2.4: Appendix

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2.4.1 Scales of measurement

All variables must take on at least two different possible values (otherwise they would be a constant rather than a variable), but different values of the variable can relate to each other in different ways, which we refer to as scales of measurement. There are four ways in which the different values of a variable can differ.

Identity: Each value of the variable has a unique meaning.
Magnitude: The values of the variable reflect different magnitudes and have an ordered relationship to one another — that is, some values are larger and some are smaller.
Equal intervals: Units along the scale of measurement are equal to one another. This means, for example, that the difference between 1 and 2 would be equal in its magnitude to the difference between 19 and 20.
Absolute zero: The scale has a true meaningful zero point. For example, for many measurements of physical quantities such as height or weight, this is the complete absence of the thing being measured.

There are four different scales of measurement that go along with these different ways that values of a variable can differ.

Nominal scale. A nominal variable satisfies the criterion of identity, such that each value of the variable represents something different, but the numbers simply serve as qualitative labels as discussed above. For example, we might ask people for their political party affiliation, and then code those as numbers: 1 = “Republican”, 2 = “Democrat”, 3 = “Libertarian”, and so on. However, the different numbers do not have any ordered relationship with one another.

Ordinal scale. An ordinal variable satisfies the criteria of identity and magnitude, such that the values can be ordered in terms of their magnitude. For example, we might ask a person with chronic pain to complete a form every day assessing how bad their pain is, using a 1-7 numeric scale. Note that while the person is presumably feeling more pain on a day when they report a 6 versus a day when they report a 3, it wouldn’t make sense to say that their pain is twice as bad on the former versus the latter day; the ordering gives us information about relative magnitude, but the differences between values are not necessarily equal in magnitude.

Interval scale. An interval scale has all of the features of an ordinal scale, but in addition the intervals between units on the measurement scale can be treated as equal. A standard example is physical temperature measured in Celsius or Farenheit; the physical difference between 10 and 20 degrees is the same as the physical difference between 90 and 100 degrees, but each scale can also take on negative values.

Ratio scale. A ratio scale variable has all four of the features outlined above: identity, magnitude, equal intervals, and absolute zero. The difference between a ratio scale variable and an interval scale variable is that the ratio scale variable has a true zero point. Examples of ratio scale variables include physical height and weight, along with temperature measured in Kelvin.

There are two important reasons that we must pay attention to the scale of measurement of a variable. First, the scale determines what kind of mathematical operations we can apply to the data (see Table 2.2). A nominal variable can only be compared for equality; that is, do two observations on that variable have the same numeric value? It would not make sense to apply other mathematical operations to a nominal variable, since they don’t really function as numbers in a nominal variable, but rather as labels. With ordinal variables, we can also test whether one value is greater or lesser than another, but we can’t do any arithmetic. Interval and ratio variables allow us to perform arithmetic; with interval variables we can only add or subtract values, whereas with ratio variables we can also multiply and divide values.

Table 2.2: Different scales of measurement admit different types of numeric operations
	Equal/not equal	>/<	+/-	Multiply/divide
Nominal	OK
Ordinal	OK	OK
Interval	OK	OK	OK
Ratio	OK	OK	OK	OK

These constraints also imply that there are certain kinds of statistics that we can compute on each type of variable. Statistics that simply involve counting of different values (such as the most common value, known as the mode), can be calculated on any of the variable types. Other statistics are based on ordering or ranking of values (such as the median, which is the middle value when all of the values are ordered by their magnitude), and these require that the value at least be on an ordinal scale. Finally, statistics that involve adding up values (such as the average, or mean), require that the variables be at least on an interval scale. Having said that, we should note that it’s quite common for researchers to compute the mean of variables that are only ordinal (such as responses on personality tests), but this can sometimes be problematic.