Guttman Scaling

An Example of a Perfect Guttman Scale

Suppose that we test a set of children and that we assess their mastery of the following types of mathematical concepts: (a) counting from 1 to 50, (b) solving addition problems, (c) solving subtraction problems, (d) solving multiplication problems, and (e) solving division problems.

Some children will be unable to master any of these problems, and these children do not provide information about the problems, so we will not consider them. Some children will master counting but nothing more; some will master addition and we expect them to have mastered counting but no other concepts; some children will master subtraction and we expect them to have mastered counting and addition; some children will master multiplication and we expect them to have mastered subtraction, addition, and counting. Finally, some children will master division and we expect them to have mastered counting, addition, subtraction, and multiplication. What we do not expect [Page 559]to find, however, are children, for example, who have mastered division but who have not mastered addition or subtraction or multiplication. So, the set of patterns of responses that we expect to find is well structured and is shown in Table 1. The pattern of data displayed in this table is consistent with the existence of a single dimension of mathematical ability. In this framework, a child has reached a certain level of this mathematical ability and can solve all the problems below this level and none of the problems above this level.

When the data follow the pattern illustrated in Table 1, the rows and the columns of the table both can be represented on a single dimension. The operations will be ordered from the easiest to the hardest, and a child will be positioned on the right of the most difficult type of operation solved. So the data from Table 1 can be represented by the following order:

This order can be transformed into a set of numerical values by assigning numbers with equal steps between two contiguous points. For example, this set of numbers can represent the numerical values corresponding to Table 1:

This scoring scheme implies that the score of an observation (i.e., a row in Table 1) is proportional to the number of nonzero variables (i.e., columns in Table 1) for this row.

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