Following Mishra and Desai and McKee et al., the current study employed the approximate transformation provided by Abramowitz and Stegun  to transform the cumulative probability distribution into a standardized normal distribution
, which are given as follows:
where X is the sample mean, [Z.sub.[alpha]/2] is the value corresponding to an upper-tail probability of [alpha]/2 from the standardized normal distribution
, a is the population standard deviation, and n is the sample size.
where [u.sub.[alpha]]--the value of the random variable U with a standardized normal distribution
, as designated for a given confidence coefficient 1-[alpha] from an array of a normal distribution, s--sample standard deviation as calculated according to the relation [6-8]
This statistic follows a standardized normal distribution
when calculated with true item and subject parameters (Reise, 1990).
The distribution of person-fit statistics on each test was very close to the standardized normal distribution
, and both tests accurately assessed the intended student's ability.
The variable Z as expressed in equation 6.5 has a standardized normal distribution
As with the quality adjustment, cost values are subtracted from 1.5 (the mean of a standardized normal distribution
), permitting up to a 50 percent shift in direction.