Difficulty: Beginner | Prerequisites: None
Statistics sits at the base of nearly every quantitative course you will take. This topic covers what statistics is, why we need it, and the vocabulary you will use for the rest of the semester. If you have missed the first few lectures, start here: every later topic (sampling methods, hypothesis testing, regression) assumes you can tell a population from a sample, a parameter from a statistic, and a categorical variable from a quantitative one. Getting these distinctions right now saves confusion later.
Statistics is the science of learning from data when you cannot measure everything. You collect a sample, summarise it, and use the patterns to draw conclusions about the larger population. The whole course builds on a handful of distinctions covered here: population vs sample, parameter vs statistic, and the different types of variables.
Statistics
A set of scientific principles and techniques for reaching conclusions about populations and processes when information is limited and variable.
In simple terms, this means: the toolkit you use whenever you need to make sense of data and cannot look at every single case.
Population
The complete collection of units or cases about which you seek information.
Think of it as: the entire group you care about, such as all students at your university or every product off a factory line.
Sample
A subset of the population selected to represent it.
Think of it as: the manageable slice you can actually study, chosen so its results say something about the whole.
Variable
A characteristic or attribute measured on each unit in a population or sample.
In simple terms, this means: whatever you are recording about each case, such as height, test score, or eye colour.
Qualitative (categorical) variable
A variable that produces non-numerical data and places units into groups.
Think of it as: labels or categories, such as breed of dog or satisfaction level, where arithmetic (adding, averaging) makes no sense.
Nominal variable
A categorical variable whose categories have no inherent order.
Examples: gender, colour, country of birth.
Ordinal variable
A categorical variable whose categories carry a meaningful ranking.
Examples: satisfaction levels (satisfied, neutral, dissatisfied), letter grades (A, B, C).
Quantitative (numerical) variable
A variable that produces numerical data on which mathematical operations are meaningful.
Think of it as: anything you can add, subtract, or average, such as height, income, or number of defects.
Discrete variable
A quantitative variable that takes countable values, often whole numbers.
Examples: number of students enrolled, number of defects in a batch.
Continuous variable
A quantitative variable that can take any value within a range or interval.
Examples: height, weight, temperature, GPA.
Explanatory variable (independent variable, input)
The variable used to explain or predict changes in another variable.
Response variable (dependent variable, output)
The variable that responds to, or is affected by, the explanatory variable.
Parameter
A numerical characteristic of an entire population (e.g. population mean µ, population standard deviation σ). Usually unknown.
Think of it as: the true answer you would get if you could measure every single unit.
Statistic
A numerical summary computed from sample data (e.g. sample mean x̄, sample standard deviation s). Used to estimate parameters, but varies from sample to sample.
Think of it as: your best estimate of the parameter, based on the slice of data you collected.
Descriptive statistics
The branch of statistics concerned with collecting, organising, visualising, and summarising data.
Inferential statistics
The branch of statistics concerned with using sample data to make decisions, predictions, or generalisations about a population.
Statistics is the science of collecting, analysing, and interpreting data to reach conclusions about populations and processes. It matters most when direct measurement of every unit is impractical or impossible, which is nearly always the case in the real world.
The field splits into two interconnected branches:
Descriptive statistics – organising, visualising, and summarising data (tables, graphs, means, medians).
Inferential statistics – using sample data to make predictions or generalisations about a population (hypothesis testing, confidence intervals, regression).
A population is the entire group you want to draw conclusions about.
A sample is the subset you actually collect data from.
Because studying every unit in a population is rarely feasible, you work with samples. The quality of your conclusions depends on how well the sample represents the population.
Identifying the variable type is one of the first steps in any analysis, because it determines which summary measures and graphs are appropriate.
Qualitative (categorical) variables sort units into groups.
Nominal: no natural order (e.g. blood type, colour).
Ordinal: categories have a meaningful rank (e.g. satisfaction rating, education level).
Quantitative (numerical) variables produce numbers you can do arithmetic with.
Discrete: countable values, usually whole numbers (e.g. number of siblings).
Continuous: any value in a range (e.g. height in centimetres).
Categorical variables are sometimes coded with numbers for convenience (1 = male, 2 = female), but adding or averaging those codes is meaningless.
When studying a relationship between two variables:
The explanatory variable (independent/input) is used to predict or explain.
The response variable (dependent/output) is what you measure as the outcome.
