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WASpecialist MathematicsSyllabus dot point

How does the sample mean vary from sample to sample, and what are its mean and standard deviation?

Describe the sampling distribution of the sample mean, including its mean and the standard error

WACE Specialist Unit 4 sampling distribution: the sample mean as a random variable, its expected value equal to the population mean, the standard error sigma over root n, and why larger samples cluster more tightly, with a worked example.

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  1. What this dot point is asking
  2. The sample mean as a random variable
  3. Its mean and standard error
  4. Why precision improves with sample size

What this dot point is asking

SCSA wants you to treat the sample mean as a random variable, state its mean and standard error, and explain the effect of sample size, before bringing in the central limit theorem.

The sample mean as a random variable

Take a random sample of size nn from a population with mean μ\mu and standard deviation σ\sigma. The sample mean xˉ\bar{x} depends on which units happen to be drawn, so it changes from sample to sample. Thinking of all possible samples, the sample mean is a random variable Xˉ\bar{X} with its own distribution, called the sampling distribution.

Its mean and standard error

The first says the sample mean is unbiased: on average it equals the population mean. The second, the standard error, measures how much the sample mean typically deviates from μ\mu. Crucially it has n\sqrt{n}, not nn, in the denominator.

Why precision improves with sample size

This explains why large samples give sample means tightly clustered around μ\mu, while small samples scatter widely. The population spread σ\sigma is fixed; only the standard error shrinks.