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In educational statistics, a normal curve equivalent (NCE), developed for the United States Department of Education by the RMC Research Corporation, is a way of normalizing scores received on a test into a 0-100 scale similar to a percentile rank, but preserving the valuable equal-interval properties of a z-score.
It is defined as:
or, approximately
where z is the standard score or "z-score", i.e. z is how many standard deviations above the mean the raw score is (z is negative if the raw score is below the mean). The reason for the choice of the number 21.06 is to bring about the following result: If the scores are normally distributed (i.e. they follow the "bell-shaped curve") then
This relationship between normal equivalent scores and percentile ranks does not hold at values other than 1, 50, and 99. It also fails to hold in general if scores are not normally distributed.
The number 21.06 was chosen because
Normal curve equivalents are on an equal-interval scale. This is advantageous compared to percentile rank scales, which suffer from the problem that the difference between any two scores is not the same as that between any other two scores (see below or percentile rank for more information).
The major advantage of NCEs over percentile ranks is that NCEs can be legitimately averaged.