A normal curve is not a picture of data — it is a picture of where the probability lives. The height of the curve means almost nothing on its own; the area beneath a stretch of it is the whole point. Shade a region and you have asked a question. Measure that region and you have the answer.
Newborn birth weight
the shaded region — your questionsimulated samplelanded insidelanded outsideμ
How the number is actually found
No table and no calculator knows how to look up a middle slice directly. Every one of them stores a single thing: the area to the left of a point. Everything else is built from that by subtraction — which is why the picture below always has the same three panels, whatever region you chose.
Shade a regiondrag the boundaries
The variable
Which region do you want?
Or jump to a standard region
Shaded area—probability
As a percent—of the population
In 1,000 draws—on average
Everything else—unshaded
Area is the answer, height is not. On a density curve the vertical axis carries units of “per gram” or “per mmHg,” so a height by itself answers nothing; only height × width — an area — comes out as a plain proportion. The entire curve encloses an area of exactly 1, so any slice of it is automatically a number between 0 and 1, and the slice you shade is simultaneously two things: the probability that one randomly drawn value lands in that range, and the proportion of the whole population sitting there. Those are the same number read two ways.
Three ways this goes wrong. First, a single point has no width, so the area above it is exactly 0 — the model gives P(X = any one exact value) = 0, and “less than” and “less than or equal to” therefore return identical answers. Slide a and b together and watch the area collapse to nothing. Second, areas add only when regions do not overlap: the middle plus the two tails is 1, but P(X < b) and P(X > a) double-count everything between a and b whenever a < b. Third, and most important, every percentage here comes out of the model, not out of the world. It is exactly right about the curve and only approximately right about real newborns, whose weights are somewhat left-skewed. A number this precise is still only as good as the assumption that the variable is normal.
Does the area really predict the count?
An area is a claim you can check. Each draw generates observations from this exact distribution, and every one is tested against the region you shaded — the dots in the strip below the curve are those observations, sorted by the only thing that matters: in the band, or out of it.