Key formulas
CLT Result
x̄ ~ N(μ, σ²/n) approximately, when n is sufficiently large
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 4
Central Limit Theorem
CLT Statement. For independent random observations from a population with mean μ, standard deviation σ, and finite variance, the sampling distribution of x̄ approaches a normal distribution as n increases.
How Large Is “Large Enough”?. A common rule of thumb is n ≥ 30. However, if the population is nearly normal, smaller samples work. If the population is highly skewed, larger samples may be needed.
Worked example
CLT in Action
Suppose household income is right-skewed with μ = $60,000 and σ = $20,000. With samples of n = 100, the sampling distribution of x̄ is approximately normal with mean $60,000 and SE = $20,000/√100 = $2,000.
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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