### Using Normal Curve Equivalents to Determine the Mean

**Introduction**

Normal curve equivalents, commonly referred to as NCEs, are values associated with a test score that relate a student's performance on an exam to the mean in a normal distribution of scores. This value is actually an interpretation of the standard score (or z-score) on a larger scale (1-99). A z-score is simply the number of standard deviations a score is from the mean, and, therefore, represents a performance level associated with other scores in a distribution. Normal curve equivalents are favorable performance measures because the scale they are measured on is one of equal intervals. This attribute of NCEs makes direct averaging and comparing between two NCEs possible. Therefore, two previously unrelated test scores can be directly related, in terms of performance against the mean, on the NCE scale.

In this blog, I will explain how NCEs, along with their corresponding test scores, can be used to calculate the mean of the data from which the NCEs were determined. This calculation is important because it presents us with the ability to easily calculate mean state test scores given two NCE, test score pairs. Using the state average, any student that falls into the distribution from which the mean was calculated could then be associated with their own NCE for that test. Because NCEs can be directly related, performance from year to year (or content area to content area) can be viewed—this opens up an avenue for determining an accurate measure of growth within school districts.

**Determining Z-Scores**

NCEs were developed specifically for educational statistics as a way of making standard scores more convenient. Mathematically altering z-scores to be on a scale ranging from values 1 to 99 results in a performance score that can be quickly and more easily interpreted. With the end goal of calculating the mean of a distribution of student scores, the first logical step is to simplify the NCEs. Standard scores are the building blocks of NCEs and contain within them the standard deviation of the data. Therefore, dealing directly with z-scores will make this process easier. The equation to find a z-score from an NCE is

**Using Z-Scores to Find Standard Deviation**

Z-scores are the number of standard deviations a score is from the mean. As a result of this, two z-score, test score pairs can be used to calculate the standard deviation of the data from which these test scores came by finding what fraction of a z-score a change in test score corresponds to. The equation to do this is

**Finding the Mean**

After finding the standard deviation, only one variable remains unknown—the mean. Using one of the z-scores from the data, as well as the recently determined standard deviation, the mean can be found by using

**Conclusion**

When a minimum of two test score, NCE pairs are known for a set of data, the data's mean and standard deviation are easily calculable. In addition to this, the process is made more simple when the NCEs are first transformed into standard scores. Overall this process makes finding information about test score distributions fast and straightforward.

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