10.02 Categorical and numerical data

In statistics, a 'variable' refers to a source (of data) that is measurable or observable. A variable could be something like temperature, mass, height, make of car, type of animal or goals scored. In most cases, we would expect a variable to change between each observation.

Types of data

Data variables can be defined as either numerical or categorical.

• Numerical data is where each data point is represented by a number. Examples include: number of items sold each month, daily temperatures, heights of people, and ages of a population. The data can be further defined as either discrete (associated with counting) or continuous (associated with measuring). Numerical data is also known as quantitative data.
   
• Categorical data is where each data point is represented by a word or label. Examples include: brand names, types of animals, favourite colours, and names of countries. The data can be further defined as either ordinal (it can be ordered) or nominal (un-ordered). Categorical data is also known as qualitative data.

 

Discrete numerical data

Discrete numerical data involve data points that are distinct and separate from each other. There is a definite 'gap' separating one data point from the next. Discrete data usually, but not always, consists of whole numbers, and is often collected by some form of counting.

Examples of discrete data:

Number of goals scored per match	$1$1, $3$3, $0$0, $1$1, $2$2, $0$0, $2$2, $4$4, $2$2, $0$0, $1$1, $1$1, $2$2, ...
Number of children per family	$2$2, $3$3, $1$1, $0$0, $1$1, $4$4, $2$2, $2$2, $0$0, $1$1, $1$1, $5$5, $3$3, ...
Number of products sold each day	$437$437, $410$410, $386$386, $411$411, $401$401, $397$397, $422$422, ...

In each of these cases, there are no in-between values. We cannot have $2.5$2.5 goals or $1.2$1.2 people, for example.

This doesn't mean that discrete data always consists of whole numbers. Shoe sizes, an example of discrete data, are often separated by half-sizes. For example, $8$8, $8.5$8.5, $9$9, $9.5$9.5. Even still, there is a definite gap between the sizes. A shoe won't ever come in size $8.145$8.145.

 

Continuous numerical data

Continuous numerical data involves data points that can occur anywhere along a continuum. Any value is possible within a range of values. Continuous data usually consists of decimal numbers, and is often collected using some form of measurement.

Examples of continuous data:

Height of trees in a forest (in metres)	$12.359$12.359, $14.022$14.022, $14.951$14.951, $18.276$18.276, $11.032$11.032, ...
Times taken to run a $10$10 km race (minutes)	$55.34$55.34, $58.03$58.03, $57.25$57.25, $61.49$61.49, $66.11$66.11, $59.87$59.87, ...
Daily temperature (degrees C)	$24.4$24.4, $23.0$23.0, $22.5$22.5, $21.6$21.6, $20.7$20.7, $20.2$20.2, $19.7$19.7...

In practice, continuous data will always be subject to the accuracy of the measuring device being used.

 

Ordinal categorical data

The word 'ordinal' basically means 'ordered'. Ordinal categorical data involves data points, consisting of words or labels, that can be ordered or ranked in some way.

Examples of ordinal data:

Product rating on a survey	good, satisfactory, good, excellent, excellent, good, good, ...
Exam grades	A, C, A, B, B, C, A, B, A, A, C, B, A, B, B, B, C, A, C, ...
Size of fish in a lake	medium, small, small, medium, small, large, medium, large, ...

Nominal categorical data

The word 'nominal' basically means 'name'. Nominal categorical data consists of words or labels, that name individual data points.

Examples of nominal data:

Nationalities in a sporting team	German, Austrian, Italian, Spanish, Dutch, Italian, ...
Make of car driving through an intersection	Toyota, Holden, Mazda, Toyota, Ford, Toyota, Mazda, ...
Hair colour of students in a class	blonde, red, brown, blonde, black, brown, black, red, ...

Nominal data is often described as 'un-ordered' because it can't be ordered in a way that is statistically meaningful.

 

Practice questions

QUESTION 1

QUESTION 2

Question 3