There are four levels of measurement. Sequentially from the lowest to the highest, they are: 1) nominal measurement, 2) ordinal measurement, 3) interval measurement, 4) ratio measurement. Data with nominal or ordinal measurement are qualitative, while that of interval or ratio measurement are quantitative.

**Nominal or Categorical Scale**

Formal definition

Consider a set of objects x_{1}, x_{2}, …, x_{N}. Suppose the object x_{i} has a real attribute A(x_{i}). Then for each pair of attributes in the collection:

A(x_{i}) = A(x_{j}) if x_{i} and x_{j} are in the same class, and

A(x_{i}) ≠ A(x_{j}) if x_{i} and x_{j} are in different classes.

A nominal scale is a * labeling system *(or labeling), L(x), of objects such that:

L(x

_{i}) = L(x

_{j}) if and only if A(x

_{i}) = A(x

_{j}) and

L(x

_{i}) ≠ L(x

_{j}) if and only if A(x

_{i}) ≠ A(x

_{j})

What does the formal definition tell us? With the phrases “same class” and “different classes” appearing in the formal definition above, it is clear that nominal scale only classifies the objects being examined into several classes/groups and gives labels to each class, so that the objects with equal attributes have equal labels, but the objects with different attributes get different labels. [Note: The label may be a number or other symbols.]

**Example 1**

Suppose that there are five students, namely Anwar, Budi, Cynthia, Dewi, and Erwin. It is also known that Anwar, Budi, and Erwin are males while Cynthia and Dewi are females. Thus, Anwar, Budi, and Erwin are in the same class, and so are Cynthia and Dewi. Suppose that we have the following labeling system: The sex of male students is labeled “M” and that of female students is labeled “F”. It can be easily seen that with such labeling, the data on the sex of the students are of nominal scale.

Alternatively, if the sex of male students is labeled 1 and that of female students is labeled 2, the list changes to:

Note that in either list, two students of different sex (i.e. in different classes) get different labels. Other examples of data with nominal scales include hobbies, blood types, favorite colors, brands of automobiles. Remember that nominal scale only classifies objects into categories. Compare this with ordinal scale, which in addition to classification, it also implies a certain sequence.

**Ordinal Scale**

Formal definition

Consider a set of objects x_{1}, x_{2}, …, x_{N}. Suppose the object x_{i} has a real attribute A(x_{i}). Then for each pair of attributes in the collection:

A(x_{i}) = A(x_{j}) if x_{i} and x_{j} are in the same class,

A(x_{i}) ≠ A(x_{j}) if x_{i} and x_{j} are in different classes, and

A(x_{i}) > A(x_{j}) if x_{i} exceeds x_{j} in terms of the amount of attribute it has.

An ordinal scale is a *labeling system* (labeling), L(x), of objects such that:

L(x_{i}) = L(x_{j}) if and only if A(x_{i}) = A(x_{j}),

L(x_{i}) ≠ L(x_{j}) if and only if A(x_{i}) ≠ A(x_{j}), and

L(x_{i}) > L(x_{j}) if and only if A(x_{i}) > A(x_{j})

In ordinal data, not only there is a classification but there is also a specific ordering in the attributes possessed by the objects under study.

**Example 2**

Clothing size, expressed as S, M, L, XL, is an example of data with ordinal level of measurement. Clothes with size S are smaller than clothes with size M. Clothes with size M are smaller than clothes with size L, and clothes with size L are smaller than clothes with size XL. In this case,. S < M < L < XL.

**Example 3**

The level of satisfaction as perceived by the customers of a service company is another example of data with ordinal level of measurement. Suppose that Ari, Benny, and Candra were all served by a company’s customer service department. Regarding the service, Ari felt very satisfied, Benny was not satisfied (or dissatisfied), while Candra was satisfied. In terms of level of satisfaction, “very satisfied” is better than “satisfied”. “satisfied” is better than “dissatisfied”. In this case, we see that there is an ordering in the levels of satisfaction. If “very dissatisfied” is labeled VD, “dissatisfied” is labeled D, “satisfied” is labeled S, and “very satisfied” is labeled VS then we can state VS > S > D > VD.

Remember that an ordinal scale, in addition to classifying objects into classes, defines some ordering on the classes.

**Interval Scale**

Formal definition

Consider a set of objects x_{1}, x_{2}, …, x_{N}. Suppose the object x_{i} has a real attribute A(x_{i}). Then for each pair of attributes in the collection:

A(x_{i}) = A(x_{j}) if x_{i} and x_{j} are in the same class,

A(x_{i}) ≠ A(x_{j}) if x_{i} and x_{j} are in different classes, and

A(x_{i}) > A(x_{j}) if x_{i} exceeds x_{j} in terms of the amount of attribute it has.

