# What is the Standard Error of the Mean?

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.

Nathan Thompson, PhD, is CEO and Co-Founder of Assessment Systems Corporation (ASC). He is a psychometrician, software developer, author, and researcher, and evangelist for AI and automation. His mission is to elevate the profession of psychometrics by using software to automate psychometric work like item review, job analysis, and Angoff studies, so we can focus on more innovative work. His core goal is to improve assessment throughout the world.

Nate was originally trained as a psychometrician, with an honors degree at Luther College with a triple major of Math/Psych/Latin, and then a PhD in Psychometrics at the University of Minnesota. He then worked multiple roles in the testing industry, including item writer, test development manager, essay test marker, consulting psychometrician, software developer, project manager, and business leader. He is also cofounder and Membership Director at the International Association for Computerized Adaptive Testing (iacat.org). He’s published 100+ papers and presentations, but his favorite remains https://scholarworks.umass.edu/pare/vol16/iss1/1/.

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