Sample means, sample proportions and the central limit theorem
See that the means of samples are far less spread out than single values, work out the standard error of a mean and of a proportion, and use the central limit theorem to find a probability.
📖 Start here: read the steps of the “Lesson” in order (1, 2, 3 …) and answer each question. Then continue with “Practice”.
- UK: Year 13
- France: grade 12
- Afghanistan: grade 12
- Russia: grade 11
Guide
This studio follows Afghan grade 12 chapter 7 (the distribution of the sample mean, the central limit theorem, the distribution of the sample proportion) and the UK A-level sampling distributions.
Take many random samples of size n from a population with mean μ and standard deviation σ. The sample means centre on μ and their standard deviation, the standard error, is σ ÷ √n. For a proportion p it is √(p(1 − p) ÷ n). By the central limit theorem, for large n the sample mean is nearly normal, whatever the shape of the population.
Common mistakes worked on here: using σ instead of the standard error, dividing by n instead of √n, thinking doubling n halves the standard error, and using σ instead of the standard error when finding z.