Normal distribution

The normal — or Gaussian — distribution shows up everywhere averages do. Two numbers fully describe it: the mean $\mu$ tells you where it centres, the standard deviation $\sigma$ tells you how wide it spreads.

What to notice

• Shifting $\mu$ slides the whole curve left or right without changing shape.
• Shrinking $\sigma$ makes the bell taller and narrower — the total area is always 1, so height and width trade off.
• Resampling redraws the sample. The histogram wiggles, but it hugs the PDF more tightly as you raise $n$. That convergence is a preview of the Law of Large Numbers.

Why it matters

The Central Limit Theorem says the sum of many independent small effects tends toward a normal, even if none of the individual effects are normal themselves. That’s why measurement error, heights, and test-score noise all look roughly Gaussian — and why it’s the default null assumption in most of statistics.