# Standard Error of Measurement

## Definition

*from: Chatterji, 2003 refs*

A statistical estimate of the amount of random error in the assessment of results or scores.

*contributed by Frank LaBanca, Ed.D.*

## Non-technical definition

*based on Texas Education Agency, 2003*

The simplest, most non-technical way to think of the standard error of measurement is the following:

*If a single student were to take the same test repeatedly (with no new learning taking place between testings and no memory of question effects), the standard deviation of his or her repeated test scores is denoted as the standard error of measurement.*

What is the difference between the "standard deviation" and the "standard error of measurement"?

When describing standard deviation of scores on an instrument, it usually refers to the standard deviation of the test scores obtained by a *group of students* on a single testing. It is a measure of the "spread" of scores between students. When describing the standard error of measurement on an instrument, that refers to the standard deviation of test scores that would have been obtained from a single student had that student been tested multiple times. Basically, it is the measure of the "spread" of scores within a student had the student been tested repeatedly.

Since it is highly unlikely that each student would be tested repeatedly on an instrument in order to estimate the standard error of measurement, how is the standard error of measurement estimated?

Fortunately, the standard error of measurement can be estimated from a single testing of a population of subjects. From the test scores of a population of subjects on a single instrument. It is straight-forward to compute estimates from the instrument's score mean, test score standard deviation, and the test score reliability.

*contributed by Frank LaBanca, Ed.D.*

## Formula

From these estimates, an estimate of the standard error of measurement is computed using the following formula:

*Standard error of measurement = standard deviation x sqrt(1-reliability)*

*contributed by Frank LaBanca, Ed.D.*