A reanalysis of Lord’s statistical treatment of football numbers
Section snippets
Admissible statistics and the measurement-statistics debate
In typical introductory statistics classes, psychology students are taught that the level of measurement should be taken into account when choosing a statistical test. For example, a test should not be performed on data that are of a nominal or ordinal level. Exactly why this rule should be followed is rarely explained and not widely known among psychologists; therefore we reiterate the rationale for it. Suppose mathematical proficiency of children was measured on an ordinal level. In such a
What do the football numbers measure?
The professor in Lord’s thought experiment repeatedly emphasizes that the numbers are nominal representations of the uniqueness of the players. Now, the numbers can certainly be used to distinguish players on the field; but this is not the property for which the statistician uses the numbers. Instead, the professor asks a question and draws a conclusion about the machine—namely that it was unlikely to be in its original state (randomly shuffled by the professor) when the freshman numbers were
Measuring machines
Lord’s statistician uses the statistical results to make an inference about the state of the vending machine, and decides that the freshman mean did not come from the machine in its original state. Thus, Lord’s inference concerns the state of the machine relative to another (possible) state of the machine. His reference class is not a set of football players, but a set of possible states of the machine (e.g., fair and biased states). Insofar as measurement is taking place in the thought
Conclusion
We have examined extensively why the test in Lord’s thought experiment appears to be inadmissible, while at the same time it leads to a scientifically useful and informative conclusion. In doing so we found that Lord’s argument depends on the assumption that the football numbers represent a property on the nominal level. Not only was it shown that it is immaterial to the argument that the numbers represent nominal uniqueness of the players, it was also shown that another property can be
Acknowledgments
We would like to extend our thanks to professors Willam H. Batchelder and R. Duncan Luce for several helpful suggestions on earlier versions of this manuscript.
References (24)
Applications of the theory of meaningfulness to psychology
Journal of Mathematical Psychology
(1985)- et al.
A theory of appropriate statistics
Psychometrika
(1965) Scales and statistics: Parametric and nonparametric
Psychological Bulletin
(1961)- et al.
Weak measurements vs. strong statistics: An empirical critique of S.S. Stevens’ proscriptions on statistics
Educational and Psychological Measurement
(1966) - et al.
Football numbers (continued)
The American Psychologist
(1954) On the statistical mistreatment of index numbers
The American Psychologist
(1954)Additive scales and statistics
Psychological Review
(1953)Scale classification and statistics
Psychological Review
(1960)Measurement scales and statistics: Resurgence of an old misconception
Psychological Bulletin
(1980)Measurement theory and practice
(2004)
Rescaling ordinal data to interval data in educational research
Review of Educational Reseach
The ordinal controversy revisited
Quality & Quantity
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