= Quick formula for the expected value of a Normal order statistic =

The expected value of  the i-th Normal Order Statistic from a sample of size, n, may be computed as the absolute value of

$$\Phi^{-1}(U(i,n))$$

where $$\Phi^{-1}$$ is the inverse Normal cumulative density function which converts a probability, such as U(i,n), to a z-score.

where the U(i,n) are the U(i) taken from [[http://en.wikipedia.org/wiki/Normal_probability_plot|here]]
and reproduced in the table below.

 ||<50%> '''U(i,n)''' ||<50%> '''i''' ||
 ||<50%> $$ 1- 0.5^{1/n} ||<50%>i=1||
 ||<50%> $$\frac{i-0.3175}{n+0.365}$$ ||<50%> $$i=2, \ldots, n-1$$ ||
 ||<50%> $$0.5^{1/n}$$ ||<50%> i=n||

For example U(1,3)=0.82 (using above approximation) compared to an exact value of 0.85.

An exact version written in the C language programming language (which may be run on Unix) is obtained using a version of Algorithm AS 177
of the Royal Statistical Applied Statistics algorithm page. The C program code is listed [[FAQ/Corder| here.]]

The algorithm is labelled Expected Normal Order Statistics (Exact and Approximate) and is based on the original FORTRAN code by Royston, 1982.

A table of expected values of normal order statistics (2.4(a)) are in Neave HR (1978).

__References__

Neave HR (1978) Statistics tables for mathematicians, engineers, economists and the behavioural and management sciences. Unwin Hyman:London.

Royston, JP (1982), Algorithm AS 177, ''Applied Statistics'', '''31(2)''':161-165.