Labels or amounts

An observational unit is the item or individual a datum is collected from. A variable is a characteristic that may change from one observational unit to another. In a veterinary clinic's records the units are the dogs; breed, weight, and vaccination status are three variables recorded about each one. Confusing the two scrambles every later question, because summaries are computed across units, one variable at a time.

Data do not have to arrive as numbers. Photographs, sounds, videos, and text collected on observational units convey meaningful information too: an audio clip of a sparrow is data, and from it a researcher can measure song length in seconds (quantitative) and record the region it was captured in (categorical). The clip is the raw material; the variables are what get measured from it.

A categorical (also called qualitative) variable takes values that are category names or group labels. A quantitative (numerical) variable takes numerical values for a measured or counted quantity, and generally carries units of measure.

The trap is that plenty of labels are written in digits: zip codes, jersey numbers, area codes, ID numbers. One test settles every case:

1. Ask what the mean of the values would tell you.
2. An average temperature of 71.6 degrees Fahrenheit describes a real amount: quantitative.
3. An average zip code of 61311.3 locates nothing: the digits are labels, so the variable is categorical.

The test also runs in reverse. Shoe size supports arithmetic - a size 10 foot is a real half inch longer than a size 9 - so it is quantitative, while shoe brand is categorical. Two variables about the same shoe, two different types.

A discrete quantitative variable can take on a countable number of values. The number of values may be finite or countably infinite, like the whole numbers: a package can hold 3 items or 4, never 3.6. Counts are the standard example.

A continuous quantitative variable can take on any value within an interval: between any pair of possible values sits another possible value. Flight time, mass, and height are measured rather than counted; the number of values they could take is measurable but not countable.

Two cautions keep the split clean. Recording precision does not change the variable: a flight time logged as 94 minutes is still a continuous quantity, rounded. And whole-number values do not make a variable categorical: 2 pets is a genuine count with a genuine mean (a neighborhood can average 1.4 pets per household, and the fraction is informative even though no household has 1.4 pets).

A parameter is a numerical attribute or summary of the variable of interest for a population. A statistic is the same kind of summary for a sample. The mean mass of every apple in an orchard's harvest is a parameter; the mean mass of the 40 apples pulled for a display is a statistic. Same variable, same computation, different group, different name.

The value of a statistic from a certain sample is often not equal to the unknown value of the population parameter, but it can provide the basis for making inferences about it. That gap is not a defect; it is the reason inference exists as a subject.

Notation keeps the scopes apart from the start: $p$ is a population proportion, $\hat{p}$ is a sample proportion. When a poll of 500 app users finds 38% listening daily, $\hat{p} = 0.38$; the $p$ it estimates belongs to all users and is not known. Writing $p = 0.38$ quietly claims the population was measured.