# What Is The Standard Error of Measurement?

The standard error of measurement (SEM) is one of the core concepts in psychometrics. One of the primary assumptions of any assessment is that it is accurately and consistently measuring whatever it is we want to measure. We, therefore, need to demonstrate that it is doing so. There are a number of ways of quantifying this, and one of the most common is the SEM.

The SEM can be used in both the classical test theory (CTT) perspective and item response theory (IRT) perspective, though it is defined quite differently in both.

## The Standard Error of Measurement in Classical Test Theory

In CTT, it is defined as

**SEM = SD*sqrt(1-r),**

where SD is the standard deviation of scores for everyone who took the test, and *r* is the reliability of the test. It is interpreted as the standard deviation of scores that you would find if you had the person take the test over and over, with a fresh mind each time. A confidence interval with this is then interpreted as the band where you would expect the person’s true score on the test to fall.

This has some conceptual disadvantages. For one, it assumes that SEM is the same for all examinees, which is unrealistic. The interpretation focuses only on this single test form rather than the accuracy of measuring someone’s true standing on the trait. Moreover, it does not utilize the examinee’s responses in any way. Lord (1984) suggested a conditional standard error of measurement based on classical test theory, but it focuses on the error of the examinee taking the same test again, rather than the measurement of the true latent value as is done with IRT below.

The classical SEM is reported in *Iteman* for each subscore, the total score, score on scored items only, and score on pretest items.

## The Standard Error of Measurement in Item Response Theory

The weaknesses of the classical SEM are one of the reasons that IRT was developed. IRT conceptualizes the SEM as a continuous function across the range of student ability, which is an inversion of the test information function (TIF). A test form will have more accuracy – less error – in a range of ability where there are more items or items of higher quality. That is, a test with most items of middle difficulty will produce accurate scores in the middle of the range, but not measure students on the top or bottom very well. The example below is a test that has many items below the average examinee score (θ) of 0.0 so that any examinee with a score above 1.0 has a relatively inaccurate score, namely with an SEM greater than 0.25.

This is actually only the *predicted* SEM based on all the items in a test/pool. The observed SEM can differ for each examinee based on the items that they answered, and which ones they answered correctly. If you want to calculate the IRT SEM on a test of yours, you need to download *Xcalibre* and implement a full IRT calibration study.

#### Nathan Thompson, PhD

#### Latest posts by Nathan Thompson, PhD (see all)

- What is Psychometrics? Definition: Improve exams with science. - July 5, 2023
- Certification Management System: Streamline Credential Management - May 19, 2023
- ANSI/ISO 17024 Accreditation - May 16, 2023