# The introduction of a model of statistical data matrices encompassing original variate population methodologies

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Purpose: The purpose of this study was to examine the history, rationale, and sources of confusion of existing models and definitions of two and three dimensional statistical data matrices, and to lay a foundation for the acceptance of a standard model by suggesting clear definitions, recommendations, and modifications for a more robust model of statistical data matrices. Methods: The methodology used was documentary research concerning statistical data matrix models, an insight into the problems of these, and innovation in the introduction of a modified model. Findings: 1. A major source of confusion in the field of data matrices is the lack of standard definitions for basic concepts. 2. A major cause of the lack of standard definitions of data matrices was a transition in the definition of Q technique by its strongest proponent, William Stephenson, and his failure to acknowledge this transition. (a) The lack of a standard definition of Q contributed to the controversy of whether or not R technique and Q technique may exist in one and the same matrix. By using Stephensonâ€™s original definition of Q, R and Q may exist in the same matrix; by using Stephensonâ€™s revised definition of Q, restricting Q to a population of statements, R and Q may not exist in the same matrix. (b) Although Stephenson claimed to have divided the six techniques of Raymond Cattell into two methodologies, the properties of the techniques were not consistent within each methodology. Stephensonâ€™s revised definition of Q contributed to the inconsistencies of the properties between the technique within each methodology. 3. Stephensonâ€™s transition was precipitated by the reciprocity principle of Sir Cyril Burt. 4. Measurement scales may be unique to a variate or common at least two variates. 5. In terms of Cattellâ€™s techniques four equivalent descriptions exist for any re-standardization. Recommendations: 1. Cattellâ€™s six techniques should be divided into three methodologies grouped according to the dimension of the original variate population sampled. The three original methodologies were designated as follows: (a) PPV- - the statistical analysis of person population variates; (b) APV- - the statistical analysis of attribute population variates; and (c) OPV- - the statistical analysis of occasion population variates, respectively. 2. The basis of each methodology would not be the two variates of analysis, as in Cattellâ€™s techniques, but rather the population of the individual original variate. 3. The properties of measurement scale, conduciveness to a Sort, and population are consistent within each population methodology. 4. Different population methodologies may exist in the same matrix without re-standardization if common scaling exists through each dimension. 5. Although in terms of Cattellâ€™s techniques four equivalent descriptions exist for any re-standardization, a re-standardization can be expressed in but one manner in the original population model. 6. Re-standardization should be clearly distinguished in terms of its original and resulting variate populations. 7. In the case of an APV Sort, an additional axis was found necessary for the population of instructions. An additional designation, IPV, the statistical analysis of instruction population variates, can only be the result of re-standardization from a previous APV Sort.