The Magic of Averages: Even if a population is wildly skewed or chaotic, the distribution of SAMPLE MEANS ($\bar{x}$) will always form a perfect bell curve!
The $n \geq 30$ Rule: For the Central Limit Theorem to work and smooth out the chaos into a Normal curve, your sample size $n$ must be sufficiently large.
Standard Error: Notice how the $\bar{x}$ curve is skinny? Averages have much less variance than individuals: $\sigma_{\bar{x}} = \sigma / \sqrt{n}$.
Tags
CLTSamplingMeansStandard Error
STANDARD ERROR OF THE MEAN
$\mu_{\bar{x}} = \mu \quad ; \quad \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$
Population Distribution
Sample Means ($\bar{x}$)
Select Population Shape
Sample Size ($n$)
Individuals per Sample $n$ = 2
TOTAL SIMULATED MEANS: 0
Pop Mean $\mu \approx$50.0
Sim Mean $\bar{x} \approx$-
Sim Spread $s_{\bar{x}}$:-
Small $n$: When taking samples of just 2 people, the averages still reflect the chaotic, skewed nature of the original population. It's too small to stabilize! Try increasing $n$.