Difficulty: Beginner | Prerequisites: None, though reading the companion notes on populations and samples first will help.
Statistics divides into two broad branches: descriptive (organising and summarising data you already have) and inferential (using sample data to draw conclusions about a larger population). Alongside that division, you need to know what kind of data you are working with, because the type of variable determines which summaries and tests are appropriate. This topic is covered in the very first week and underpins every method taught later in the course.
Descriptive statistics summarise and display data (tables, graphs, averages). Inferential statistics use sample data to make predictions or generalisations about a population. Variables come in three flavours: categorical (labels or categories), quantitative discrete (countable numbers), and quantitative continuous (measurable values that can take any number in a range).
Descriptive statistics
The branch of statistics concerned with organising, summarising, and presenting data. Think of it as describing what you can see in the data you have, without trying to say anything about data you have not collected.
Inferential statistics
The branch of statistics that uses sample data to draw conclusions about a broader population. In simple terms, this is where you go beyond what the data literally says and make educated guesses (with measurable uncertainty) about the bigger picture.
Variable
Any characteristic that can differ from one observation to the next. Height, eye colour, number of siblings, and latitude are all variables.
Categorical variable
A variable whose values are labels or group names rather than numbers. Examples: blood type, country of origin, favourite colour. You cannot meaningfully add or average these values.
Quantitative variable
A variable whose values are numbers on which arithmetic operations make sense. Subdivides into discrete and continuous.
Discrete variable
A quantitative variable that takes countable values, typically whole numbers. Think of it as values you can list out: number of children, number of cars owned, number of defective items in a batch.
Continuous variable
A quantitative variable that can take any value within a range, including decimals. Think of it as values you measure rather than count: height, weight, temperature, latitude.
Descriptive statistics is used to summarise and describe samples (or populations, if you have all the data).
Tools: tables, graphs, measures of centre (mean, median, mode), measures of spread (range, standard deviation, variance).
Goal: make raw data understandable at a glance.
Inferential statistics is the procedure of using a sample to learn about and form conclusions relating to the population.
Tools: confidence intervals, hypothesis tests, regression models.
Goal: generalise findings beyond the data you collected.
Descriptive always comes first. You summarise your sample, then use inferential methods to say something about the population.
Every variable falls into one of three categories:
Categorical: values are labels or group names. No meaningful arithmetic. Examples: gender, marital status, postcode (used as a label, not a number).
Quantitative discrete: values are countable numbers with gaps between possible values. Examples: number of students in a class, number of goals scored, number of apps on your phone.
Quantitative continuous: values can take any number in a range, including decimals. Examples: height (1.73 m), weight (68.2 kg), temperature (22.7°C), latitude (44.9778°N).
Variable: Latitude.
Latitude is quantitative continuous. It is a numerical value measured on a scale, and between any two latitudes there are infinitely many possible decimal values. You measure latitude rather than count it, which is the hallmark of a continuous variable.
Descriptive statistics power every dashboard and summary report you see: batting averages, GDP figures, average house prices. Inferential statistics are behind clinical drug trials (testing a sample of patients to decide whether a treatment works for everyone) and quality control in manufacturing (testing a batch of components to decide whether the whole production run is acceptable).
Knowing your variable type matters because it dictates your analysis. You would not compute a mean for blood types (categorical), and you would not draw a bar chart for heights (continuous). Choosing the wrong summary or graph is one of the most common early mistakes in applied work.
Students often think descriptive and inferential statistics are separate courses or unrelated activities. They are not. Descriptive summaries are the input to inferential methods.
A common error is classifying zip codes or jersey numbers as quantitative because they look like numbers. If you cannot meaningfully add or average the values, the variable is categorical regardless of whether the labels happen to be digits.
Students sometimes assume "discrete" means "small numbers." A discrete variable can take very large values (e.g. the number of grains of sand on a beach). What matters is that the values are countable with gaps between them.
Confusing "continuous" with "has decimals in the data." A variable is continuous if it could theoretically take any value in a range, even if the recorded data happens to be rounded to whole numbers.
⚠️ "Name the two branches of statistics and describe each" is a classic short-answer question on early exams. Have the two-sentence description of each branch memorised.
⚠️ Variable classification questions appear regularly. You will be given a variable and asked to classify it as categorical, quantitative discrete, or quantitative continuous, with a brief justification.
⚠️ The justification matters as much as the label. Saying "latitude is continuous" without explaining why (measurable, infinitely many values between any two points) will cost marks.
True or False: Descriptive statistics uses sample data to make predictions about a population. (False, that is inferential statistics.)
Fill in the blank: A variable whose values are group names is called a ______ variable. (Categorical.)
True or False: The number of pets someone owns is a continuous variable. (False, it is discrete: you count pets in whole numbers.)
True or False: Temperature measured in degrees Celsius is a continuous variable. (True.)
Fill in the blank: The two branches of statistics are ______ and ______. (Descriptive and inferential.)
Q: What is the difference between descriptive and inferential statistics?
A: Descriptive statistics summarises and organises data from a sample or population (using graphs, tables, and summary measures). Inferential statistics uses sample data to draw conclusions, make predictions, or test hypotheses about a larger population.
Q: Classify the following variable and explain: "Number of text messages sent per day."
A: Quantitative discrete. It is a numerical value that you count in whole numbers. You cannot send 14.7 text messages.
Q: Classify the following variable and explain: "Colour of a car."
A: Categorical. The values are labels (red, blue, silver) rather than numbers. You cannot perform arithmetic on colours.
Q: A weather station records the daily high temperature in degrees Fahrenheit. Is this variable categorical, discrete, or continuous? Explain.
A: Quantitative continuous. Temperature can take any value within a range (e.g. 72.3°F, 72.31°F). It is measured, not counted.
This connects to populations, samples, parameters, and statistics: descriptive statistics summarises the sample (calculating statistics), while inferential statistics uses those summaries to estimate population parameters. Variable type also determines which later methods apply: categorical data leads to proportion tests and chi-square tests, while continuous data leads to t-tests, ANOVA, and regression.
Descriptive statistics, inferential statistics, branches of statistics, categorical variable, qualitative variable, quantitative variable, discrete variable, continuous variable, variable classification, data types statistics, nominal, ordinal, interval, ratio, STAT 3021, introduction to probability and statistics, University of Minnesota