Key formulas
From the Sampling & CLT formula sheet
- Standard Error of xbar:
SE_xbar = sigma / sqrt(n)— Standard deviation of the sampling distribution of the sample mean.
Module 4 · Sampling & CLT · Topic 3
Sampling Distributions
Sampling Distribution. The sampling distribution of a statistic is the distribution of that statistic across all possible samples of the same size from the same population.
Key Properties for x̄. Mean of the sampling distribution: μ_x̄ = μ (unbiased). Standard deviation of the sampling distribution (standard error): σ_x̄ = σ / √n.
Worked example
Conceptual Example
If you draw 1,000 samples of size 50 from a population and compute x̄ for each, the histogram of those 1,000 means forms the sampling distribution. It will be centered at μ and be narrower than the population distribution.
When to use it
Why It Matters
The sampling distribution is the bridge between a single sample and the population. It tells us how much sample statistics vary and is the foundation for confidence intervals and hypothesis tests.
Related glossary terms
- Parameter: A number that describes an entire population. Example: The true mean height of all students at a university.
- Statistic: A number computed from a sample. Example: The mean height of 80 sampled students.
- Standard Error: The standard deviation of a statistic across repeated samples. Example: The standard error of xbar is sigma divided by sqrt(n).
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