Distribution of Sample Mean
If X ~ N(μ, σ²), then the sample mean X̄ from n independent observations follows X̄ ~ N(μ, σ²/n). The standard error of the mean is SE = σ/√n.
Real World
The Office for National Statistics estimates average weekly earnings by sampling 9,000 workers: because of the σ/√n rule, their sample mean is far more precise than a single worker's wage, with uncertainty shrinking by a factor of 95.
Exam Focus
Replace σ² with σ²/n (not σ/n) when writing the distribution of X̄ — confusing variance and standard error loses the notation mark.
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