WJ-IV W_score functions.xls

	A	B	C	D	E	F
1					WJ-IV W_score functions
2		Variable names	Variable values (editable)
3		age	9.5		"W score" is a measure of intelligence used in the Woodcock-Johnson IQ tests, specifically the W-score for "General Intellectual Ability" in the WJ-IV test. (Each of the many subtests and scales in the WJ has it's own W-score). The Stanford-Binet IQ tests, also published by Riverside Publishing, use "CSS" scores, which are essentially equivalent to WJ W-scores. The W and CSS scores are defined so that an average 10 year old scores 500 (in their American norming samples). W and CSS scores are "ratio scales" in that all arithmetic operations including division are meaningful, that is, one can say that one person's score is some percentage higher or lower than another person's score, which is not valid for IQs. In this regard, W and CSS scores are like temperatures measured in Kelvin, while IQ and other test scores are like temperatures measured in degrees Celsius or degrees Fahrenheit. W and CSS scores are also comparable across people of different ages, including between children and adults, which is not true for IQ scores. W and CSS scores provide a direct measure of ability and of the difficulty of problems which a person can answer correctly, unlike other measures such as IQ scores, percentiles and most other test scores. The following equations were derived from empirically fitting curves to the data in Appendix C of: McGrew, K. S., LaForte, E. M., & Schrank, F. A. (2014). Technical Manual. "Woodcock- Johnson IV." Rolling Meadows, IL: Riverside. (Table C-1, pp. 279-280)
4		IQ	125
5		IQ_adult	75
6		W	485
7		percentile	99.00%
8		child_percentile	99.00%
9		child_age	7
10		class_age	10
11		W_age_1	520
12		age_1	10
13		age_2	7
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23	Copied variable values - edit only above			Calculation		Formula
24	9.5			497.2		mean_W[age] = 547 - 473 / age
25						[valid for age 5 to 17, hereafter ranges written in brackets: [5, 17], this curve fit is essentially perfect]
26						This gives the average W-score for ages between 5 and 17.
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28	9.5			10.5		SD_W[age] = 22.2452 - 1.919505 * age + 0.07165335 * age^2
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31	9.5			10.5		SD_W[age] = 23/321 * age^2 - 167/87 * age + 89/4 // rational approximation
32						[valid for age [5, 17], the actual standard deviations differ by an average of -0.28 points, but in two cases (ages 9 and 11) the measured s.d. is over 1 point less than the rational approximation curve fit. This may be an artifact of the particular norming sample; the data is noisy, the s.d.s vary from one age to the next year by up to 2 points, and vary from the curve fit by nearly that much year-to-year. The curve fit was selected assuming that low s.d. numbers compared to neighboring years were likely to be in error. For the actual s.d. numbers refer to the WJ-IV Technical Manual pp. 279-80.]
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39						This gives the standard deviation of W-scores at a given age. The graph of s.d.s is a parabola, with high s.d.s at younger ages, falling to a minimum at about age 13.4 and then rising again.
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41				13.4		Min_SD_age = 17869/1334 (approx. age 13.395, s.d. 9.394)
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44				520.0		W_adult_average = 520
45				10.5		W_adult_SD = 10.5
46						This is the adult average W-score and standard deviation.
47						[It is within the margin of error of the data to use 520, s.d. 10.5 for ages above 17. The data for ages 18 and 19 are noisy: 515.8, s.d 10.73 age 18 (2.3 points lower than the expected 518.1); and 519 s.d. 11.2 age 19. More accurate figures for higher ages: 521, s.d. 10 in the 20s; 520, s.d. 11 in the 30s; 517.5, s.d 12 in the 40s; 516.5 s.d. 12 in the 50s]
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52	75			502.5		W[IQ_adult] = (IQ_adult - 100) / 15 * 10.5 + 520 =
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54	75			502.5		W[IQ_adult] = 0.7 * IQ_adult + 450
55						[IQ mean 100, s.d. 15; IQ_adult preferably age [20, 40], but not too far off for ages [18, 70]]
56						This gives the W-score eqivalent to an adult IQ.
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58	485			7.6		mental_age[W] = 473 / (547 - W)
59						[valid for W between [451, 517], and since W = 0.7 * IQ_adult + 450 :]
60						This gives the mental age equivalent of a W-score, that is, the age for which that W-score is the average score.
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62	75			10.6		mental_age[IQ_adult] = 473 / (547 - ( 0.7 * IQ_adult + 450))
63						This gives the mental age equivalent of an adult IQ score, that is, the age for which that adult IQ’s raw test score would be the average score.
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65	75			10.6		mental_age[IQ_adult] = 473 / (97 - ( 0.7 * IQ_adult))
66						[for adult IQ between about 40 to 97]
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68	125			1.7		Z_score[IQ] = (IQ - 100)/15
69						This gives the Z-score equivalent of an IQ. Z-score is the number of standard deviations a score is above or below average for that age.
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71	99.00%			2.3		Z_score[percentile] = NORMSINV(percentile) // OpenOffice or Excel function
72						["percentile" must be a fraction between 0 and 1 (to be technical, to keep Z_score in the range [-4,4]: [(1/31574), (1 - 1/31574)] ); divide conventional percentile by 100 if needed to make it between 0 and 1]
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75						This gives the Z-score equivalent of a percentile ranking. Z-score is the number of standard deviations a score is above or below average for that age.
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77						Z_score[W, age] = (W - mean_W[age]) / SD_W[age]
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79	485	9.5		-1.2		Z_score[W, age] = (W - (547 - 473 / age))/(23/321 * age^2 - 167/87 * age + 89/4)
80						[Valid for ages [5, 17].]
81						This gives the Z-score equivalent of a W score at a given age. Z-score is the number of standard deviations a score is above or below average
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83		using Z of last line		12.20%		percentile[Z_score] = NORMDIST(Z_score,0,1)
84						["percentile" will be a fraction between 0 and 1; multiply by 100 to get a conventional percentile]
85						This gives the percentile equivalent to a given Z-score, that is, equivalent to that number of standard deviations above or below the average, depending on the sign of the score.
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87						W[percentile, age] = Z_score[percentile] * SD[age] + mean_W[age]
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89	99.00%	9.5		521.6		W[percentile, age] = ( NORMSINV(percentile) * (23/321 * age^2 - 167/87 * age + 89/4) + (547 - 473/age) )
90						[Valid for ages [5, 17].]
91						This gives the W-score eqivalent to a percentile score for that age.
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94						W[IQ, age] = Z_score[IQ] * SD_W[age] + mean_W[age]
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96	125	9.5		514.7		W[IQ, age] = ((IQ - 100)/15) * (23/321 * age^2 - 167/87 * age + 89/4) + (547 - 473 / age)
97						[For ages above 17 use: W[IQ_adult] = 0.7 * IQ_adult + 450.]
98						This gives the W-score equivalent to an IQ and age.
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