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Foundations of Statistics

Manual: General · Subject: Statistics

Build the core vocabulary and workflow used throughout statistical analysis.

What Statistics Studies

Statistics in practice

Statistics is the science of collecting, organizing, analyzing, and interpreting data to support decisions under uncertainty. It helps answer questions like: What is typical?, How variable is the data?, Is a difference real or just random noise?

Core terms

Population
The full set of individuals or outcomes of interest.
Sample
A subset of the population used to draw conclusions.
Parameter
A numerical summary of a population, such as μ\mu or σ\sigma.
Statistic
A numerical summary computed from a sample, such as xˉ\bar{x} or ss.
Variable
A measurable characteristic that can take different values.
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Why statistics matters

The goal is rarely to describe data alone; it is to make reliable inferences, quantify uncertainty, and communicate conclusions clearly.

A standard statistical workflow

  1. 1

    Step 1: Define the question and identify the population of interest.

  2. 2

    Step 2: Collect data using an appropriate sampling or experimental design.

  3. 3

    Step 3: Summarize the data with graphs and numerical measures.

  4. 4

    Step 4: Model uncertainty and test hypotheses or estimate parameters.

  5. 5

    Step 5: Interpret the results in context and state limitations.

Descriptive vs inferential statistics

Descriptive statistics

  • Summarize observed data
  • Use tables, graphs, averages, and spread measures
  • Does not generalize beyond the data set

Inferential statistics

  • Use sample data to learn about a population
  • Includes confidence intervals and hypothesis tests
  • Quantifies uncertainty from sampling

Which statement best describes a statistic?

In one sentence, distinguish a population from a sample.