What is a Variable; Definition
A variable is defined based on how its elements are measured. This can define how will be analyzed statistically. The form that its elements can take are described below. A variable can contain information for example on age, height, or Gender, or the type of Cars. Variables can be divided in two main categories based on the form that its elements are.
i) Discrete (Non-Continuous) or Qualitative or Categorical and
ii) Continuous or Quantitative
Explanation of Terminology
The multiple terminology that is presented in (a) and (b) points try to explain the different properties that a variable of the 1st and 2nd type can have by using different “view angles”. These terminologies are compared and analyzed below.
Discrete (Non-Continuous) and Continuous variables
i) Discrete Variables
The variables that can be characterized as Discrete, and therefore as Non-Continuous variables, are those variables that have elements which are integers (1, 2, 3, 4, 5, 6, ….) and the decimal digits have no meaning. The number of students in a class is an example. Note that the value of 2.43 or 3.456 cannot be identified to a single student, it does not exist! Also, such variables have a finite set of numbers. That is, no infinite students can exist. Another such example is the Gender. It is not possible to measure 2.373 males or females but you can measure 100 or 34 males or females. Moreover, the number of males or females are very specific, it is not an infinitive number.
ii) Continuous Variables
The variables that are characterized as Continuous Variables, have elements which can take all the value of the arithmetic scale such as decimal digits: 1.2, 3.4, 5.6. Weight, Age, or Height are examples of such variables. You can have a height of 1.74.
Quantitative and Qualitative Variables
i) Qualitative Variables
These variables that its elements were not measured / did not measure something, that is, they cannot be added, subtracted, divided, or multiplicated, then, these variables can be called “Qualitative Variables”. Qualitative or Discreet variables can be the Gender, the height of your wages in ranges, or your eye color.
ii) Quantitative Variables
These variables that its elements are measured somehow / have measured something, and therefore, they can be quantified, then these variables can be called “Quantitative Variables”. When, some mathematical operations such as addition or subtraction can be performed on such elements, then they have measured on an Interval Scale. When addition or subtraction, and Multiplication or division can be performed on such elements, then they have measured on a Ratio Scale. Quantitative variables can be age, or height or distance.
i) The Discrete or Qualitative or Categorical variables can be divided into:
a) Nominal variables (Gender, Religion) and
b) Ordinal variables (Relative Size of a Country)
Nominal variables can be characterized those variables which its elements are “names” / “categories” and therefore, the concept of “Greater than” or “Lower than” cannot be applied. A such example is Gender. The only logical concept that can be extracted from such type of variables are that its categories are amiable exclusive: e.g. “males” cannot be “females” and “females” cannot be “males” in the same time. Similar examples are the Religion and the eye color. Someone can claim that is both “Muslim” (this person who worship Islam) and “Christian”. Then, a new category must be used in the related variable.
Ordinal variables can be characterized those variables which its elements can be ordered e.g. from the lowest to the highest element, or from the largest to the smallest element. A such example can be the relative size of a country such as America which is larger than Italy which is larger than Malta. Note that the position of an element here plays a significant role while e.g. in Gender, the position of its elements cannot affect the results. So, if America is identified by “1”, Italy by “2”, and Malta by “3”, this does not mean that these numbers can be subtracted or added or multiplicated or divided. Their differences are qualitative, not quantitative.
ii)The Quantitative variables can be divided into:
a) Variables of Interval scale (Temperature in Celsius degrees, students’ Grades) and
b) Variables of Ratio scale (Weight, Height, Age)
A Variable can be characterized as Interval if the elements that includes are measured in an interval scale. In that scale: a) the difference of distance between two values must be the same with the difference of distance between two other values in the same scale of reference, and b) therefore, the starting point is set arbitrarily and thus, the zero “0” has no the meaning of “nothingness”. This happens because a such scale can be arbitrarily applied on events / situations in order these events / situations can be partly quantified.
Example Ι: Temperature in Celsius degrees: The temperature of zero “0” does not mean that there is no at all “temperature”. Moreover, the difference that exists in the distance between the 5 and 6 Celsius degrees is the same difference that exists in the distance between the 0 and 1 Celsius degrees. Note that here you cannot say that “40 Celsius degrees” is the double of 20 Celsius degrees. The results of multiplications or divisions have no true meaning here. This can become clearer when these Celsius degrees are transformed into Kelvin scale. A Kelvin scale has a true zero “0” point which means total absence of temperature. So, the 50 Celsius degrees equals 313.15 Kelvin degrees while the 20 Celsius degrees equals 293.15 Kelvin Degrees. That is, the first Kelvin value is not the double of the latter one.
Example ΙΙ: The grades in school. A grade of zero “0” does not mean that your knowledge on e.g. maths or literature are also “zero”. Moreover, if a student has a grade of 8 in comparison with another student that has a grade of 4, this does not mean that the knowledge of the first student in a topic is double from the knowledge on the same topic from the other student. Moreover, if the best degree is assigned to a student in a topic, this does not mean that this student also has absolute knowledge of this topic.
A Ratio variable can be characterized the variable that its elements have measured in a ratio scale, that is, that the zero “0” point has its true meaning, nothingness. Age or Height or Weight are measured in a ratio scale. That is, zero height or zero weight means the total absence of height or weight. Similar examples are the speed and the distance.
The following graph shows the classification system of variables depending on their measurement scale as well some examples per variable type and what mathematical concepts can apply to each case.
Age: Continuous Ratio
When age is measured in full depth, that is, decimal digits have not been prohibited to exist, a ratio scale has been used. That is, an Age of 22.50 indicates that this person is 22 years old plus 6 months (6/12=0.50).
Age: Continuous Interval
When age is measured in rounding years such as 20, 37, 64, then its values contain interval information BUT not ratio information. That is, some information on age is lost. Based on the previous example, when the age of 22.5 has been rounded to 23 years, then, we cannot retrieve anymore these 6 months. This information has been lost. The attribute of the age variable has been downgraded from Continuous Ratio to Continuous Interval.
Age: Discrete Ordinal
When age is measured in Ranges such as 21-30, 31-40, 41-50, it includes a nature of order, the first age range is lower than the second one etc.Here, we know only the Age Range that a person belongs and not the exact age of this person. That is, some information on age has been lost. Based on the previous example, we will only know that this person -aged 22.50- can have an age that is ranged from 21 years to 30 years, and nothing else. The attribute of this variable has been downgraded from Continuous Interval to Discrete Interval.
Age: Discrete Nominal
When the Age has been categorized into Young (e.g. below or equal to 40 years) and Old (e.g. above 40 years) and these categories / labels are used in the same way that someone treats the labels in Gender variable (Males / Females), then, this Age variable can be called “Nominal / Categorical variable”. That is, the only type of information that can be extracted from such variables is that the elements of one category is mutually exclusive from the elements of the other category/ies, e.g. those that were characterized as Young cannot be in the same time Old.
Conclusion based on the Example given
Note that the elements of a variable that has been measured on a Continuous Ratio scale include all the possible information that can exist from a measurement / scale while when these elements have been collected using a Discrete Nominal scale, then these elements include the least possible information that can exist from a measurement / scale. Notice that sometimes it is appropriate the measurement scale of the elements to be downgraded for some Research aims and designs.