Levels of Measurement

Measurement is a process in which numbers do the job of summarising, describing, or quantifying some aspect (typically a variable or underlying attribute) of an object, person or philosophical concept. There are many different ways in which we can measure a variable. Some approaches convey more detailed information than others. Whatever the type, measurements can be divided into two main types depending on the type of information they convey. These two types of measurement are qualitative and quantitative.

Qualitative

Qualitative measurements convey descriptive, textual, or language based information about a variable. A qualitative variable describes a known characteristic, e.g. gender, eye colour, ethnicity, home suburb, political party preference, clothing colour, hair colour etc. All of these examples individually vary and are thus variables, however the type of information that is measured is descriptive or categorical.

Quantitative

Quantitative variables on the other hand, convey meaningful numerical information. These types of variables are considered to be more sophisticated because numbers can be mathematically interpreted and manipulated in meaningful ways. For example, someone who weighs 100 kg is twice as heavy than someone who weighs 50 kg. We also know that if the person who weighs 100 kg lost 25% of their weight, they would now weigh 75 kg. In this example, weight is the variable and kg is the unit of measurement. Weight is quantitative as it uses numbers in a meaningful way to convey information about a variable.

Is It Qualitative or Quantitative?

You must be careful when deciding if a measurement of a variable is qualitative or quantitative. There are many situations when the distinction can get tricky. Consider the assignment of the number 1 for males and a number 2 for females. This is often done in large data sets where it is easier to write a 1 or 2 for gender as opposed to typing "male" or "female". Is the data quantitative or qualitative? In this situation the data is still qualitative. The underlying variable "gender" is qualitative and cannot be changed to quantitative with the assignment of a number. In this context, the numbers 1 and 2 are just labels. There is a plethora of examples where numbers are used in qualitative ways, e.g. credit card numbers, phone numbers, Australian Business Number, car number plate, postcode and tax file numbers.

There are also times when qualitative measurements can be combined to become quantitative measurements. For example, gender is a qualitative variable, but the total number of males and females in a class is quantitative. If you are ever in doubt about whether a variable is measured in a qualitative or quantitative way, ask the question:"Can the measurement be used in a mathematically meaningful way? For example, can I tell if one observation was regarded as being higher or lower than another observation?" Using this question in relation to counting males and females in the classroom, we would quickly realise that the count can be used in a mathematically meaningful way, e.g. to determine whether there are more males or females in a class.

In the previous sections we have discussed how measurements of variables can be loosely broken down into two broad categories — qualitative and quantitative. We will now break down these categories into further detail. We will use the four categories proposed by S. S. Stevens in the 1940s. Stevens believed that variables are "measured" in dramatically different ways, and because the statistical analyses that validly can be performed depend on the type of variable being measured, he found it useful to classify by name the different types of variables. At times there has been spirited debate about whether he was right with all his ideas. Nevertheless his notion of four different "levels" of measurement has endured and it is generally acknowledged to be of great help in understanding the nature of measurement.

The table below shows the four scales of measurement proposed by Stevens (1946). As can be seen from the table above the four different measurement scales each differ in the degree of sensitivity of the measurement yielded. They range in order from the nominal scale that is the "crudest" to the ratio scale that is the "most sensitive" or sophisticated.

Nominal

In published literature the nominal level is often referred to as "the labeling scale", "the naming scale", "the categorical scale", and sometimes "the qualitative scale". The nominal scale represents the first, and "lowest" or "weakest" level of measurement. The three higher levels of measurement are, in relative order, the ordinal, interval, and ratio levels. In any nominal measurement, people or objects are placed into separate classes, groups, categories or sets. Each person or object in the set shares at least one common property with all others in that set. If people or objects can be differentiated on the basis of a characteristic being observed – that is, they can be separated on the basis of some underlying trait or characteristic – this would constitute a "measurement" at the nominal level. Some people have taken exception to this process being described as measurement, because any numbers involved cannot be meaningfully added, subtracted, and so on. However, nominal "measurement" in this discussion is accepted as a worthwhile idea.

Some examples of nominal variables include Visa® card numbers, blood group, gender, marital status, occupation, political party, postcode, religious affiliation or type of car driven.

Nominal measurement involves the assignment of numbers to represent classes, groups, categories, or sets of things. Numbers used for group membership are arbitrary and are used merely as labels. Such numbers cannot be added or subtracted or placed in order. No meaningful arithmetic operation can be performed on them. Because categories are mutually exclusive, individuals can belong to one category only for the variable under consideration. Notice that two of the examples given above (postcode and Visa® card number) appear to be more like "real" numbers than some of the others. However, these two do not possess any more numerical meaning than any of the others in the normal sense.

Ordinal

Data that can be rank-ordered (as well as placed into categories) are located at the ordinal level of measurement. The numbers obtained indicate the order rather than any precise quantity, or amount, of the variable. For example, the winners of a marathon race are given in the table below

Here the numbers 1, 2, 3, 4, 5 and 6 do not behave like "proper" numbers. The difference between 5 and 6 is 1 and the difference between 1 and 2 is 1. However the apparent equality of the two differences is not real. For example, James Jones may have thrashed Henry Smith, whereas Jessica Todd may have only just beaten Lee Ning. So differences between adjacent ranks on an ordinal scale are not necessarily equal, even though 1 is just better than 2, which is just better than 3, and so on.

Interval

This scale of measurement (also referred to as the equal interval scale) possesses all the features of the nominal and ordinal scales and possesses the property that the differences or intervals are meaningful. Examples of variables that are measured on an interval scale include temperature in Celsius, calendar time and scores on a test.

The intervals (or differences) between measurements are readily calculated and represent "real" margins. or in other words are measured on a real number line. For example, the difference between 17°C and 24°C is 7°C (or -7°C depending upon the order of the subtraction), however, absolute amounts are unknown. A temperature of 0°C does not mean the absence of temperature; there is no real zero point on the scale. Notice, also, that it should take the same amount of heat to change the temperature, say, in a boiler from 80°C to 87°C as it would from 17°C to 24°C. This is because the temperature changes or differences are each equal to 7°C. For a real zero point the zero measurement is absolute; that is, the amount of the variable being measured should be nothing. For a variable measured at the interval level there is no absolute zero or true "vanishing" point. There is a temperature scale, the Kelvin scale, which is an absolute one in the sense that it does have a true zero. At 0°K (the equivalent of roughly -273°C) the movement of the molecules in the object being measured stops altogether. Measurements on a Kelvin scale temperature gauge comprise ratio measurements that are discussed shortly.

Ratio

This scale possesses all the features of the nominal, ordinal, and interval scales and in addition possesses an absolute or "real" zero. If a measured value of zero arises on such a scale it means that the amount of that variable present is nothing. The ratio level of measurement lends itself to the greatest range of numerical and statistical analyses. Examples of ratio level variables include temperature (measured in Kelvin), pressure, density, height, mass ¹ and speed.

1. Note that we use the term mass not weight. An astronaut in space has mass, but is weightless, implying that weight has an arbitrary zero.