Data, Variables, and Where Statistics Begins

Every day you are handed numbers and asked to believe something: a poll says a candidate leads by four points, a headline claims a new diet cuts risk in half, an app tells you your average screen time rose 12%. Statistics is the discipline that lets you decide which of those claims deserve your trust. Before we can compute anything, though, we need a shared vocabulary for the raw material of the subject — data — and for the questions we ask of it.

Populations, samples, and why we bother

A population is the entire group we want to learn about: every registered voter in a country, every bolt produced by a factory this year, every user of an app. A sample is the subset we actually observe. We almost never measure the whole population — it is too expensive, too slow, or literally impossible — so we measure a sample and generalize carefully.

This split gives statistics its two central objects. A number that describes a population is a parameter (for example, the true mean height μ of all adults in a city). A number computed from a sample is a statistic (the mean height x̄ of the 250 adults we measured). Parameters are fixed but unknown; statistics are known but vary from sample to sample. Most of this course is about the disciplined leap from statistic to parameter.

Types of variables

A variable is any characteristic recorded for each individual in a study. Variables come in two broad families:

• Categorical (qualitative) variables sort individuals into groups: major, blood type, favorite streaming service. The values are labels, and arithmetic on them is meaningless — you cannot average blood types.

• Quantitative (numerical) variables are genuine measurements or counts: height, income, number of siblings. Quantitative variables split further into discrete (countable values, like number of pets — you cannot own 2.4 cats) and continuous (any value in an interval, like body temperature or commute time).

One subtlety trips up beginners: numbers are not automatically quantitative. A zip code and a jersey number are written with digits, but they are labels — categorical data in numeric costume. Ask yourself whether averaging the values would mean anything. The average of zip codes 90210 and 10001 is not a place. Some categorical variables do carry a natural order (poor, fair, good, excellent); we call these ordinal, and the order matters even though the gaps between levels are not defined.

A worked example: the campus coffee survey

Suppose the manager of a university coffee shop wants to know how much the roughly 18,000 students on campus spend on coffee per week. Interviewing all 18,000 is impractical, so one Tuesday she surveys 120 students chosen at random from the registrar's list. Each student reports four things: their class year (first-year, sophomore, junior, senior), whether they own an espresso machine (yes/no), the number of coffee drinks they bought last week, and their total spend in dollars.

Let us classify everything. The population is all 18,000 enrolled students; the sample is the 120 surveyed. Class year is categorical and ordinal (the years have an order). Espresso-machine ownership is categorical (two labels). Number of drinks is quantitative and discrete — you buy 0, 1, 2, … drinks, never 3.7. Weekly spend is quantitative and continuous for practical purposes, since it can land on any value like $11.35. If the 120 students spent an average of x̄ = $14.20, that figure is a statistic. The unknown average across all 18,000 students, μ, is the parameter the manager actually cares about.

Notice one more design decision hiding in plain sight: the students were chosen at random. Had she surveyed only the people standing in her own shop, espresso enthusiasts would be wildly overrepresented and every later calculation would inherit that bias. No formula in this course can rescue data collected badly — a lesson worth learning before any formulas at all.

Why this matters

Every technique you will meet — histograms, confidence intervals, hypothesis tests — assumes you know what kind of variable you hold and whether your numbers describe a sample or a population. Choosing a tool that mismatches the data type is the single most common statistical error in the wild, from business dashboards averaging satisfaction ratings to news stories confusing one sample's result with the truth about everyone. Get the vocabulary right and the rest of the course clicks into place.

Curriculum aligned with OpenStax's Introductory Statistics 2e; all lesson text is original to Syllabus.