= Rankit =

A '''rankit''' is an order statistic. It reflects the expected value of a given rank in a set of observations.

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== Description ==

Consider a set of 20 observations drawn from a [[Analysis/NormalDistribution|normal distribution]], and then sort them low to high. (For simplicity, assume standard normal.) These observations are now ranked 1 to 20.

For a given ordered position ''i'', there is an expected value ''p'' calculated as:

{{attachment:rankit1.svg}}

This gives the rankit for position ''i''. The rankits for a sample of 20 are:
 1.  -1.96
 2.  -1.44
 3.  -1.15
 4.  -0.93
 5.  -0.76
 6.  -0.60
 7.  -0.45
 8.  -0.32
 9.  -0.19
 10. -0.06
 11. -0.06
 12.  0.19
 13.  0.32
 14.  0.45
 15.  0.60
 16.  0.76
 17.  0.93
 18.  1.15
 19.  1.44
 20.  1.96

More generally, the set of observations does not need to be sorted, only ranked. Furthermore, the formula can be abstracted to alternative constants as ''a'' (thereby requiring adjustments to the denominator as well). For observation ''i'', the rank is ''r,,i,,'' and the rankit is ''p,,i,,'':

{{attachment:rankit2.svg}}

General guidance for ''a'':
 * for ''n ≤ 10'', use 3/8
 * for ''n > 10'', use 0.5
   * Note that ''1 - 2a'' cancels out in this case



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