Source: STA 270-012 Lecture Slides, Fall 2025 (Section 1.1)
Difficulty: Introductory | Prerequisites: None. This is the first section of the course.
This section lays the groundwork for everything else in STA 270. Before you can analyse data, you need a shared vocabulary: what counts as a population, what counts as a sample, what types of variables exist, and how variables relate to each other. These distinctions matter because they determine which statistical methods you are allowed to use later in the course. If you are joining late or missed the first week, start here.
Statistics is about drawing conclusions from data when information is limited and uncertain. Data is structured around populations (the full group you care about), samples (the subset you actually observe), and variables (the characteristics you measure). Variables split into categorical (nominal and ordinal) and quantitative (discrete and continuous), and understanding which type you have dictates the analysis you can run.
Statistics
A set of scientific principles and techniques for reaching conclusions about populations and processes when available information is limited and variable. It encompasses designing data collection, making inferences from samples, and analysing numerical data under uncertainty.
In simple terms, statistics is the toolkit you use to make sense of data and draw conclusions when you can't observe everything.
Descriptive statistics
The branch of statistics concerned with collecting, processing, visualising, summarising, and describing sampled data.
Think of it as the "what happened" side of statistics: you are organising and presenting data you already have.
Inferential statistics
The branch of statistics concerned with making decisions and drawing conclusions about populations based on sample data.
Think of it as the "what can we conclude" side: you are using a sample to say something about a larger group.
Unit (case)
A single entity (usually a person or an object) whose characteristics are of interest.
In simple terms, one row in your dataset. Each person surveyed, each apartment listed, each country measured is a unit.
Population
The complete collection of units (cases) about which information is sought.
Think of it as the entire group you care about, whether or not you can actually reach all of them.
Finite population
A population containing a finite (countable, limited) number of units. Example: all students admitted to a university for a given term.
Infinite population
A population containing an infinite number of units. Example: all books to be published after 2025 (the total is unknown and ever-growing).
Sample
A subset of a population, selected to act as a representative of the whole.
In simple terms, the slice of the population you actually collect data from.
Sample of size n
A sample that contains exactly n units.
Variable
A characteristic measured or recorded about each unit (case) in a population or sample.
Think of it as a column in your dataset: age, rent, ZIP code, number of employees.
Qualitative (categorical) variable
A variable that yields non-numerical information. Arithmetic operations (adding, averaging) are not meaningful for categorical data, even when the categories are coded as numbers.
In simple terms, a variable whose values are labels or group names rather than quantities.
Nominal variable
A categorical variable that places a unit into one of several groups or categories with no inherent order.
Examples: gender, breed of animal, colour, brand name.
Ordinal variable
A categorical variable that incorporates an ordered position or ranking.
Examples: level of preference (low, medium, high), level of satisfaction, class ranking.
Quantitative (numerical) variable
A variable that yields numerical measurements for which arithmetic operations are meaningful.
In simple terms, a variable you can meaningfully add, subtract, or average.
Discrete variable
A quantitative variable that can assume only a countable number of values. Counts are discrete variables.
Examples: number of defects, number of students registered in a course.
Continuous variable
A quantitative variable that can take any one of an uncountable number of values within an interval. Measurements are continuous variables.
Examples: height, weight, temperature, GPA, wind speed.
Explanatory variable (independent variable, input variable, X)
The variable used to help understand or predict values of another variable.
Response variable (dependent variable, output variable, Y)
The variable whose values are being understood or predicted.
Statistics sits at the intersection of data collection and decision-making. The formal definitions all converge on the same idea: you have limited, variable information and you need to reach a conclusion anyway.
Three formal angles on the same discipline:
Scientific principles and techniques for reaching conclusions about populations when information is limited and variable
Principles for designing data collection and making inferences about a population from sample information
Analysis of numerical data for the purpose of reaching a decision or communicating information under uncertainty
The field splits into two broad areas:
Descriptive statistics covers collection, processing, visualisation, summarisation, and description of the data you have in hand. You are looking at your sample and organising what you see.
Inferential statistics covers making decisions and drawing conclusions about a larger population. You are using your sample to say something about the group it came from.
Descriptive statistics comes first in the workflow. You collect and describe your data before you attempt to generalise from it.
A population is the complete collection of units you want to learn about. A sample is the subset you actually observe.
A finite population has a countable, limited number of units (all students admitted to a university for a given term).
An infinite population has an unlimited number of units (all books to be published after 2025).
A sample of size n contains exactly n units drawn from the population.
The sample's job is to be representative of the population, so conclusions drawn from the sample can reasonably extend to the whole group.
Population examples from the lecture:
All daily newspapers published in the United States (finite)
All companies listed on the New York Stock Exchange (finite)
All students admitted to NAU for Fall 2025 (finite)
All books to be published after 2025 (infinite)
A variable is a characteristic recorded about each unit. In a dataset, each case (unit) is a row and each variable is a column.
Variables divide into two main families:
Qualitative (categorical) variables yield non-numerical information. Even when categories are coded as numbers (tested positive = 1, negative = 0; ZIP codes), arithmetic operations on those numbers are meaningless.
Two sub-types:
Nominal: categories with no inherent order (gender, breed of animal, colour, brand name)
Ordinal: categories with a meaningful order or ranking (level of satisfaction, preference ranking)
Quantitative (numerical) variables yield numerical measurements where arithmetic is meaningful.
