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 7
CLT and Sample Size
The n ≥ 30 Guideline. The “n ≥ 30” rule is a convenient guideline, not a strict law. The required sample size depends on the population’s shape: symmetric populations need fewer observations, while heavily skewed or multimodal populations need more.
When n < 30. If the population is known to be normal, the sampling distribution of x̄ is exactly normal for any n. Otherwise, non-parametric methods or bootstrap techniques may be needed for small samples from non-normal populations.
Worked example
Visualizing the CLT
Start with a uniform population (flat). With n = 5, the histogram of x̄ starts looking mound-shaped. By n = 30, it is very close to a smooth bell curve. This convergence is the CLT at work.
When to use it
Practical Advice
When in doubt about sample size, larger is better. Also check for outliers and extreme skewness in your data - these slow the CLT’s convergence.
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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