# What is the conditional standard error of measurement (CSEM)?

The conditional standard error of measurement (CSEM) is a concept from psychometrics which seeks to characterize error in the process of measuring examinees on a test or assessment.

## What is measurement error?

We can all agree that assessments are not perfect, from a 4th grade math quiz to a Psych 101 exam at university to a driver’s license test.  Suppose you got 80% on an exam today.  If we wiped your brain clean and you took the exam tomorrow, what score would you get?  Probably a little higher or lower.  Psychometricians consider you to have a true score which is what would happen if the test was perfect, you had no interruptions or distractions, and everything else fell into place.  But in reality, you of course do not get that score each time.  So psychometricians try to estimate the error in your score, and use this in various ways to improve the assessment and how scores are used.

## The Original Approach: Classical Test Theory

Classical test theory (CTT) is a psychometric paradigm that is extremely useful in many situations, but generally oversimplifies things.  Its approach to measurement error certainly qualifies.  CTT assumes that measurement error – called the standard error of measurement – is the same for every examinee.  It is calculated as SEM=SD*sqrt(1-r) where SD is the standard deviation of raw scores on the exam, and r is the reliability of the exam.

## Why conditional standard error of measurement?

Early researchers realized that this assumption is unreasonable.  Suppose that a test has a lot of easy questions.  It will therefore measure low-ability examinees quite well.  Imagine that it is a Math placement exam for university, and has a lot of Geometry and Algebra questions at a high school level.  It will measure students well who are at that level, but do a very poor job of measuring good students.

There was an initial suggestion of calculating a conditional standard error of measurement in classical test theory, but at that time, a new paradigm called item response theory was being developed.  It considered error to be a function of ability, not a single number.  In the previous example, the standard error for low students should be much less than the standard error for high students.

An example of this is shown below.  On the right is the conditional standard error of measurement function, and on the left is its inverse, the test information function.  Clearly, this test has a lot of items around -1.0 on the theta spectrum, which is around the 15th percentile.  Students above 1.0 (85th percentile) are not being measured well.

## How is CSEM used?

A useful way to think about conditional standard error of measurement is with confidence intervals.  Suppose your score on a test is 0.5 with item response theory.  If the CSEM is 0.25 (see above) then we can get a 95% confidence interval by taking plus or minus 2 standard errors.  This means that we are 95% certain that your true score lies between 0.0 and 1.0.  For a theta of 2.5 with an CSEM of 0.5, that band is then 1.5 to 2.5 – which might seem wide, but remember that is like 94th percentile to 99th percentile.

You will sometimes see scores reported in this manner.  I once saw a report on an IQ test that did not give a single score, but instead said “we can expect that 9 times out of 10 that you would score between X and Y.”

There are various ways to use the CSEM and related functions in the design of tests, including the assembly of parallel linear forms and the development of computerized adaptive tests. To learn more about this, I recommend you delve into a book on IRT, such as Embretson and Riese (2000).  That’s more than I can cover here.

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#### Nathan Thompson, PhD

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.