# What Is The Standard Error of Measurement?

The standard error of measurement 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 perspective and item response theory perspective, though it is defined quite differently in both.

## The Standard Error of Measurement in Classical Test Theory

In classical test theory, 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.

The classical SEM – and an attempt to conditionalize it – are reported in Iteman.

## 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.  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 mostly 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 above the average examinee score (θ) of 0.0, so that any examinee with a score of less than 0.0 has a relatively inaccurate score, namely with an SEM greater than 0.50.

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.  To see this concept in live action, download my IRT Scoring Tool, which implements IRT scoring and lets you watch a graph change in real time as you change the items (or their IRT parameters) or the responses.  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.

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#### nthompson

Nathan Thompson earned his PhD in Psychometrics from the University of Minnesota, with a focus on computerized adaptive testing. His undergraduate degree was from Luther College with a triple major of Mathematics, Psychology, and Latin. He is primarily interested in the use of AI and software automation to augment and replace the work done by psychometricians, which has provided extensive experience in software design and programming. Dr. Thompson has published over 100 journal articles and conference presentations, but his favorite remains https://pareonline.net/getvn.asp?v=16&n=1.