## Tuesday, February 03, 2009

### Standard Error of Mean vs. Standard Error of Measurement

Everybody with basic statistical knowledge should understand the differences between the standard deviation (SD) and the standard error of mean (SE or SEM). However, people may be confused with the terms of Standard Error of Mean (SEM) vs. Standard Error of Measurement (SEM). While both shares the same acronym, the meaning and the calculation are quite different. At least, this is the situation when I saw the term 'standard error of measurement'.

I first saw this term in a literature discussing various approaches to identify the minimal clinically important difference (MCID). In an article by Copay et al, SEM (standard error of measurement) was quoted as one of the many approaches in evaluating the MCID. This method was also discussed in a paper by Wyrwich et al. Initially, I mistakenly thought that SEM was for standard error of mean. After further exploration, I realized that this SEM is quite different from that SEM.

The standard error of the mean (SEM) is the standard deviation of the sample mean estimate of a population mean. (It can also be viewed as the standard deviation of the error in the sample mean relative to the true mean, since the sample mean is an unbiased estimator.) SEM is usually estimated by the sample estimate of the population standard deviation (sample standard deviation) divided by the square root of the sample size (assuming statistical independence of the values in the sample).

The standard error of measurement (SEM) estimates how repeated measures of a person on the same instrument tend to be distributed around his or her "true" score. The true score is always an unknown because no measure can be constructed that provides a perfect reflection of the true score. SEM is directly related to the reliability of a test; that is, the larger the SEm, the lower the reliability of the test and the less precision there is in the measures taken and scores obtained. Since all measurement contains some error, it is highly unlikely that any test will yield the same scores for a given person each time they are retested.

Ar article by Dr. James Brown at University of Hawai'i at Manoa gave an good comparison of these two concepts. Also, an free paper by Harvill LM from East Tennessee State University explained in detail how the standard error of measurement is calculated.