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Distinguish qualitative and quantitative variables, discrete and continuous measurements, and the underlying quantity from its rounded record. Original examples and worked practice.
Before you startUnderstand a variable as something recorded for each unit.
01 / Meaning before appearance
Qualitative: categories or attributes. Quantitative: numerical amounts.
For quantitative variables, distinguish discrete allowed values from a continuous range.
Route 302 is a label for a bus service; 30 passengers is a count. Both contain digits, but they represent different kinds of information. Look at what is recorded and whether arithmetic describes a meaningful amount.
Qualitative
A route code identifies a category. Arithmetic on route numbers does not measure an amount.
Classify the meaning of the variable, not just whether its recorded value contains digits. The notes below cover the distinctions without needing these controls.
02 / Categories and labels
Eye colour, method of travel and type of fuel are categories. A database may code them as 1, 2 and 3 without turning them into measured quantities. Nominal categories have no inherent ranking. Ordinal categories, such as poor/fair/good/excellent, have an order, but the gaps between categories are not necessarily equal.
A survey records car colour as 1 = blue, 2 = red, 3 = white. Is car colour quantitative?
The code stands for a category.
No. Colour is qualitative. The numerical coding does not make the difference between red and white a measured quantity.
Classify satisfaction recorded as very dissatisfied, dissatisfied, neutral, satisfied or very satisfied.
There is an order, but no measured numerical spacing.
Qualitative, ordinal data. The categories are ranked, but equal gaps between categories are not established.
03 / Numerical quantities
Counts, lengths, masses and durations are quantitative. Units matter: a journey time of 30 minutes and a journey distance of 30 kilometres measure different variables. Ask what one data value means before calculating summaries.
Route code 302: qualitative identifier
Adding route labels has no useful quantitative interpretation.
Passenger count 30: quantitative, discrete
Passengers are counted in whole units.
Journey time 23.5 minutes: quantitative, continuous
Duration is measured; the recorded precision may be limited.
Pause, replay or seek freely. The notes explain the same idea and stay in view.
A parcel record contains “delivery zone 4” and “mass 4 kg”. Classify each variable.
One four is a label; the other is a measured amount.
Delivery zone is qualitative categorical data. Mass is quantitative continuous data, measured in kilograms.
04 / Discrete quantitative variables
Numbers of customers, defects or messages are integer counts. A discrete variable can also use fixed fractional steps, such as shoe sizes in a system allowing only whole and half sizes. Discrete does not mean “must be an integer” or “has only a few possible values”.
Classify the number of emails received in a day. Could 4.5 be a possible observation?
Each email contributes one to the count.
Quantitative discrete. An individual daily count is a nonnegative integer, so 4.5 is not a possible count.
The mean number of children per household in a sample is 1.7. Does this make the variable continuous?
An average need not itself be an attainable observation.
No. Each household’s number of children is an integer count, so the variable is discrete. The average can be fractional.
A shoe-size system allows 6, 6.5, 7, 7.5 and so on. Is shoe size continuous?
Not every value between neighbouring sizes is allowed.
No. The allowed sizes are discrete, despite some being fractional.
05 / Continuous quantitative variables
Lengths, masses, durations and temperatures are normally modelled as continuous. In practice instruments round or limit precision, but the underlying quantity is still continuous in the statistical model. “Continuous” does not mean that every imaginable real number is physically possible; there may be bounds.
Relative humidity is recorded as a percentage. Does its range from 0% to 100% make the underlying variable discrete?
A bounded interval can still contain continuously varying values.
No. Relative humidity is modelled as continuous; it can vary within that interval. Recording it to whole percentages limits the recorded precision.
06 / Underlying versus recorded values
Rounding a continuous measurement produces a record on a discrete grid.
It does not change the nature of the underlying variable.
A time recorded as 12 seconds to the nearest second represents an underlying duration around 12 seconds, not necessarily exactly 12. In a question asking for the type of “time”, continuous is appropriate; if asked about the set of possible rounded records, those occur at integer-second steps. Explain the distinction.
Student heights are recorded to the nearest centimetre. Classify the underlying variable and the possible recorded values.
Separate height from its rounded representation.
Underlying height is continuous quantitative. The rounded records lie on a whole-centimetre grid and so form a discrete set of recorded values.
Compare exact age as elapsed time since birth with age in completed whole years.
The measurement definitions differ.
Exact elapsed age is continuous. Age in completed whole years takes integer values and is discrete. State which definition the study uses.
07 / Classify and justify
A delivery study records vehicle registration, number of stops and exact route distance. Classify each with a reason.
One is an identifier, one a count and one a measurement.
Registration is a qualitative identifier. Number of stops is quantitative discrete because it is a count. Exact route distance is quantitative continuous because it is measured on a continuous length scale.
Section 1 of 7 · Meaning before appearance