*50*

A **standard error of measurement**, often denoted **SE _{m}**, estimates the variation around a “true” score for an individual when repeated measures are taken.

It is calculated as:

**SE _{m}** = 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:

SE_{m} = s√1-R = 3.17√1-.88 = **1.098**

**How to Use SE**_{m} to Create Confidence Intervals

_{m}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*– SE_{m},*x*+ SE_{m}] - 95% Confidence Interval = [
*x*– 2*SE_{m},*x*+ 2*SE_{m}] - 99% Confidence Interval = [
*x*– 3*SE_{m},*x*+ 3*SE_{m}]

For example, suppose an individual scores a 92 on a certain test that is known to have a SE_{m} 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:

- SE
_{m}= 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:

- SE
_{m}= 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.