly used in fact a random variable itself - and not their realizations - to zero mean and unit variance is transformed. It is rather the case that is most often spoken by standardizing, although in statistical evaluations actually Studentisieren is meant.

Example

The accompanying table contains 10 realizations of a random variable. In this case, once the original values ​​and the corresponding values ​​are indicated studentized.

For the original values ​​apply:

  • Arithmetic mean:
  • Sample variance:

Consequently, the associated studentized values ​​are calculated as follows:

Actually applies to these values ​​thus obtained:

  • Arithmetic mean:
  • Sample variance:

With the studentized values ​​you can now judge whether an associated original value is conspicuous far away from the mean of all the data very easily. So you can see that the value of number 5 is very low, because the associated studentized value is. This says that the original value of two sample standard deviations is less than the mean.

Swell

  • Bortz, Statistics for Human and Social Sciences, 6th edition, 2005, Springer
  • Falk et al., Foundations of statistical Analyses and applications with SAS, 2002, Birkhäuser
  • Descriptive Statistics
Standard score
752593
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