An interval scale is a labeling system (labeling), L(x), of objects such that:

L(x_{i}) = L(x_{j}) if and only if A(x_{i}) = A(x_{j}),

L(x_{i}) ≠ L(x_{j}) if and only if A(x_{i}) ≠ A(x_{j}),

L(x_{i}) > L(x_{j}) if and only if A(x_{i}) > A(x_{j}), and

L(x) = c⋅A(x) + b for some c > 0.

**Example 4**

The size of women’s clothing by American standards is data with interval scale. Consider the following list.

If the attribute of interest is the waist size, the relationship between the dress size L(x) and the waist size A(x) can be expressed as L(x) = A(x) – 24, where A(x) = waist size in inches. But if the attribute that we are observing is the size of the thigh, then the relationship between the size of the thigh and the size of the dress can be expressed as L(x) = A(x) – 27.

Other examples of data with interval scale are temperature and IQ scores. It is important to note that in interval scale, 0 (zero) is not absolute. This means that if the label is zero, it does not mean that the object has no attribute under study. For example, if the temperature of a solution is zero, it does not mean that the solution does not have the heat. Different from data with ratio scale, the zero in interval-scaled data is relative.

**Ratio Scale**

Formal definition

Consider a set of objects x_{1}, x_{2}, …, x_{N}. Suppose the object x_{i} has a real attribute A(x_{i}). Then for each pair of attributes in the collection:

A(x_{i}) = A(x_{j}) if x_{i} and x_{j} are in the same class,

A(x_{i}) ≠ A(x_{j}) if x_{i} are x_{j} are in different classes, and

A(x_{i}) > A(x_{j}) if x_{i} exceeds x_{j} in terms of the amount of attribute it has.

A ratio scale is a labeling system (labeling), L(x), of objects such that:

L(x_{i}) = L(x_{j}) if and only if A(x_{i}) = A(x_{j}),

L(x_{i}) ≠ L(x_{j}) if and only if A(x_{i}) ≠ A(x_{j}),

L(x_{i}) > L(x_{j}) if and only if A(x_{i}) > A(x_{j}), and

L(x) = c⋅A(x) for some c > 0.

Data with ratio scale can be viewed as a special case of interval-scaled data. By examining the formal definitions of the scales, L(x) = c⋅A(x) is nothing but L(x) = c⋅A (x) + b where b = 0. In ratio-scaled data, the zero is absolute. If the label of an object is zero, it means that the object has no attribute under study. To illustrate this, if the number of students attending a lecture is zero, then it indicates that there is no student attending the lecture. If the current in a wire is zero, we can conclude that there is no electricity flowing in the wire. Furthermore, in the ratio scale, if the label of the first object L(x_{1}) is equal to k times that of the second object L(x_{2}), then the attribute of the first object A(x_{1}) is also equal to k times that of the second object A(x_{2}). Mathematically, it can be written as L(x_{1}) = k⋅L(x_{2}) ⇔ A(x_{1}) = k⋅A(x_{2}). Look at the following examples.

**Example 5**

Suppose that wires A and B are 6 meters and 2 meters long, respectively. From this, we can conclude that the length of wire A is 3 times that of wire B.

**Example 6**

Suppose that weight plates A and B have the mass of 10 kg and 2 kg, respectively. Then, we can conclude that the mass of weight plate A is 5 times that of B.

In interval scale, such conclusion is not valid. For example, suppose that solution A has a temperature of 400 degrees centigrade and the temperature of solution B is 100 degrees centigrade. Solutions A and B, if measured in Fahrenheit, are 752 ^{0}F and 212 ^{0}F, respectively. But note that 752 ^{0}F ≠ 4 x 212 ^{0}F, while 400 ^{0}C = 4 x 100 ^{0}C. There is some inconsistency regarding the ratio, if the temperature is measured in centigrade and Fahrenheit. In the ratio scale, consistency will always be obtained, regardless of the units being used. Suppose that wire A has a length of 10 meters = 1000 cm and wire B has a length of 2 meters = 200 cm. The length of wire A is 5 times that of wire B, no matter they are measured in meters or centimeters!

Examples of data with ratio scales include the length (measured in meters or other units of length), mass (measured in kilograms or other units of mass), time (measured in seconds or other units of time), (electric) current (measured in Amperes or other units of current) , area (measured in m^{2} or other units of area), volume (measured in m^{3} or other units of volume), pressure (measured in Pascals or other units of pressure).

AN IMPORTANT NOTE ABOUT THE SCALES

Levels of measurement are determined by the labeling applied to the attributes being examined. Consider the following example. Suppose that we label the temperature of liquids according to the rule represented by the table below.

Suppose that the temperatures of 5 liquids are as follows.

By the qualitative labeling “cold”, “warm”, and “hot” specified above, the data can be recorded as follows.

Note that if the temperature data are recorded with the qualitative labeling, they are of ordinal scale. But if the they are recorded in degrees centigrade, they are of interval scale.

So, **the level of measurement of data depends not only on the attribute under study, but also on how the labeling system is applied**.