Understanding the Numbers

I have been thinking about "numbers," "testing," and "test interpretation" after attending recent parent/teacher conferences and I thought it might be a good time to begin an ongoing discussion of what some of those numbers mean. When attending conferences, I always flashback to a few years ago to when I heard an instructor interpret a test to a parent that a "70" standard score was actually ok because it was like making a "C" when, in effect, it was two standard deviations below the mean and on an intelligence test, this would clearly fall in the mentally retarded range of ability.

Anyway, I know many of you interpret a wide range of assessments and measurement instruments so this may seem like a review, but feel free to post and tell us about some of the tests you use and what you like/dislike about the administration, scoring, and interpretation procedures of each instument.

Most of the information that you will read today and in the future on statistics are from Jerome Sattler's "Measurement of Children" and I thought it would be good to start with norm-referenced measurement (an indication of average or typical performance of the specified group).

A norm group should be representative of the various demographic populations as a whole, the number of subjects in the the norm group (size) should also be large enough to ensure stability of the test scores, and it is always important to consider how relevant the norms are to the evaluation of the examinee.

The first derived score we will look at is Age-Equivalent and Grade-Equivalent Scores.

"Age-equivalent and grade-equivalent scores are derived by determining the average score obtained on a test by individuals of various ages or grade placements. For example, if the average score of a 17-year old student on a test is 15 items correct out of 25, then any other student obtaining a score of 15 receives an age-equivalent score of 17. An age-equivalent score is found by computing the mean raw score of a measure for a group of children with a specific age. Similarly, a grade-equivalent score is found by computing the mean raw score obtained by children in each grade. If the mean score of seventh graders on an arithmetic test is 30, then a child obtaining a score of 30 is said to have arithmetical knowledge at the seventh grade level. A grade-equivalent score is expressed in tenths of a grade (for example, 10.5 refers to average performance at the middle of the tenth grade). A grade-equivalent score, therefore, refers to the level of test performance of an average student at that grade level. It does not mean that the student is performing at a level consistent with curricular expectations at his or her particular school. (Other terms for age-equivalent scores are mental age (MA) and test age.)

Age- and grade-equivalent scores require careful interpretation, for the following reasons:

1. Within an age-equivalent (or grade-equivalent) distribution of scores, the scores may not represent equal units. The difference between second and third grade-equivalent scores may not be the same as the difference between eleventh and twelfth grade-equivalent scores.
2. Many grade equivalents are obtained by means of interpolation and extrapolation. Consequently, the scores may not actually have been obtained by children.
3. Grade equivalents encourage comparison with inappropriate groups. For example, a ninth grader who obtains a grade equivalent of 11.1 in arithmetic should not be said to be functioning like a eleventh grader at the beginning of the school year; this is the wrong comparison group. The ninth grade student shares with the average eleventh grader the number of items right on the test—not other attributes associated with eleventh grade mathematical skills. The grade equivalent of 11.1 should be thought of in reference to only the student’s ninth grade group, not any other group.
4. Identical grade-equivalent scores on different tests may mean different things.
5. Grade equivalents assume that growth is constant throughout the school year; this assumption may not be warranted.
6. At upper levels, grade or age equivalents have little meaning for school subjects that are not taught at those levels or for skills that reach their peak at an earlier age.
7. Grade equivalents exaggerate small differences in performance—a score slightly below the median may result in a grade level equivalent one or two years below grade level.
8. Grade equivalents vary from test to test, from subtest to subtest within the same battery, and from percentile to percentile, thereby greatly complicating any type of comparison.
9. Grade-equivalent scores depend on promotion practices and on the particular curriculum in different grades.
10. Age- and grade-equivalent scores tend to be based on ordinal scales that are too weak to support the computation of important statistical measures, such as the standard error of measurement.

The preceding discussion indicates that grade- and age-equivalent scores are psychometrically impure; however, they still may be useful on some occasions. Grade- and age-equivalent scores place performance in a developmental context, provide information that is easily understood by parents and the public, and reduce misinterpretations. (Percentile ranks, for example, are often misinterpreted as indicating the percentage of questions that the child answered correctly.) Instead of abandoning grade- and age-equivalent scores, we should better educate people in their use."