Tuesday, December 9, 2008

Understanding the Numbers - III

I wanted to continue our discussion on statistics and interpreting test scores. Again, most of the information that you will read on this topic are from Jerome Sattler's "Measurement of Children."

If you will remember (or review previous posts found in the archives under "Statistics" heading), "derived scores vary in their usefulness." "The major types of derived scores used in norm-referenced testing are age and grade equivalent scores, ratio IQs, percentile ranks, standard scores, and stanines." We have already discussed age and grade equivalent scores and percentile ranks and I would like to begin this post by discussing ratio intelligent quotients, standard scores, stanines, and finally the relationship among derived scores.

Ratio Intelligence Quotients
"In order to interpret age-equivalent or grade-equivalent scores, we must know the student's chronological age (CA). Knowing the student's MA (Mental Age) and CA allows us to make a judgment about the child's relative performance. For example, a student with a CA of 16-0 and an MA of 18-0 has performed at an above average level, whereas a child with a CA of 16-0 and an MA of 14-0 has performed at a below average level.

When IQs were first introduced, they were defined as ratios of mental age to chronological age, multiplied by 100 to eliminate the decimal: IQ = MA/CA X 100. Substituting an MA of 12 and a CA of 10 into the formula yields a ratio IQ of l20 (IQ = 12/10 X 100 = 120). Unfortunately, because the standard deviation of the ratio IQ distribution does not remain constant with age, IQs for different ages are not comparable: The same IQ has different meanings at different ages. The use of the Deviation IQ, which is a standard score, effectively avoids this problem."

Standard Scores
"Standard scores are raw scores that have been transformed to have a given mean and standard deviation. They express how far an examinee's score lies from the mean of the distribution in terms of the standard deviation.

A z score is one type of standard score, with a mean of 0 and a standard deviation of 1. On many standardized tests z scores range from -3.0 to +3.0. Frequently, z scores are transformed into other standard scores in order to eliminate the + and — signs. For example, a T score is a standard score based on a distribution with a mean of 50 and a standard deviation of 10. The Deviation IQ is another standard score; it has a mean of 100 and a standard deviation of 15 or 16, depending on the test used."

"Stanines (a contraction of standard nine) provide a single-digit scoring system with a mean of 5 and a standard deviation of 2. The scores are expressed as whole numbers from 1 to 9. When scores are converted to stanines, the shape of the original distribution is changed into a normal curve. The percentages of scores at each stanine are 4, 7, 12, 17, 20, 17, 12, 7, and 4, respectively."

Relationship Among Derived Scores
"It should be evident from the preceding discussion that the various types of derived scores are all derived from raw scores. The different derived scores are merely different expressions of a student's performance. One type of derived score can be converted to another type. The most frequently used conversion in the area of intelligence testing is from standard scores (for example, scaled scores or Deviation IQs) to percentile ranks. Although standard scores are the preferred derived scores, percentile ranks — and, on occasion, age equivalents—are also useful. The latter two scores may be helpful in describing the student's performance to parents or teachers.

A Normal Curve shows the relationships among various derived scores. If a test has a Deviation IQ of 100, a standard deviation of 15, and scores that are normally distributed, the percentile ranks associated with each IQ can be determined precisely. As an example, let us see how percentile ranks associated with IQs at various standard deviation points are computed.

An IQ of 100 represents the 50th percentile rank, because an IQ of 100 has been set as the mean of the distribution. In this example an IQ of 115 represents the point that is +1 SD away from the mean. The percentile rank associated with this IQ—the 84th percentile rank— is obtained by adding 50 to 34 percent. The 50 percent represents the proportion of the population below the mean of 100, and the 34 percent represents the proportion of the population between the mean and +1 SD away from the mean. The key here is to recognize that an IQ of 115 is +1 SD above the mean because 15 is the standard deviation of the distribution.

Using the same rationale, we can readily compute the percentile rank associated with an IQ of 130. An IQ of 130 is +2 SD away from the mean. We know that the area below the mean represents 50 percent of the population, the area from the mean to +1 SD represents approximately 34 percent of the population, and the area from +1 SD to +2SD represents approximately 14 percent of the population. To arrive at the percentile rank for an IQ of 130, we add the following percentages: 50 + 34 + 14 = 98th percentile rank.

What is the percentile rank associated with an IQ of 85? The answer you should obtain is the 16th percentile rank. You subtract 34 from 50, because an IQ of 85 corresponds to the point that is -1 SD away from the mean. The percentile rank associated with an IQ of 70 is the 2nd percentile rank (50 - 34 - 14 = 2).

The above examples hold only for tests that have a Deviation IQ with a mean of 100 and an SD of 15. For tests that have a mean of 100 and an SD of 16 the percentile ranks associated with the various IQs are slightly different except at the mean. The IQ of 100 is still at the 50th percentile rank, but an IQ of 116 (not 115) is at the 84th percentile rank because the 5D is 16."

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