Standard Error of Measurement

    The standard error of measurement (SEM) is an estimate of error to use in interpreting an individual’s test score. A test score is an estimate of a person’s  “true” test performance. Using a reliability coefficient and the test’s standard deviation, we can calculate this value:

SEM  =  s  Ö( 1 – r)

Where:

    S = the standard deviation for the test

    r = the reliability coefficient for the test

For example,  

A Wechsler test with a split-half reliability coefficient of .96 and a standard deviation of 15 yields a SEM of 3

 SEM  =  s  Ö( 1 – r ) =  15 Ö ( 1-.96)   =   15 Ö.04   =  15 x .2   = 3

Now that we have a SEM of 3 we can apply it to a real life situation.

Example: Joe took the Wechsler test and received a score of 100.

Let’s build a “band of erroradd a definition here around Joe’s test score of 100, using a 68% interval.   A 68% interval is approximately equal to 1 standard deviation on either side of the mean.

For a 68% interval, use the following formula:

  Test score  ± 1(SEM) 

          Where:  The test score is Joe’s score  

                =  100  ± (1 x 3)  =  100 ± 3

Why the “± 1”? Because we are adopting the normal distribution for our theoretical distribution of error, and 68% of the values lie within the area between 1 standard deviation below the mean and 1 standard deviation above the mean.  

Chances are 68 out of 100 that Joe’s true score falls within the range of 97 and 103.

What about a 95% confidence interval?  A 95 percent interval is approximately equal to with area within 2 standard deviations on either  side of the mean.

Test score  ± 2(SEM) =  100  ± (2 x 3)  = 100 ± 6

Chances are 95 out of 100 that Joe’s true score falls within the range of 94 and 106.

The higher a test’s reliability coefficient, the smaller the test’s SEM.  The larger the SEM the less reliable the test is.

Try another example

Jane has a score of 110.  Find the "band of error" for a 68% interval.

With a 68% interval the true score will fall between 107 and 113. ± 3 points on either side of her score of 110.

Now try a 95% interval

With a 95% interval the true score will fall between 104 and 116.  ± 6 points on either side of her score

 

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