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Krus, D.J., & Ceurvorst, R.W. (1978)
Computer assisted construction of variable norms. Educational and Psychological Measurement, 38, 135-137.
COMPUTER ASSISTED CONSTRUCTION OF VARIABLE NORMS
David J. Krus and
An algorithm for updating the means and variances of a norm group after each computer-assisted administration of a test is described. The algorithm does not require storage of the whole data set and provides for unlimited, continuous expansion of the test norms.
The desirability of local norms for virtually any type of measuring instrument has been stressed repeatedly. Thus Angoff (1971, p. 536) has stressed that
“The most useful norms are the local norms collected by the user.”
In a similar vein, Cronbach (1970, p. 107) recommended that
“Whenever he can, the test interpreter should prepare norms for the local groups with which he deals.”
The advent of computer-administered tests has increased the attractiveness of local norms, updated after each test administration. However, updating the mean(s) and variance(s) is usually expensive in terms of computer time and storage space when an entire set of data is stored in order to re-compute these parameters. The purpose of this paper is to present a procedure for updating means and variances of a norm group which does not require storage of an entire set of data. The only values stored after each update are the new means, variances, and groups sizes. Knowledge of these parameters is sufficient for a subsequent update following the next administration of the test.
Updating the Mean
Consider a distribution of norm scores with a mean and variance where n is an index of the number of times a test was administered. The next administration of the test, yielding score permits one to update the mean of the norm group as
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Accepting convention to symbolize the n / (n + 1) as a and the 1 / (n + 1) expression as b, the above equation can be written as
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For an instance of , addition of a new norm value 5
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where a equals .80 and b equals .20, the new mean can be calculated as (.80) 2.50 + (.20) 5.00 which equals 3.00, the updated mean.
Updating the Variance
The procedure for updating the variance necessitates the consideration of some additional parameters, as schematized in Figure 1.
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Figure 1.
Schematic representation of relationships
necessary to consider for updating variance of variable norms.
For the instance of the discussed norm values,
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When a new norm value is added to the distribution, the mean of the norm group is changed by the amount
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The deviation of the new norm value from the new mean also has to be considered. It is written as follows:
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Adding to the variance between the old scores and the new mean the new deviation score , the new variance can be computed as
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Expanding the binomial,
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Since equals zero and is a constant
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The first term on the right side of the above equation is the sum of the n squared deviation scores from their original mean, divided by n + I, which equals n / (n + 1) times the old variance. Thus, the above equation can be further simplified to
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thus
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Accepting convention introduced above that a equals n / (n+1) and b equals 1 / (n+1), the above equation can be written as
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or
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For the instance of the discussed norm values where a equals .80 and b equals .20, the variance can be updated as which equals 2.00, the new variance.
A subroutine for updating the means and variances of a norm group is presented in Table 1.
Table 1. Subroutine to Update Means and Variances of Variable Norms.
SUBROUTINE
NORM(K, SCORE, OLDM, OLDV, UMEAN, UVAR, LU)
DIMENSION
SCORE(K), OLDM(K), OLDV(K), UMEAN(K), UVAR(K)
C READ IN OLD NORMS
REWIND LU
READ(LU,1O) OLDN, (OLDM(I),OLDV(I),
I = 1,K)
10 FORMAT(
)
C UPDATE N
UN=OLDN+I.0
C UPDATE MEANS
DO 20 I= 1,K
20 UMEAN(I
)=(OLDN*OLDM(I)+SCORE(I ))/UN
C UPDATE VARIANCES
A =OLDN/UN
B= 1.0/UN
DO 30 I=1,K
D1 =OLDM(I)—UMEAN(I)
D2=SCORE(I)- UMEAN(I)
30 UVAR(I)=A*(OLDV(I)+D1**2)+B*D2**2
C RESTORE THE NORMS
REWIND LU
WRITE(LU,10) UN, (UMEAN(I),UVAR(I),I
= 1,K)
END
The formal parameters pertain to the number of norm means and variances to be updated (K), new scores obtained from an administration of the test (SCORE), old norm means (OLDM), old norm variances (OLDV), updated norm means (UMEAN), updated norm variances (UVAR), as well as to the logical unit (LU) involved in the input-output communications with the peripheral mass storage file. Parameters K and LU arc integer constants and SCORE, OLDM, OLDV, UMEAN, UVAR are real arrays of size K. The number of norm updates (OLQN, UN), the norm means (OLDM, U MEAN) and the norm variances (OLDV, UVAR) are stored in a mass storage file associated by the LU parameter with the actually accessed peripheral unit. Arrays OLDM, U MEAN, OLDV, and UVAR are referenced as formal parameters of the NORM subroutine to permit their variable dimensioning, the input of new means and variances for norm update is via the SCORE array.
Discussion
The described procedure allows for transfer of a minimal amount of mass-storage based information, necessary for norm-referenced interpretations of a test and for continuous updates of the norm group. It precludes the mass storage of the whole norm-data set and provides for an unlimited expansion of the test norms. As such, it represents a useful technique for construction of norm-interpretative computerized measurement instruments.
References
Angoff, W. H. (1971) Scales, norms, and equivalent scores. In R. L. Thorndike (Ed.) Educational Measurement. Washington, D. C.: American Council on Education.
Cronbach, L. J. (1970) Essentials of psychological testing
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