Shape of the Sampling Distribution of x̄
When you repeatedly take random samples of size n from a population and compute the sample mean
x̄ each time, the collection of those means forms the sampling distribution of
x̄. Its shape depends mainly on the population’s shape and on the sample size.
Mean and spread of x̄
The value \(\sigma_{\bar{x}}\) is also called the standard error of the mean. As n increases,
\(\sigma_{\bar{x}}\) decreases, so the sampling distribution becomes narrower.
Finite population correction (when sampling without replacement)
If sampling is without replacement from a finite population of size N and the sample is not small compared to the population,
adjust the standard error using the finite population correction factor.
A common “small sample fraction” guideline is n / N ≤ 0.05. If this holds, the correction is usually not needed.
How to decide the shape
Case 1: The population is normal
If the population distribution is normal, then the sampling distribution of x̄ is normal for any sample size.
Case 2: The population is not normal
If the population distribution is not normal, then the sampling distribution of x̄ is not necessarily normal for small n.
However, for a sufficiently large sample size, the central limit theorem implies that the sampling distribution becomes approximately normal.
- For small n, the shape of x̄ can reflect the population’s skewness or multimodality.
- As n increases, the distribution of x̄ tends to look more bell-shaped and the spread shrinks.
A common rule of thumb is that “large” means n ≥ 30, but highly skewed populations may require a larger n.
What the simulation in the calculator shows
The calculator can simulate a population (normal or a chosen non-normal shape), draw many random samples of size
n, and plot a histogram of the resulting sample means x̄.
This makes it easy to see:
- Narrowing spread: increasing n reduces \(\sigma_{\bar{x}}\).
- Normality effect: for non-normal populations, larger n produces a more bell-shaped sampling distribution.
Tip: Try a non-normal population and compare n = 5 versus n = 30 in the simulation.
