Skip to content
The Lumenary
Grades 9–12Probability & StatisticsLesson available

Sampling and statistical reasoning

Populations and samples, random sampling and bias, sampling variability, the normal distribution as a model, margin of error, correlation and regression lines, and why correlation is not causation.

math.prob.statistical-inference-intro

Open lesson Show everything needed to reach this topic

Builds on

Statistics: centre, spread and distributionsProbability

Required prerequisites, all the way down: 12

You will learn to

  • Identify sources of bias in surveys and explain why random sampling helps.
  • Use simulation to see how sample statistics vary, and interpret a margin of error.
  • Fit and interpret a least-squares line and the correlation coefficient, distinguishing correlation from causation.

Definition

  • A population is the whole group of interest; a sample is the part that is observed.
  • In a simple random sample of size n, every subset of the population of size n is equally likely to be chosen.
  • The correlation coefficient r measures the strength and direction of a linear association; −1 ≤ r ≤ 1.

Theorem

  • Larger random samples give less variable estimates; for a sample mean, the spread shrinks in proportion to 1/√n.
  • Correlation, however strong, does not by itself establish causation; well-designed randomised experiments can provide evidence of causation.
  • r = 0 means no linear association, not necessarily no relationship.

Common misconceptions

  • Thinking a large biased sample is better than a smaller random one.
  • Misreading a 95% confidence interval: in the frequentist interpretation, 95% describes the long-run success rate of the method, not the probability that one computed interval contains the true value.

Leads to

Statistical inference

Optional discoveries