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Cruise Scientific ¨ Visual Statistics Studio ¨ Table of Contents |
Krus, D. J. & Liang, K. S. (1984) Wanted—central tendency and variability of test norms: how to recover them? Educational and Psychological Measurement, 44, 123-129.
WANTED --.CENTRAL TENDENCY AND VARIABILITY
OF
TEST NORM TABLES: HOW TO RECOVER THEM?
David J. Krus and
An algorithm for obtaining means and standard deviations or obtained (raw) scores, underlying table values of standard test scores, is described. This algorithm is crucial for obtaining initial values of continuously updated test norms.
Implementation of standardized tests in a computer environment rapidly becomes a favorite project of educational researchers. Unfortunately, some lack technical knowledge necessary for such an undertaking. Recently, the authors encountered a graduate research assistant seeking advice for a minor problem pertaining to a format of data to be entered manually from a keyboard terminal. The research assistant had been hired and paid for a whole semester for a sole purpose to enter extensive norm tables of an educational achievement test into a computer data file. When advised that this whole-semester project could have been accomplished by a set of simple linear transformations from means and standard deviations of the distributions of the original test scores, both the graduate assistant and his advisor were dismayed. However, that experience was not the end of the story. Later on, the authors were again approached by this team of educational researchers for advice on how to convert a stranger computer tape to a format compatible with the tape drive of the university mainframe computer. Inquiring about the nature of this project, the authors learned that the researchers had initiated a lengthy process of requesting computer tapes from the test publisher with an intent to compute means and standard deviations of the original distributions of the test scores, not listed at the bottom of the norm tables in the test manual where these indices customarily used to be found.
Cursory inspection of recent test manuals showed that means and standard deviations of raw data used for construction of norm test tables are often not reported. The present communication pertains In part to the problem of how to reconstruct these indices if they are missing. Specifically, the three major purposes of this paper were (a) to present an algorithm for reconstruction of means and standard deviations underlying table values of standard test scores, (b) to suggest computer implementation of the algorithm, and (e) to validate empirically the computer implementation of the algorithm by comparing known and estimated values of means and standard deviations underlying sample MMPI table values of the standard T scores.
ALGORITHM
A linear transformation in a general form Y= BX + A where B is the slope of the transformation line and A its intercept can be written for a set of scores and with the respective means and standard deviations as
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Typically, the mean and the standard deviation of the target distribution Y are selected as some preferred combination of numbers such as (100,15) for the distribution of the IQ scores, or (50,10) for the distribution of the T scores. Simple algebraic manipulation of Equation (1) gives the mean of the source distribution X as
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(2) |
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and its standard deviation as
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(3) |
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However, these simple algebraic manipulations do not provide for the desired solutions, as more than a single pair of points is called for. Also, the transformed scores (Y) are customarily rounded to whole numbers. Thus, the measures of the central tendency and variability of the source distribution (X), if not given, cannot be determined, but only estimated. To facilitate the solution of this problem, one may define both the source and target distributions in standard scores where
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The set of standard scores to be estimated from the distribution of scores will be written as . The general form of linear transformation will be reversed; thus X’ = BY + A, in standard scores,
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where the Greek letter beta symbolizes the slope of the line, its coordinates expressed in standard score units with an intercept equal to zero. To estimate distribution of the scores, to minimize error, one has to differentiate the loss function
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Substituting the right hand side of the Equation (5) for the
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while expanding the binomial in the Equation (7), note that the variance of standard scores is one and that the middle term on the right hand side is the Pearson’s product moment coefficient of correlation, i.e.,
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(8) |
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Thus the above function can be simplified as
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Differentiating the loss function in Equation (9) with respect to beta to find its theoretical minimum,
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(10) |
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allows one to determine its theoretical minimum as the Pearson’s product moment coefficient of correlation, i.e.,
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where the Pearson’s product moment coefficient of correlation, expressed as the arc tangent of its numerical value, definies the slope of the regression line. Substituting the right hand expression from Equation (11) for the first right hand term of Equation (5) and interchanging its z-terms with the right hand sides of equations in (4)
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(12) |
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The missing mean of the original distribution on which the norms are based can be computed as
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After the missing mean is retrieve, the missing standard deviation of the original distribution can be obtained as
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The correlation , even though theoretically equal to 1.00, can fluctuate due to errors introduced by rounding. However, the problem is that, initially, in Equation (13) the is not known and in Equation (14) the is unknown. This circulus vitiosus is solved by iterations, as described in the following section.
COMPUTER CODE
A subroutine for the suggested computational operations leading to reconstruction of missing means and standard deviations of generic distributions of standardized test scores is listed in Table 1.
