# How to compute the standard error of measurement (SEM) from a reliability estimate?

SPSS returns lower and upper bounds for Reliability. While calculating the Standard Error of Measurement, should we use the Lower and Upper bounds or continue using the Reliability estimate.

I am using the formula :

$$\text{SEM}\% =\left(\text{SD}\times\sqrt{1-R_1} \times 1/\text{mean}\right) × 100$$

where SD is the standard deviation, $R_1$ is the intraclass correlation for a single measure (one-way ICC).

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You seem to be calculating the coefficient of variation of the measurement, not the standard deviation or standard error. –  GaBorgulya Apr 7 '11 at 14:47
@GaBorgulya Usually, SEM is computed in a different way; contrary to SD or SE, it is supposed to account for scores reliability, specific to the measurement instrument. –  chl Apr 8 '11 at 1:10
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## 2 Answers

You should use the point estimate of the reliability, not the lower bound or whatsoever. I guess by lb/up you mean the 95% CI for the ICC (I don't have SPSS, so I cannot check myself)? It's unfortunate that we also talk of Cronbach's alpha as a "lower bound for reliability" since this might have confused you.

It should be noted that this formula is not restricted to the use of an estimate of ICC; in fact, you can plug in any "valid" measure of reliability (most of the times, it is Cronbach's alpha that is being used). Apart from the NCME tutorial that I linked to in my comment, you might be interested in this recent article:

Tighe et al. The standard error of measurement is a more appropriate measure of quality for postgraduate medical assessments than is reliability: an analysis of MRCP(UK) examinations. BMC Medical Education 2010, 10:40

Although it might seem to barely address your question at first sight, it has some additional material showing how to compute SEM (here with Cronbach's $\alpha$, but it is straightforward to adapt it with ICC); and, anyway, it's always interesting to look around to see how people use SEM.

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There are 3 ways to calculate SEM. Also it is important if you want to have SEM agreement or SEM consistency. I will show you the SEM calculaton from reliability.

First you should have ICC (intra-class correlation) and the SD (standard Deviation). Then you calculate SEM as follows:
$$SEM= SD*(1-\sqrt{ICC})$$ It is SEM consistency.

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Welcome to the site, @amin. I took the liberty of editing your post to clean it up slightly & display the formula with $\LaTeX$. Please make sure everything still says what you want. –  gung Feb 17 at 3:39
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