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The Standard Error of Norms and the Standard Error of Measurement

Lord, Frederic M.
Publication Year:
Report Number:
ETS Research Memorandum
Document Type:
Page Count:
Subject/Key Words:
Equated Scores, Error of Measurement, Norms, Test Norms


The number of schools required to provide adequate norms for new, widely administered tests is addressed. The average standard error of measurement (averaged over all examinees) is first considered. The approximate standard error appropriate for examinees at a given ability level may also be estimated; this average is equal to the average standard error of measurement computed from the Kuder Richardson formula-21 reliability coefficient. Norms are obtained from a sample of schools from the population the norms are supposed to represent. Extensive and reliable data for a certain aptitude test showed that the standard deviation of school means in the population was about four tenths of the standard deviation among all individuals. It is possible to compute the discrepancy caused by the standard error of measurement of the score, and the standard error of the normative information. The standard deviation of this difference is thus a measure of the joint effect of errors of measurement and of sampling errors in the norms. These two sources of error are independent. It is important to note that equating two test forms introduces an additional sampling error. Sampling errors that afflict the norms table do not cancel out over large numbers of individual examinees. When a group mean is computed for a particular test, the result tends to be in error by an amount equal to the norming error plus the equating error. Problems will occur when more than one norms table (such as for geographic regions or type of school) is prepared for any test. A description of stratified sampling for norms purposes is appended.

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