Measurement scales

Five different scales are used to classify measurements based on how much information each measurement conveys.

The different levels of measurement involve different properties of the numbers or symbols that constitute the measurements and also an associated set of permissible transformations.

Measurement scale Properties Permissible transformation
(click to enlarge) Nominal Two things are assigned the same symbol if they have the same value of the attribute.

For example, Gender (Male, Female); Religion (coded as 0=None, 1=Christian, 2=Buddhist).

Permissible transformations are any one-to-one or many-to-one transformation, although a many-to-one transformation loses information.
(click to enlarge) Ordinal Things are assigned numbers such that the order of the numbers reflects an order relation defined on the attribute. Two things x and y with attribute values a(x) and a(y) are assigned numbers m(x) and m(y) such that if m(x) > m(y), then a(x) > a(y).

For example, Moh's scale for the hardness of minerals; academic performance grades (A, B, C, ...).

Permissible transformations are any monotone increasing transformation, although a transformation that is not strictly increasing loses information.
(click to enlarge) Interval Things are assigned numbers such that differences between the numbers reflect differences of the attribute. If m(x) - m(y) > m(u) - m(v), then a(x) - a(y) > a(u) - a(v).

For example, Temperature measured in degrees Fahrenheit or Celsius.

Permissible transformations are any affine transformation t(m) = c * m + d, where c and d are constants; another way of saying this is that the origin and unit of measurement are arbitrary.
(click to enlarge) Ratio Things are assigned numbers such that differences and ratios between the numbers reflect differences and ratios of the attribute.

For example, Temperature measured in degrees Kelvin scale; Length in centimeters.

Permissible transformations are any linear (similarity) transformation t(m) = c * m, where c is a constant; another way of saying this is that the unit of measurement is arbitrary.
(click to enlarge) Absolute Things are assigned numbers such that all properties of the numbers reflect analogous properties of the attribute.

For example, Number of children in a family, Frequency of occurrence.

The only permissible transformation is the identity transformation.

While the measurement scale cannot determine a single best statistical method appropriate for data analysis, it does define which statistical methods are inappropriate. Where possible the use of a variable is restricted when its measurement scale is not appropriate for the analysis. For example, a nominal variable cannot be used in a t-test.

When the measurement scale of a variable is unknown, the scale is inferred from its role in the analysis and the type of data in the variable. If the measurement scale cannot be inferred, you must set the measurement scale.