Studentization

Under studentization or studentization (after the pseudonym “student” of statistician William Sealy Gosset ) one understands in mathematical statistics a transformation of the realizations of a random variable , so that the resulting values ​​have the arithmetic mean zero and the empirical variance one. Since the empirical standard deviation is the square root of the sample variance, it is also equal to one.

Studentizing is e.g. B. necessary to be able to compare differently distributed random variables with one another.

If the realizations of a random variable have an arithmetic mean , the corresponding Studentized values ​​are obtained by subtracting the arithmetic mean and dividing it by the sample standard deviation: ${\displaystyle x_{i}}$ ${\displaystyle n}$ ${\displaystyle {\overline {x}}}$ ${\displaystyle z_{i}}$

${\displaystyle z_{i}={\frac {x_{i}-{\overline {x}}}{\sqrt {{\frac {1}{n}}\sum \limits _{k}\left(x_{k}-{\overline {x}}\right)^{2}}}}}$

The following applies to the values ​​obtained in this way:

• arithmetic mean: ${\displaystyle {\overline {z}}={\frac {1}{n}}\sum _{i}{z_{i}}=0}$
• Sample variance: . ${\displaystyle {\frac {1}{n}}\sum \limits _{i}\left(z_{i}-{\overline {z}}\right)^{2}=1}$

In many statistical programs such as  SPSS and Statistica , the possibility of studying the measurement results is already built in. The term “ standardization ” is often used incorrectly , in which actually a random variable itself - and not its realizations - is transformed to expected value zero and variance one. Most of the time, standardization is used, even when statistical evaluations actually mean studentization.

example

Number (i)	Original value ( ) ${\displaystyle x_{i}}$	Studentized value ( ) ${\displaystyle z_{i}}$
1	3	0.5
2	−1	−0.5
3	2	0.25
4th	4th	0.75
5	−7	−2
6th	7th	1.5
7th	2	0.25
8th	5	1
9	−2	−0.75
10	−3	−1

The adjacent table contains 10 realizations of a random variable. The original values ​​and the associated studentized values are given once . ${\displaystyle x_{i}}$ ${\displaystyle z_{i}}$

The following applies to the original values:

• Arithmetic mean: ${\displaystyle {\overline {x}}:={\frac {1}{n}}\sum _{i}{x_{i}}=1}$
• Sample variance: ${\displaystyle {\frac {1}{n}}\sum \limits _{i}\left(x_{i}-{\overline {x}}\right)^{2}=16}$

Consequently, the corresponding studentized values ​​are calculated as follows: ${\displaystyle z_{i}={\frac {x_{i}-1}{\sqrt {16}}}={\frac {x_{i}-1}{4}}}$

For these values ​​obtained in this way, the following actually applies: ${\displaystyle z_{i}}$

• Arithmetic mean: ${\displaystyle {\overline {z}}:={\frac {1}{n}}\sum _{i}{z_{i}}=0}$
• Sample variance: ${\displaystyle {\frac {1}{n}}\sum \limits _{i}\left(z_{i}-{\overline {z}}\right)^{2}=1}$

With the studentized values ​​it is now very easy to assess whether an associated original value is noticeably far away from the mean value of all data. So you can see that the value number 5 is very low, since the associated studentized value is. This means that the original value of two sample standard deviations is smaller than the mean. ${\displaystyle -2}$ ${\displaystyle -7}$

swell

• Bortz, Statistics for Human and Social Scientists , 6th edition, 2005, Springer
• Falk et al., Foundations of statistical analyzes and applications with SAS , 2002, Birkhäuser