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 here and reproduced in the table below.

• U(i,n)	i
  $$ 1- 0.5^{1/n}	i=1
  $$\frac{i-0.3175}{n+0.365}$$	$$i=2, \ldots, n-1$$
  $$0.5^{1/n}$$	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 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.