# Standard Error of Measurement: Definition & Example

A standard error of measurement, often denoted SEm, estimates the variation around a “true” score for an individual when repeated measures are taken.

It is calculated as:

SEm = s√1-R

where:

• s: The standard deviation of measurements
• R: The reliability coefficient of a test

Note that a reliability coefficient ranges from 0 to 1 and is calculated by administering a test to many individuals twice and calculating the correlation between their test scores.

The higher the reliability coefficient, the more often a test produces consistent scores.

### Example: Calculating a Standard Error of Measurement

Suppose an individual takes a certain test 10 times over the course of a week that aims to measure overall intelligence on a scale of 0 to 100. They receive the following scores:

Scores: 88, 90, 91, 94, 86, 88, 84, 90, 90, 94

The sample mean is 89.5 and the sample standard deviation is 3.17.

If the test is known to have a reliability coefficient of 0.88, then we would calculate the standard error of measurement as:

SEm = s√1-R = 3.17√1-.88 = 1.098

### How to Use SEm to Create Confidence Intervals

Using the standard error of measurement, we can create a confidence interval that is likely to contain the “true” score of an individual on a certain test with a certain degree of confidence.

If an individual receives a score of x on a test, we can use the following formulas to calculate various confidence intervals for this score:

• 68% Confidence Interval = [x – SEmx + SEm]
• 95% Confidence Interval = [x – 2*SEmx + 2*SEm]
• 99% Confidence Interval = [x – 3*SEmx + 3*SEm]

For example, suppose an individual scores a 92 on a certain test that is known to have a SEm of 2.5. We could calculate a 95% confidence interval as:

• 95% Confidence Interval = [92 – 2*2.5, 92 + 2*2.5] = [87, 97]

This means we are 95% confident that an individual’s “true” score on this test is between 87 and 97.

### Reliability & Standard Error of Measurement

There exists a simple relationship between the reliability coefficient of a test and the standard error of measurement:

• The higher the reliability coefficient, the lower the standard error of measurement.
• The lower the reliability coefficient, the higher the standard error of measurement.

To illustrate this, consider an individual who takes a test 10 times and has a standard deviation of scores of 2.

If the test has a reliability coefficient of 0.9, then the standard error of measurement would be calculated as:

• SEm = s√1-R = 2√1-.9 = 0.632

However, if the test has a reliability coefficient of 0.5, then the standard error of measurement would be calculated as:

• SEm = s√1-R = 2√1-.5 = 1.414

This should make sense intuitively: If the scores of a test are less reliable, then the error in the measurement of the “true” score will be higher.