Variables: What Does Each Record Describe?

A spreadsheet can contain names, numbers, photographs, and comments. Before calculating anything, ask what each row describes and what each entry measures. That small pause prevents mistakes such as treating a postal code as a measured distance or calling a sample average a population fact.

Units, Variables, and Summaries

An observational unit is the item or individual from which a datum is collected. A variable is a characteristic that can change from one unit to another. In a study of apartment energy use, an apartment is the unit. Monthly electricity use, floor area, heating type, and a photograph of the meter can all supply information about that apartment.

Apartment	Electricity, kWh	Floor area, square metres	Heating type	Meter photograph	Resident comment
A01	340	65.4	Electric	Image of A01 meter reading	“Usually empty in daytime”
A02	280	54.8	Gas	Image of A02 meter reading	“Someone works from home”
A03	410	79.2	Electric	Image of A03 meter reading	“Three residents”

The apartment identifier helps join records; it is not electricity consumption. The observations in the photograph and text columns can be coded or inspected for useful information. A sound recording might supply a noise-level measurement or a category such as traffic noise. A video might show an event count. Photographs, sounds, videos, and text can therefore contribute meaningful data, but the study still needs a clear definition of what is measured from each source.

A parameter describes a population. It can be a numerical attribute or a numerical summary of the variable of interest. If we want the mean monthly electricity use of every apartment in a particular building, that full-building mean is a parameter. A statistic describes a sample: the mean for 20 measured apartments is a statistic. In symbols, a population mean is often $\mu$ and a sample mean is $\bar{x}$.

The sample mean is usually not exactly the unknown population mean. A different sample may produce a different statistic. Nevertheless, a well-designed sample can provide information for making an inference about the population parameter. “Sample” and “population” refer to the study’s intended group; they cannot be inferred from whether a value has many decimal places.

Worked Example: Two Averages

A school wants the mean height of all 600 current students. It measures a random sample of 30 students and obtains a mean of 166.2 cm. The observational unit is a student, height is the variable, and 166.2 cm is a statistic. The mean height of all 600 students is the parameter. We may use the sample statistic to estimate that parameter, but we cannot rename 166.2 cm as the exact population mean simply because it was calculated correctly.

Categorical or Quantitative?

A categorical, or qualitative, variable records a category name or group label: heating type, transport mode, or whether a lamp works. A quantitative, or numerical, variable records a measured or counted quantity and generally has units: kilometres traveled, grams of leftovers, or number of occupants. Arithmetic on quantitative values has a contextual meaning. Doubling a 20-minute trip gives 40 minutes; averaging apartment labels A01 and A03 has no comparable meaning.

Digits alone do not make a variable quantitative. A bus route number identifies a route; subtracting route 7 from route 9 does not measure a two-unit difference in transportation. A satisfaction response coded 1 through 5 is a category with an order unless the study provides a justified quantitative scale. Always return to what the recorded values mean.

Worked Example: A Cafeteria Record

For each tray, a student records the meal type, the number of unopened milk cartons, and leftover food mass in grams. Meal type is categorical. The carton count and food mass are quantitative because they record a count and a measurement. The tray is the observational unit, even though people chose and ate the meals.

Discrete and Continuous Quantitative Variables

A discrete quantitative variable has a countable set of possible values. The set may be finite, such as the number of functioning bulbs in a box of 12, or countably infinite in an idealized setting, such as a nonnegative whole-number count with no stated upper limit. “Countable” means the possible values can be listed in sequence; it does not require that the list end.

A continuous quantitative variable is modeled as able to take any value in a specified interval. A length between 10 and 20 cm could be 12 cm, 12.5 cm, 12.51 cm, and values between those values. The ideal possible values are not a countable list, and every pair of distinct possible lengths has other possible lengths between them. The quantity is measured rather than counted.

Recorded precision is a separate issue. A ruler may round length to the nearest millimetre, producing a finite list of recorded values. That does not make the underlying length a count of objects. State whether you are discussing the underlying quantity or the rounded records. Likewise, a discrete count can have a very large range and still be discrete.

Variable	Type	Reason
Number of unopened cartons on a tray	Quantitative, discrete	Possible values are whole-number counts
Underlying leftover mass	Quantitative, continuous	The model permits all masses within an interval
Meal type	Categorical	Values name groups
Tray identifier	Categorical identifier	Digits or labels identify a tray, not an amount

Independent Practice

Use the information in each question to make your own calculation, representation, or argument. Explain what your result means in its setting.

Original generated practice for the Fall 2026 course.

Read the rows

A table has one row for each city bus and columns for route label, passenger count at noon, a sound recording of the engine, and a driver comment.

Part a

Identify the observational unit and two variables. Explain how the sound recording and text could convey meaningful data.

Your reasoning

Worked answer and rubric

Parameter or statistic?

The mean age of every bus in the city fleet is unknown. A sample of 15 buses has a mean age of 6.4 years.

Part a

Identify the parameter and the statistic. Explain why 6.4 years need not equal the population mean and why it may still be useful.

Your reasoning

Worked answer and rubric

Numbers can be labels

A delivery record gives postcode 40712, distance 8.6 km, number of parcels 4, and vehicle category “van.”

Part a

Classify each recorded variable as categorical (qualitative) or quantitative (numerical). For each quantitative variable, state its unit or counted item. Explain why the postcode remains categorical even though it contains digits.

Your reasoning

Worked answer and rubric

Countable is not the same as small

Consider (i) the number of occupied seats in a 40-seat bus, (ii) an idealized whole-number event count with no upper bound, and (iii) the underlying time, in seconds, for a bus trip.

Part a

Classify the quantitative variables as discrete or continuous. Explain finite and countably infinite possibilities and what values are possible between two distinct trip times.

Your reasoning

Worked answer and rubric

Design a delivery data sheet

A cooperative has 240 delivery trips in a week and samples 24 trips. For every selected trip it records driver ID (a numerical identifying code), distance in kilometres, number of stops, vehicle type, a photograph of the loaded vehicle, a 10-second audio clip, a video of unloading, and a driver comment. The sample mean distance is 18.5 km. For this exercise, an independently known full-week mean distance is 19.1 km.

Part a

Identify the observational unit and four variables, including at least one quantity and one category. Explain why the driver ID does not change the observational unit.

Your reasoning

Part b

For each of photograph, audio, video, and text, propose one clearly defined piece of information that could be extracted.

Your reasoning

Part c

Identify 18.5 km and 19.1 km as statistic or parameter. Explain their difference and the role the sample value could have if the full-week mean were unknown.

Your reasoning

Part d

Classify distance, stop count, vehicle type, and driver ID as categorical (qualitative) or quantitative (numerical). State the quantitative units. Explain why the numerical driver ID remains categorical despite being written with digits.

Your reasoning

Part e

Classify the underlying distance and stop count as continuous or discrete. Explain why rounding distance to 0.1 km does not turn the underlying quantity into a count.

Your reasoning

Part f

Give a finite discrete trip variable and an idealized countably infinite discrete trip-related variable. Specify their possible values.

Your reasoning

Worked answer and rubric