5.3 – Comparing Individual Scores from Different Scales

Let’s say we wanted to see if our client, Grover, has a learning disability. This would entail giving Grover some sort of aptitude test along with an achievement test. Aptitude tests are intended to measure a person’s aptitude, their potential ability, or what they can learn. Achievement tests are intended to measure a person’s achievement, or what they have actually learned. We expect that most people’s achievement test scores will be very similar to their aptitude test scores because they have learned as much as they can learn. However, with learning disabilities, there is often a gap between what the person has learned and what they can learn. The idea is that a learning disability, such as dyslexia, might be making it very difficult for the person to achieve all that they can achieve. If it takes a person much longer to read because they have dyslexia, then it can be more difficult to actually learn all that you can learn.

So let’s say we are working with Grover, and we believe that he might have a learning disability, so we will need measurements of both his aptitude and achievement. Typically, psychologists will administer an intelligence test to measure aptitude, but for our purposes let’s say that Grover had taken the Verbal subtest of the SAT (the Scholastic Aptitude Test), a common test for entrance into college. Grover earned a score of X = 600 on the Verbal subtest. We then administered an achievement test (the Wide-Ranging Achievement Test version 4, WRAT-4) to Grover, and he earned a score of X = 95.

Here’s what we know about Grover and the tests he has taken:

In order to explore the possibility of a learning disability, we need to look for a discrepancy between Grover’s aptitude and achievement test scores. Clearly, there is a big difference between X = 600 and X = 95, but hopefully, you recognize that those two scores come from different tests with different scales.

Thus, we can’t directly compare the scores of 600 and 95 for Grover. In order to compare them we need to standardize them, so we will convert them both to z-scores:

SAT Subtest:  [latex]z=\frac{X-\mu}{\sigma}=\frac{600-500}{100}=\frac{+100}{100}=+1.00[/latex]

WRAT-4:  [latex]z=\frac{X-\mu}{\sigma}=\frac{95-100}{15}=\frac{-5}{15}=-0.33[/latex]

We can see that Grover had a z-score of z =  +1.00 for the SAT subtest, and a z-score of z = -0.33 for the WRAT-4. In other words, Grover is “one standard deviation above the mean on the SAT”, and he is “one-third of a standard deviation below the mean on the WRAT-4.”

Because these two scores are now on the same scale because they are both z-scores and thus have the same mean (μ = 0) and standard deviation (σ = 1), and thus they can be compared directly. In fact, we can simply subtract the SAT z-score by the WRAT-4 z-score to see if there is a difference.

[latex]\text{SAT z-score} – \text{WRAT z-score} = +1.00 – (–0.33) = +1.00 + 0.33 = +1.33[/latex]

We can now see that Grover’s Aptitude score is 1.333 standard deviations higher than his Achievement score, indicating that he has not achieved the level of his ability or aptitude.

As an aside, whether or not this discrepancy rises to the level of a learning disability depends upon the state laws. Some states in the U.S. require a minimum of a 1.0 standard deviation discrepancy in order for a child to be diagnosed with a learning disability, while others require a minimum discrepancy of 1.5 standard deviations, and others require a minimum discrepancy of 2.0 standard deviations.

For now, though, we simply want to focus on the fact that we were able to directly compare two scores, which each came from different scales. Using the z-score to standardize the two scores allows us to make this comparison.