 Normal curve equivalent

In educational statistics, a normal curve equivalent (NCE), developed for the United States Department of Education by the RMC Research Corporation,^{[1]}^{[2]} is a way of standardizing scores received on a test. It is defined as (approximately)
 50 + 21.06z,
where z is the standard score or "zscore", 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 "bellshaped curve") then
 the normal equivalent score is 99 if the percentile rank of the raw score is 99;
 the normal equivalent score is 50 if the percentile rank of the raw score is 50;
 the normal equivalent score is 1 if the percentile rank of the raw score is 1.
However, this relationship between normal equivalent scores does not hold for percentile ranks other than 1, 50, and 99. It also fails to hold in general if the scores are not normally distributed.
The number 21.06 was chosen because
 99 is 49 more than 50—thus 49 points above the mean;
 it is desired that a score of 99 correspond to the 99th percentile;
 the 99th percentile or the standard normal distribution is 2.3263;
 49/2.3263 = 21.06.
The percentile rank scale is not an equalinterval scale; that is, the difference between any two scores is not the same between any other two scores (see below or percentile rank for more information). Normal curve equivalents do not suffer from this problem, since they are on an equalinterval scale (see [1] and [2] for examples).
The major advantage of NCEs over percentile ranks is that NCEs can be averaged.^{[3]} The Rochester School Department webpage describes how NCE scores change:
In a normally distributed population, if all students were to make exactly one year of progress after one year of instruction, then their NCE scores would remain exactly the same and their NCE gain would be zero, even though their raw scores (i.e. the number of questions they answered correctly) increased. Some students will make more than a year's progress in that time and will have a net gain in the NCE score, which means that those students have learned more, or at least have made more progress in the areas tested, than the general population. Other students, while making progress in their skills, may progress more slowly than the general population and will show a net loss in their NCE ranks.
Caution
Careful consideration is required when computing effect sizes using NCEs. NCEs differ from other scores, such as raw and scaled scores, in the magnitude of the effect sizes. Comparison of NCEs typically results in smaller effect sizes, and using the typical ranges for other effect sizes may result in interpretation errors.^{[4]}
Excel formula for conversion from Percentile to NCE: =21.06*NORMSINV(PR/100)+50, where PR is the percentile value.
Excel formula for conversion from NCE to Percentile: =100*NORMSDIST((NCE50)/21.06), where NCE is the Normal Curve Equivalent (NCE) value
References
 ^ AllPsych Online statistics primer.
NCE stands for Normal Curve Equivalent and was developed [for] the [US] Department of Education.
 ^ Mertler, C. A. (2002). Using standardized test data to guide instruction and intervention. College Park, MD: ERIC Clearinghouse on Assessment and Evaluation. (ERIC Document Reproduction Service No. ED470589)
Normal curve equivalent (NCE): A normalized standardized score with a mean of 50 and a standard deviation of 21.06 resulting in a near equal interval scale from 0 to 99. The NCE was developed by RMC Research Corporation in 1976 to measure the effectiveness of the Title I Program across the United States and is often used to measure gains over time. (p. 3)
 ^ Rochester School Department webpage
 ^ McLean, J. E., O'Neal, M. R., & Barnette, J. J. (2000, November). Are all effect sizes created equal? Paper presented at the Annual Meeting of the MidSouth Educational Research Association, Bowling Green, KY. (ERIC Document Reproduction Service No. ED448188)
External links
 Norm Scale Calculator (Utility for the Transformation and Visualization of Norm Scores)
 CTB/McGrawHill FAQ about NCEs.
 Advanced Math, Subject: Normal Curve Equivalents (educational orientation).
 Scholastic Testing Service, a glossary of terms related to the bell or normal curve.
 Another page comparing different types of scores.
 A webpage including calculations for various scores.
 UCLA webpage describing how to convert percentile ranks to NCEs with Stata.
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