Why you cannot say a 20°C day is twice as hot as 10°C
In short: In statistics, not all numbers are created equal. Depending on the scale of measurement, a simple multiplication might be mathematically meaningless. Understanding these four levels of measurement is the difference between precise scientific insight and making fundamental errors in data analysis.
In 1946, psychologist Stanley Smith Stevens introduced a landmark classification for scientific measurement: nominal, ordinal, interval, and ratio scales. This framework was designed to distinguish between qualitative data, such as categories like nationality or genre, and quantitative data. While Stevens's typology remains ubiquitous in psychology, it has faced significant scrutiny. For instance, mathematical psychologists like R. Duncan Luce argued that Stevens's definition of measurement lacked the necessary rigor, particularly regarding the use of scales to define empirical laws.
The hierarchy functions through increasing mathematical complexity. A nominal scale merely classifies items into distinct categories, such as biological species or parts of speech, where numbers act only as labels without numerical value. An ordinal scale introduces rank order, such as a student's class position, but it does not guarantee equal intervals between ranks. This is why an IQ score—an ordinal measure—cannot be used to claim one person is twice as intelligent as another.
The distinction becomes most critical at the interval and ratio levels. Interval scales, like the Celsius temperature scale or dates, allow us to define the degree of difference between points, but they lack a true zero. Consequently, you can say a 10-degree difference is twice as large as a 5-degree difference, but you cannot say 20°C is 'twice as hot' as 10°C. Only ratio scales, used extensively in physical sciences for properties like mass or length, possess an absolute zero, allowing for meaningful multiplication and division.
Modern theorists have expanded these boundaries. Mosteller and Tukey proposed a seven-level system in 1977 to capture complexities like percentages, while Nicholas R. Chrisman introduced a ten-level model to account for cyclical measurements, such as clock time, and graded membership categories.
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- Summary by Intellect · source: Level of measurement
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