Topic 1.1 Notes – Introducing Statistics: What Can We Learn from Data?

What Statistics Is and What Data Tell Us

Statistics is the science of learning from data in the presence of variation.

That last word matters. If every measurement were identical, we wouldn’t need statistics. But test scores differ. Heights differ. Reaction times differ. Variation is everywhere.

Core vocabulary

• Data → recorded information (numbers or labels) about individuals in context
• Data set → the entire collection of data values
• Individuals (elements, cases) → the objects described by the data
• Observation → all recorded values for one individual

If a class of 28 students takes a quiz:

• The students are the individuals.
• Each student’s full row of information (name + score + study hours, etc.) is one observation.
• All 28 rows together form the data set.

Now the big split in statistics:

Right now in Unit 1, you’re mostly living in descriptive statistics. Inference comes later, but everything builds from here.

And here’s the enduring idea: because variation may be random or systematic, conclusions are never 100% certain.

Numbers Only Matter in Context

A number by itself tells you nothing.

“84.”

Is that:

• An exam score?
• A heart rate?
• A temperature?
• A reaction time?

Numbers become meaningful only when placed in context.

On free-response questions, writing “The mean is 84” is incomplete. Writing “The mean exam score for these 28 AP Statistics students was 84 points” earns credit because it names the variable and the individuals.

AP graders are strict about this.

The W’s of Data Context

Whenever you see a study description, mentally answer the W’s.

Who

Who are the individuals?

Vocabulary changes depending on the setting:

• Respondents → people answering a survey
• Subjects/participants → people in an experiment
• Experimental units → non-human individuals (plants, animals, objects)

Always ask: To whom do these results apply?

What

What variable was measured?

A variable is a characteristic recorded for each individual.

In experiments:

• Independent variable → what is manipulated
• Dependent variable → what is measured as the response
• Controlled variables → held constant

Clear variable names matter. “Time” is vague. “Time (in seconds) to complete a memory task” is usable.

When and Where

When and where was the data collected?

Results from:

• 2020 vs. 2024
• One school vs. a nationwide sample

may look very different. Time and location affect interpretation.

Why

What question is being asked?

Statistics always starts with a question about variation in one variable, such as:

• What is a typical value?
• How spread out are the values?
• Are there unusually high or low values?

The question shapes everything else.

How

How were the data collected?

Common methods:

• Surveys
• Experiments
• Observational studies
• Existing data sources

The method affects bias and quality. A voluntary online poll tells a different story than a randomized experiment.

Variation and Uncertainty

Even when measuring the same variable repeatedly, values differ.

That variation might be:

• Random → natural differences, measurement error
• Systematic → caused by bias or external factors

Because we can’t always tell which type we’re seeing, conclusions involve uncertainty.

That uncertainty is the reason inference procedures exist later in the course.

From Raw Data to Meaning

Large data sets are hard to interpret just by staring at numbers.

Descriptive statistics help by:

• Organizing → frequency tables
• Displaying → graphs
• Summarizing → numerical measures (mean, median, etc., coming next)

The goal is never “calculate for fun.”
The goal is to answer a question about variation in context.