Normal curve equivalent
Encyclopedia
In educational statistics, a normal curve equivalent (NCE), developed for the United States Department of Education
by the RMC Research Corporation,
where z is the standard score
or "z-score", i.e. z is how many standard deviation
s 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
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
The percentile rank scale is not an equal-interval 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 equal-interval scale (see http://www.rochesterschools.com/Webmaster/StaffHelp/rdgstudy/ncurve2.gif and http://www.citrus.kcusd.com/gif/bellcurve.gif for examples).
The major advantage of NCEs over percentile ranks is that NCEs can be averaged. The Rochester School Department webpage describes how NCE scores change:
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:
United States Department of Education
The United States Department of Education, also referred to as ED or the ED for Education Department, is a Cabinet-level department of the United States government...
by the RMC Research Corporation,
NCE stands for Normal Curve Equivalent and was developed [for] the [US] Department of Education.is a way of standardizing scores received on a test. It is defined as (approximately)
- 50 + 21.06z,
where z is the standard score
Standard score
In statistics, a standard score indicates how many standard deviations an observation or datum is above or below the mean. It is a dimensionless quantity derived by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation...
or "z-score", i.e. z is how many standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
s 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
- the normal equivalent score is 99 if the percentile rankPercentile rankThe percentile rank of a score is the percentage of scores in its frequency distribution that are the same or lower than it. For example, a test score that is greater than 75% of the scores of people taking the test is said to be at the 75th percentile....
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 equal-interval scale; that is, the difference between any two scores is not the same between any other two scores (see below or percentile rank
Percentile rank
The percentile rank of a score is the percentage of scores in its frequency distribution that are the same or lower than it. For example, a test score that is greater than 75% of the scores of people taking the test is said to be at the 75th percentile....
for more information). Normal curve equivalents do not suffer from this problem, since they are on an equal-interval scale (see http://www.rochesterschools.com/Webmaster/StaffHelp/rdgstudy/ncurve2.gif and http://www.citrus.kcusd.com/gif/bellcurve.gif for examples).
The major advantage of NCEs over percentile ranks is that NCEs can be averaged. 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.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:
External links
- Norm Scale Calculator (Utility for the Transformation and Visualization of Norm Scores)
- CTB/McGraw-Hill 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 StataStataStata is a general-purpose statistical software package created in 1985 by StataCorp. It is used by many businesses and academic institutions around the world...
.