Example: studying crop yield and rainfall. Rainfall is explanatory, crop yield is the response.
This distinction is central to inference:
A parameter describes the population (e.g. µ = population mean, σ = population standard deviation). It is fixed but almost always unknown.
A statistic describes the sample (e.g. x̄ = sample mean, s = sample standard deviation). It is calculated and varies from sample to sample.
The whole point of inferential statistics is to use a statistic to estimate its corresponding parameter.
Symbol | Meaning | Level |
|---|---|---|
µ (mu) | Population mean | Parameter |
σ (sigma) | Population standard deviation | Parameter |
x̄ (x-bar) | Sample mean | Statistic |
s | Sample standard deviation | Statistic |
n | Sample size | Statistic |
N | Population size | Parameter |
The Greek letters (µ, σ) are reserved for population parameters. The Latin letters (x̄, s) are used for sample statistics. This convention appears throughout the course, so learn it early.
Polling companies survey a sample of voters to predict election outcomes for the entire electorate. Medical researchers test a new drug on a sample of patients and use inferential statistics to decide whether the drug works for the broader population. Quality-control teams measure a sample of products from a factory line to estimate the overall defect rate. In each case, the logic is the same: you cannot measure everyone, so you use a well-chosen sample and the parameter-vs-statistic framework to draw conclusions.
Students often think a "statistic" is any number. It is not. A statistic is specifically a numerical summary computed from a sample. A parameter is the corresponding value for the whole population.
Students confuse ordinal and nominal variables. If the categories have a rank order (e.g. low, medium, high), the variable is ordinal, not nominal. If there is no ranking (e.g. red, blue, green), it is nominal.
Students assume that because ordinal categories are sometimes coded 1, 2, 3, you can average them. You cannot. The numbers are labels for rank, not quantities.
Students mix up the explanatory and response variables. A useful check: the explanatory variable is the one you think might cause or predict change. The response variable is the outcome you measure.
⚠️ Expect questions asking you to classify a variable as nominal, ordinal, discrete, or continuous. These are easy marks if you know the definitions, and easy to lose if you do not.
⚠️ The parameter-vs-statistic distinction appears in nearly every inference topic later in the course. If a question says "population mean," it wants µ. If it says "sample mean," it wants x̄.
⚠️ Identifying explanatory vs response variables is often tested through short scenarios. Read the scenario carefully: the variable being predicted or measured as an outcome is the response.
True or false: A sample is the entire group you want to study. (False. That is the population.)
Fill in the blank: A variable whose categories have no inherent order is called a ______ variable. (Nominal.)
True or false: The sample mean x̄ is a parameter. (False. It is a statistic. The population mean µ is the parameter.)
Fill in the blank: Height measured in centimetres is an example of a ______ variable. (Continuous.)
True or false: The number of siblings a person has is a continuous variable. (False. It is discrete, because you count whole numbers.)
Q: A researcher records the hair colour of 200 participants. What type of variable is hair colour?
A: It is a qualitative (categorical) nominal variable, because the categories (brown, black, blonde, red) have no inherent order.
Q: In a study examining whether hours of study affect exam scores, identify the explanatory and response variables.
A: Hours of study is the explanatory variable (independent). Exam score is the response variable (dependent).
Q: A report states that the average household income in a city is $62,000. Is this value more likely a parameter or a statistic? Explain.
A: If the figure was calculated from every household in the city (a census), it is a parameter. If it was calculated from a sample of households, it is a statistic. In practice, most reported averages come from samples, so it is more likely a statistic.
Q: A survey asks respondents to rate a product as "poor," "fair," "good," or "excellent." What type of variable is this?
A: It is qualitative, ordinal. The categories carry a meaningful rank (poor < fair < good < excellent), but the gaps between them are not necessarily equal.
Q: Give one example each of a discrete variable and a continuous variable.
A: Discrete: the number of courses a student is enrolled in (countable whole numbers). Continuous: a student's GPA (can take any value in a range, e.g. 3.47).
The population-vs-sample distinction feeds directly into every sampling method covered in the next topic (simple random sampling, stratified sampling, etc.). The parameter-vs-statistic framework is the backbone of confidence intervals and hypothesis testing later in the course. Variable types determine which visualisation and summary measures you use: categorical variables get bar charts and frequency tables, quantitative variables get histograms, boxplots, means, and standard deviations.
Related Terms / Search Tags
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