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probability density function decreases as x moves farther from . Consequently, the probability that a measurement falls far from is small, and at some distance from the probability of an interval can be approximated as zero. The area under a normal probability density function beyond 3 from the mean is quite small. This fact is convenient for quick, rough sketches of a normal probability density function. The sketches help us determine probabilities. Because more than 0.9973 of the probability of a normal distribution is within the interval 1 3 , 3 2 , 6 is often referred to as the width of a normal distribution. Advanced integration methods can be used to show that the area under the normal probability density function from x is 1. De nition A normal random variable with 0 and

likely to be the problematical WAIS-R adolescent norms. The possibility of the gain truly reflecting improved ability was given some support by Carvajal et al. s (1984) WISC/WAIS study. They analyzed data for 66 retarded individuals tested on the WISC at an average age of 11 years, 9 months and on the WAIS at 17 years, 6 months. The group gained 9 1 2 points in FS-IQ on the retest. But Carvajal et al. were able to retest 21 subjects on the WAIS at a mean age of 28 years, 6 months. This subsample showed a gain in mean IQ from 64 on the WISC at time 1 to 73 on the WAIS at time 2 to 791 2 on the WAIS at time 3; the gain from older adolescence to adulthood for this sample of individuals with mental retardation held the instrument constant. However, the sample was small and contained only those willing to be tested; conclusions are tentative and not generalizable. The only conclusion is that the WAIS-R (like the WAIS before it) tends to yield higher IQs than the WISC-R, especially on the Verbal and Full Scales, for lowfunctioning individuals who are retested on the WAIS-R after several years. WAIS-R versus WISC-R for Individuals with Mental Retardation When Zimmerman, Covin, and Woo-Sam (1986) combined data from their two subsamples of referrals, they found that the discrepancies in favor of WAIS-R were largest for mentally retarded individuals and virtually disappeared for subjects with average or near-average IQs. Based on WISC-R Full Scale IQ, they found the following discrepancies with WAIS-R IQ by IQ level:

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is called a standard normal random variable and is denoted as Z. The cumulative distribution function of a standard normal random variable is denoted as 1z2 P1Z z2

IQ Level 40 59 60 69 70 79 80 89 90 and above N 11 23 32 15 9 Mean Difference +14.4 +5.5 +2.7 +1.1 +1.3

Appendix Table II provides cumulative probability values for 1z2 , for a standard normal random variable. Cumulative distribution functions for normal random variables are also widely available in computer packages. They can be used in the same manner as Appendix Table II to obtain probabilities for these random variables. The use of Table II is illustrated by the following example. EXAMPLE 4-11 Assume Z is a standard normal random variable. Appendix Table II provides probabilities of the form P1Z z2. The use of Table II to nd P1Z 1.52 is illustrated in Fig. 4-13. Read down the z column to the row that equals 1.5. The probability is read from the adjacent column, labeled 0.00, to be 0.93319. The column headings refer to the hundredth s digit of the value of z in P1Z z2. For example, P1Z 1.532 is found by reading down the z column to the row 1.5 and then selecting the probability from the column labeled 0.03 to be 0.93699.

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These results are highly similar to the results reported by Zimmerman et al for the separate Verbal and Performance Scales Rubin, Goldman, and Rosenfeld (1985) divided their institutionalized sample of individuals with mental retardation into subgroups of trainable mentally retarded (N = 21) and educable mentally retarded (N = 20), using a WISC-R Full Scale IQ of 55 as the cutoff Like Zimmerman et al (1986), Rubin et al found the difference in Full Scale IQs for the group with lower IQs to be larger than the WAIS-R/WISC-R discrepancy for the EMRs (14 versus 7 points) Also, Carvajal et al (1984) found slightly larger gains from WISC to WAIS over a 6-year interval for retarded people with IQs below 70 (101 2 points) than for those with IQs of 70 and above (8 points).

z 0 ...

0.00 0.50000 0.93319

Again, the direction of the relationship between the size of the IQ discrepancy and level of intelligence is predictable based on the known impact of the phenomenon of regression to the mean; however, the magnitude of the relationship is more than one would anticipate from the statistical artifact The consequence of the higher WAIS-R than WISC-R IQs for low-IQ individuals is a different intelligence classification on reevaluation, which occurred frequently in Zimmerman et al s (1986) study, and dramatically in the investigation by Rubin et al (1985) The former group of researchers state: Such dramatic changes in classification may have ominous repercussions for school personnel (Zimmerman et al, 1986, p 150) Rubin et al.

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