Two sub-types:
Discrete: only a countable number of possible values. Counts are discrete (number of defects, number of students enrolled).
Continuous: any value within an interval, an uncountable number of possibilities. Measurements are continuous (height, weight, temperature, GPA, wind speed).
Distinguishing variable types matters because the appropriate statistical analysis method depends on which type of variable you are working with.
When studying the relationship between two variables, both are measured on the same cases (units, subjects, or individuals).
If one variable is used to understand or predict the other:
The explanatory variable (also called the independent variable, input variable, or X) is the one doing the predicting or explaining.
The response variable (also called the dependent variable, output variable, or Y) is the one being predicted or explained.
Examples from the lecture:
Predicting crop yield from yearly rainfall: explanatory = rainfall (X), response = crop yield (Y)
Predicting exam score from hours studied: explanatory = hours studied (X), response = exam score (Y)
Predicting change in body temperature from amount of alcohol given to mice: explanatory = amount of alcohol (X), response = change in body temperature (Y)
Not every pair of variables has a clear explanatory/response relationship. When studying SAT math scores and SAT verbal scores, neither naturally predicts the other, so it is more reasonable to simply explore the relationship between them rather than designating one as explanatory.
Credit companies use statistics to decide consumers' credit levels, combining variables such as income, payment history, and existing debt to assign a score. Political pollsters use sample surveys to predict election outcomes, drawing conclusions about millions of voters from a sample of a few thousand. Public health researchers map disease rates (such as kidney cancer death rates by county) to identify geographic patterns and allocate resources.
Students often think that because a categorical variable is coded as a number (e.g. ZIP code, or 0/1 for male/female), it becomes a quantitative variable. It does not. The coding is for convenience; arithmetic on those numbers is still meaningless.
Students sometimes confuse ordinal and nominal variables. If the categories have a natural ranking (e.g. satisfaction: low, medium, high), the variable is ordinal, not nominal.
A discrete variable can still take a very large number of values. "Discrete" means countable, not "small." The number of grains of sand on a beach is discrete (you could, in principle, count them), even though the number is enormous.
Students sometimes assume that every pair of variables must have a clear explanatory/response relationship. Some pairs (e.g. SAT math vs. SAT verbal) are better described by exploring their association without labelling one as the cause of the other.
⚠️ Be able to classify any given variable as nominal, ordinal, discrete, or continuous. This is a standard exam question format.
⚠️ Know the difference between a population and a sample, and be able to identify each in a word problem.
⚠️ Be able to identify the explanatory (X) and response (Y) variable in a study description. The question will describe a research scenario and ask you to name each.
⚠️ Remember that ZIP codes, jersey numbers, and other numerically coded categories are categorical, not quantitative. This is a common trick question.
⚠️ Understand that descriptive statistics describes the sample data you have, while inferential statistics uses that sample to draw conclusions about the population.
True or false: A ZIP code is a quantitative variable because it is made up of numbers.
Fill in the blank: The two branches of statistics are ______ statistics and ______ statistics.
True or false: A sample is the complete collection of units about which information is sought.
Fill in the blank: A variable that places units into unordered groups is called a ______ variable.
True or false: If you are predicting exam scores from hours studied, hours studied is the response variable.
Answers: 1. False (ZIP codes are categorical/nominal). 2. Descriptive and inferential. 3. False (that is the population; a sample is a subset). 4. Nominal. 5. False (hours studied is the explanatory variable; exam score is the response).
Q: Identify the variable "taste order of a chocolate pie" as categorical (nominal or ordinal) or quantitative (discrete or continuous).
A: Ordinal. A taste order implies a ranking (1st, 2nd, 3rd in preference), which is categorical with an inherent order.
Q: Identify the variable "breaking strength of a given type of string" as categorical or quantitative.
A: Quantitative, continuous. Breaking strength is a measurement that can take any value within an interval.
Q: Identify the variable "species of an insect found in a plot of cropland."
A: Categorical, nominal. Species names are unordered group labels.
Q: Identify the variable "number of children in a family."
A: Quantitative, discrete. It is a count (0, 1, 2, 3, ...) with only whole-number values.
Q: A study examines whether yearly rainfall predicts crop yield. What is the explanatory variable? What is the response variable?
A: Explanatory variable (X) = yearly rainfall. Response variable (Y) = crop yield.
Q: A researcher wants to know if hours spent studying predict exam scores. Identify X and Y.
A: Explanatory variable (X) = hours spent studying. Response variable (Y) = exam score.
Q: Researchers give different amounts of alcohol to mice and measure the change in body temperature. What are the explanatory and response variables?
A: Explanatory variable (X) = amount of alcohol. Response variable (Y) = change in body temperature.
Q: In a study of SAT math scores and SAT verbal scores, should one be treated as explanatory and the other as response?
A: Neither score naturally predicts the other, so it is more reasonable to simply explore the relationship between them rather than designating one as explanatory.
This section connects directly to Section 1.2 and beyond, where you will learn about different methods of data collection (sampling methods, observational studies, experiments). The variable types introduced here determine which summary statistics and visualisation methods are appropriate later in the course: categorical variables get frequency tables and bar charts, while quantitative variables get histograms, means, and standard deviations. The explanatory/response distinction becomes central in regression analysis later in STA 270.
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