Step 01 · Distributions
Reading the Normal Curve
The normal curve is the most reproduced shape in mathematical statistics, and for good reason: it is the fixed point toward which many independent influences drift. Reading it well means reading it from the middle outward, letting the peak set the tone and the tails set the caution.
The peak and the spread
Two numbers decide the whole picture. The first is the centre, written mu, which tells you where the highest point sits. The second is the spread, written sigma, which tells you how quickly the height falls away as you walk outward. Move sigma and the curve flattens or sharpens without ever losing its symmetry; move mu and the whole shape slides along the horizontal axis without any change of form. That two-piece description is remarkably economical, and it is the reason the normal model is so often the first one taught and the last one abandoned.
It helps to fix the area rather than the height in your mind. Whatever the spread, the total area under the curve stays at one, because the curve is a density and not a count. When the curve sharpens, it grows taller to keep the same area; when it flattens, it must reach further to the sides to do the same. The height at the peak is not a probability of anything in particular; it is only a density, and densities are read as areas, never as points.
The standard-score ladder
Distance from the centre, measured in units of sigma, is the working unit of the curve. A value one unit above the centre is written as a standard score of one; a value two units below is a standard score of minus two. On this ladder the familiar landmarks appear: roughly sixty-eight per cent of the area lies within one unit, roughly ninety-five within two, and roughly ninety-nine and a half within three. These figures are near-memorable precisely because they are near-true, and they give a reader a fast sense of how unusual an observation is.
| Interval around the centre | Approximate area | Reading |
|---|---|---|
| ± 1 unit | 0.683 | ordinary ground |
| ± 2 units | 0.954 | broad ground |
| ± 3 units | 0.997 | near the edge |
| beyond 3 units | 0.003 | rare ground |
Why the shoulders matter
Beginners stare at the peak; experienced readers watch the shoulders. The shoulder is where the curve bends from convex to concave, one unit each side of the centre, and it is there that the density changes most quickly. Many practical questions are really questions about the shoulders: how many observations fall just past the ordinary ground, and does the observed count sit near what the curve predicts? A single glance at the shoulders can tell a reader whether a sample is plausibly drawn from the shape it claims.
A caution about the tails
The tails of the normal curve fall away faster than most real records do. Observations far from the centre are therefore rarer under the curve than in many empirical collections, which is why the compendium keeps the empirical tables of the first step close at hand. The curve is a benchmark, not a promise.
Reading a picture of the curve
When you meet the curve as a diagram, ask three questions always in the same sequence. First, where is the centre? Second, how wide is the ordinary ground? Third, what share of the area lies beyond the marks you care about? Answer those three and you have read the picture completely; everything else is decoration. The remaining papers in this step add the language of moments, which summarises the shape in numbers, and the empirical ladder, which reconstructs the curve from a real sample.