TABLE 1. Subroutine to Reconstruct
Means and Standard Deviations
Underlying Table Values of Standardized Test Scores
SUBROUTINE SLEUTH (TM,TSD,RS,TS,RM,RSD)
AN = 00
AMX = 0.0
AVX = 0.0
AMY = 0.0
AVY = 0.0
XY = 0.0
R = 0.0
B = 0.0
A = 0.0
RM = 0.0
RSD = 0.0
ENTRY SLEUT1(TM,TSD,RS,TS,RM,RSD)
Update N
UN = AN + 1.0
Update means
UMX = (AN*AMX + RS)/UN
UMY = (AN*AMY + TS)/UN
Update variances
F1 = AN/UN
F2 = 1.0/UN
DI = AMX - UMX
D2 = TS - UMY
UVX = F1*(AVX + D1**2) + F2*D2**2
D1 = AMY - UMY
D2 = TS - UMY
UVY = F1*(AVY +L1**2) + F2*D2**2
Update covariance
XY = XY + RS*TS
COV = XY/UN - UMX*UMY
Compute correlation
IF(UVX.LE.0.0.OR.UVY.LF.0.0)GOTO 1
USX = SQRT(UYX)
USY = SQRT(UVY)
R = COV/(USX*USY)
Compute slope and intercept
B = R*USY/USX
A = UMY- B*UMX
Recover the mean and the standard deviation
RM = (TM - A)IB
RSD = R*TSD/B
Reset replacements
1 AN = UN
AMX = UMX
AVX = UVX
AMY = UMY
AVY = UVY
RETURN
END
The input parameters are the known mean and standard deviation of the standard distribution TM,TSD and the raw-standard score pairs RS,TS. The output parameters RM, RSD are the recovered mean and standard deviation of the distribution of the raw test scores.
The theoretical rationale for continuous update of means and variances was described by Krus and Ceurvorst (1978, p. 817). The relational update statements were adapted from a subroutine for autocorrelation analysis of secular trends (Krus and Ko, 1983, Table 3). For FORTRAN compilers not permitting multiple entry points an IF statement should be inserted at the beginning of the subroutine to test the call number counter for the first pass through the subroutine during which the zero initializations should be made.
IMPLEMENTATION
A T-score conversion scale for Wiener-Harman
subtle subscales of the MMPI (Dahlstrom, Welsh, Dahlstrom. 1972, pp. 402-403)
for a sample of
TABLE 2.
T score Conversions for Wiener-Harmon Subtle Subscales
of the .MMPI. Norms for
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Pairs of raw and standard scores were entered one at a time. Only three pairs of scores were entered for each scale. The uppermost, the lowermost, and the middle pairs of test scores were entered. The obtained results are reported in Table 3 for the means, and in Table 4 for the standard deviations of the original distributions of the raw scores. These values were compared with true values for these distributions as reported in Greene (1980, pp. 231-241). The observed differences between true and estimated scores are listed at the bottom of Tables 3 and 4.
TABLE 3
Original
and Estimated Means for Wiener-Harmon
Subtle Subscale for
TABLE 4
Original and Estimated Standard Deviations fir Wiener-Harmon Subtle Subscales
of
the MMPI. Norms for
DISCUSSION
The discrepancies between true and estimated values were small, as expected, since the source of error was the rounding of T scores to whole numbers. However, it is necessary to point out that the present algorithm is valid only for standardized scores obtained by a linear transformation from their corresponding raw scores. If the standardized-score distribution was obtained by an area transformation of the test scores, the use of this algorithm could result in spurious values of the estimated scores. For further discussion of this topic, one may consult Krus and Kennedy (1977).
Another possible source of error could be the use of degrees of freedom instead of N in the denominator of the variance formula when one initially has transformed from raw to standardized scores. Unfortunately, the practice of using “unbiased” statistical indices indiscriminately has become increasingly popular. This trend is being reinforced by calling the maximum likelihood variance “biased,” an antonym that the writers believe has been coined in poor taste indeed. Within the area of linear transformations of test scores, there is no compelling reason for using unbiased in lieu of maximum likelihood variances.
However, these possible sources of error are unlikely to occur for a majority of test table norms.. When uncertain about their possible presence, to minimize the error, one should enter as many raw-standard score pairs as feasible.
REFERENCES
Dahlstrom, G. S.,
Welsh, G. S. and Dahlstrom, L. E. (1972). An MMPI handbook.
Green, R. L. (1980).
The MMPI: An interpretative manual.
Krus, D. J. and Ceurvorst, R. W. (1978). Computer assisted construction of variable norms. Educational and Psychological Measurement, 38, 815-818.
Krus, D. J. and Ko, H. O. (1983) Algorithm for autocorrelation of secular trends. Educational and Psychological Measurement, 43, 821-828.
Krus, D. J. and Kennedy, P. H. (1977). Lost: McCall’s T scores: Why? Educational and Psychological Measurement, 37, 257-261.
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Cruise Scientific ¨ Visual Statistics Studio ¨ Table of Contents |