Area Under the Normal Curve

The total area under the normal curve is exactly 1 — and that one fact is the engine of the whole subject, because it turns areas into probabilities. Here is why the total is 1, and how to find the area for any region you are given.

Why the total area is exactly 1

A probability distribution must account for every possible outcome exactly once: something will happen, with total probability 1 (100%). The normal curve is a probability density, so its total area is set to 1 by construction — the height of the curve is whatever it needs to be for the area to come out whole. That is also why the curve's height is not a probability: the peak of the standard normal reaches about 0.3989, and the probability of any single exact value is zero. Only areas — ranges — carry probability. When someone says "the area under the curve to the left of 1.25 is 0.8944", they are saying P(Z < 1.25) = 0.8944, an 89.44% chance. The z table is nothing more than a book of pre-measured areas; the standard normal distribution table page explains how those areas are laid out.

The four area questions, and how each is answered

• Area to the left of z: read it straight from the cumulative table. Left of 1.25 → 0.8944.
• Area to the right of z: subtract from the total. Right of 0.75 → 1 − 0.7734 = 0.2266. The total being 1 is what makes this legal.
• Area between two z values: subtract the two cumulative areas. Between −1 and 2 → 0.9772 − 0.1587 = 0.8185. The between-two solver in the tool above prints these steps for any pair.
• Area outside two z values: add the two tails, or subtract the middle from 1. Outside ±1.96 → 1 − 0.95 = 0.05, the origin of the 5% significance convention.

Every probability question about a normal variable is one of these four in disguise — including questions phrased in raw units, which become area questions after standardising with z = (X − μ) / σ. The finder above shades each region as you choose it, and the blog post on finding the area under a normal curve works additional examples, including the total-area question itself, which appears verbatim in exams ("what is the total area under a normal curve?" — answer: 1).

Areas you already know: the empirical rule as area

The empirical rule is an area statement: the area between −1 and 1 is 0.6827, between −2 and 2 is 0.9545, between −3 and 3 is 0.9973. Those three areas leave 0.0027 — about 1 in 370 — outside three standard deviations, which is why a value beyond ±3 is treated as an event demanding explanation (see z score outliers). Half-areas are just as useful: the area from the mean to z = 1 is 0.3413, the "body" column of the unit normal table, and the area beyond z = 1 in the tail is 0.1587. Fluency is mostly recognising which of the four shapes a question is asking for before touching the table.

How little area lives in the far tails

The total of 1 is dominated by the middle more than intuition suggests. The area beyond ±2 is 0.0455; beyond ±3 it is 0.0027; beyond ±4 it is about 0.000063 — roughly 6 values in every 100,000. That is why the printed table stops where it does: past ±3.99 the remaining area changes nothing at four-decimal precision, and why a claimed area like "0.30 beyond z = 4" is impossible on its face. Areas are also unit-free and scale-free: the slice between the mean and one standard deviation is 0.3413 whether the units are grams, marks or metres, which is the deep reason one standard table can serve every normal distribution ever measured.

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