standard-error-of-the-mean

What is the Standard Error of the Mean?

Nathan Thompson, PhDJanuary 5, 2018

The standard error of the mean is one of the three main standard errors in psychometrics and psychology.  Its purpose is to help conceptualize the error in estimating the mean of some population based on a sample.  The SEM is a well-known concept from the general field of statistics, used in an untold number of applications.

For example, a biologist might catch a number of fish from a lake, measure their length, and use that data to determine the average size of fish in the lake.  In psychometrics and psychology, we usually utilize data from some measurement of people.

For example, suppose we have a population of 5 employees with scores below on a 10 item assessment on safety procedures.

Student  Score

1             4

2             6

3             6

4             8

5             7

Then let’s suppose we are drawing a sample of 4.  There are 5 different ways to do this, but let’s just say it’s the first 4.  The average score is (4+6+6+8)/4=6, and the standard deviation for the sample is 1.63.  The standard error of the mean says that

   SEM=SD/sqrt(n) = 1.63/sqrt(4) = 0.815.

Because a 95% confidence interval is 1.96 standard errors around the average, and the average is 6, this says we expect the true mean of the distribution to be somewhere between 4.40 to 7.60.  Obviously, this is not very exact!  There are two reasons for that in this case: the SD is relatively large 1.63 on a scale of 0 to 10, and the N is only 4.  If you had a sample N of 10,000, for example, you’d be dividing the 1.63 by 100, leading to an SEM of only 0.0163.

How is this useful?  Well, this tells us that even the wide range of 4.40 to 7.60 means that the average score is fairly low; that our employees probably need some training on safety procedures.  The other two standard errors in psychology/psychometrics are more complex and more useful, though.  I’ll be covering those in future posts.

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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://scholarworks.umass.edu/pare/vol16/iss1/1/ .
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