Continuous Distributions and the Normal

The one idea: The normal distribution is a two-parameter shape — mean μ tells you where it's centered, standard deviation σ tells you how wide it is — and once you convert any value to a z-score (how many standard deviations it sits from the mean), the same table of areas answers every "what fraction of cases fall here?" question, for any normally distributed variable, regardless of its original units.

⚠️ The trap: "Our data is roughly bell-shaped, so I'll just eyeball 95% as 'almost everyone.'" The empirical rule's 68-95-99.7% figures are for distances of exactly 1, 2, and 3 standard deviations — round to the nearest of those and you can be off by a lot. A value 1.5 SDs out is not "between the 68% and 95% bands" in some vague sense — it has a precise z-table answer (86.6% below it), and eyeballing instead of looking it up is how "roughly 95% of users" reports quietly become wrong.

A continuous variable — dollars spent, minutes to load a page, degrees of temperature — can take any value in a range, not just whole numbers. That changes the math in one specific way: for a continuous variable, the probability of hitting any single exact value is zero (there are infinitely many possible values to divide the probability among), so probability only makes sense as an area under a curve over some interval. The curve itself is called a probability density function, and by far the most important one in business statistics is the normal distribution.

Why the normal shows up everywhere

The normal distribution's formula looks intimidating —

f(x)…