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 – SEm, x + SEm]
- 95% Confidence Interval = [x – 2*SEm, x + 2*SEm]
- 99% Confidence Interval = [x – 3*SEm, x + 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.
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Standard Error of Measurement
The term standard error of measurement indicates the spread of measurement errors when estimating an examinee’s true score from the observed score. Standard error of measurement is most frequently useful in test reliability. An observed score is an examinee’s obtained score, or raw score, on a particular test. A true score would be determined if this particular test was then given to a group of examinees 1,000 times, under identical conditions. The average of those observed scores would yield the best estimate of the examinees’ true abilities. Standard deviation is applied to the average of those scores across persons and administrations to determine the standard error of measurement. Observed score and true score can be used together to determine the amount of error:
Scoretrue= Score observed+ Scoreerror.
However, this true ...
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