The Central Limit Theorem states that if you draw repeated samples of size $n$ from any population distribution (even heavily skewed ones), the distribution of those sample means will approach a normal distribution (a bell curve) as $n$ gets larger.
The standard deviation of this resulting sampling distribution is called the Standard Error (SE). It is calculated by dividing the population's standard deviation ($\sigma$) by the square root of the sample size ($n$):
This mathematically proves why larger sample sizes yield tighter, more precise estimates of the population mean: as the sample size $n$ (the denominator $\sqrt{n}$) grows, the error shrinks, clustering the sample means closer to the true population mean.
Note on Real-World Application: This simulation uses the exact mathematical truth ($\sigma$) because we defined the population. In the real world, the true population standard deviation is rarely known, so statisticians must estimate it using the standard deviation of their single sample ($s$). In practice, the formula becomes $SE \approx \frac{s}{\sqrt{n}}$.
The underlying shape of the population.
Number of observations drawn in each single sample.
Total times we draw a sample of size $n$ and record its mean.
A highly skewed Exponential distribution.
Mean ($\mu$) = 40 | Std Dev ($\sigma$) = 40.00
Histogram of sample means with theoretical normal overlay.