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A commonly used transformation of a standard normal distribution.   
A commonly used transformation of a normal distribution.   
The resulting distribution has a mean of 0 and a standard deviation of 1.  Used extensively in hypothesis testing.


Also known as the Z score.
The resulting standardised normal distribution has a [[mean]] of 0 and a [[standard deviation]] of 1.


So for example if a data point has a Z score (or Z statistic) of -1.64, then it lies 1.64 standard deviations below the mean.
 
It is used extensively in hypothesis testing.
 
 
The Z statistic is also known as the 'Z score'.
 
 
So - for example - if a data point has a Z score (or Z statistic) of -1.64, then it lies 1.64 standard deviations below the mean.


The Z score is calculated as the difference between the data point (X) and the mean E[x], all divided by the standard deviation of the population (SD).
The Z score is calculated as the difference between the data point (X) and the mean E[x], all divided by the standard deviation of the population (SD).
For example if:
For example if:
the mean (E[x]) of a population = 100;
the standard deviation (SD) = 10; and
a given observation (or data point) = 83.6;


then the Z score (Z) is calculated as:
the mean (E[x]) of a population = 100;
 
the standard deviation (SD) = 10; and
 
a given observation (or data point) = 83.6;
 
then the Z score (Z) is calculated as:
Z = (X - E[x])/SD
Z = (X - E[x])/SD
= (83.6 - 100 = -16.4)/10
 
= - 1.64 standard deviations.
= (83.6 - 100 = -16.4)/10
 
= - 1.64 standard deviations.
 


In this case the Z score is negative, indicating that the data point (83.6) lies below the mean (of 100).  
In this case the Z score is negative, indicating that the data point (83.6) lies below the mean (of 100).  


== See also ==
== See also ==
* [[Hypothesis testing]]
* [[Standardised normal distribution]]
* [[Standardised normal distribution]]
* [[Transformation]]
* [[t-statistic]]


[[Category:Risk_frameworks]]

Latest revision as of 09:08, 28 September 2022

A commonly used transformation of a normal distribution.

The resulting standardised normal distribution has a mean of 0 and a standard deviation of 1.


It is used extensively in hypothesis testing.


The Z statistic is also known as the 'Z score'.


So - for example - if a data point has a Z score (or Z statistic) of -1.64, then it lies 1.64 standard deviations below the mean.

The Z score is calculated as the difference between the data point (X) and the mean E[x], all divided by the standard deviation of the population (SD).


For example if:

the mean (E[x]) of a population = 100;

the standard deviation (SD) = 10; and

a given observation (or data point) = 83.6;

then the Z score (Z) is calculated as: Z = (X - E[x])/SD

= (83.6 - 100 = -16.4)/10

= - 1.64 standard deviations.


In this case the Z score is negative, indicating that the data point (83.6) lies below the mean (of 100).


See also