Normal Distribution
Line up the heights of a thousand adult men and they pile into a smooth hump: most crowd near `69` inches, thinning fast toward the very tall and the very short. Two numbers describe that whole pile.
- ▸Two numbers, a center and a width, redraw the whole curve, and a wider curve is shorter, not taller.
- ▸The same three percentages, 68, 95, and 99.7, hold in every bell, whatever its center or width.
- ▸The bands lie on lopsided data: a right-skewed income pile hides far more out in its long tail.
Measure the height of every adult man in a city and drop each one onto a number line. They do not scatter evenly. They pile up near the middle, around 69 inches, and thin out fast in both directions: a few very short, a few very tall, almost none at the extremes. That pile has a shape, a smooth hump high in the center tapering off into two tails, and the same shape shows up again and again in exam scores, measurement errors, and the weights of loaves from one bakery.
Two numbers pin that shape down completely. The first is the center of the pile, the mean you already know, and it decides where the hump sits along the line. Slide the mean from 69 up to 71 and the whole curve slides two inches to the right without changing form. The second is the standard deviation, the spread from the last topic, and it decides how wide the hump is. A small standard deviation packs everyone into a narrow spike; a large one flattens the pile into a broad, low mound.
Here is the part that surprises people: those are the only two dials. Fix the center and fix the width and there is no freedom left, the height of the curve at every point along the line is already determined. A curve this rigid, set by exactly two numbers, is the normal distribution, and it is often called the Gaussian after the mathematician who first put it to work on astronomical errors.