How to Find the Area Under a Normal Curve
"Area under the curve" is probability said geometrically. The whole area is exactly 1, every question asks for a slice of it, and the z table is the catalogue of pre-measured slices. Here is the method for any region you are handed.
The method: Standardise the boundaries to z scores, then take the region's slice from the cumulative table: left slices read directly, right slices subtract from 1, between slices subtract two reads, outside slices add two tails. Four shapes cover every question ever set.
The slider starts at 1.96 — the most-used value in statistics: it marks the edge of the middle 95% of values.
Values generated from the standard normal cumulative distribution · Verified November 2026
Why the total area is 1 — and why you care
The curve is a probability density: it must account for every possible outcome exactly once, so its total area is fixed at 1 (100%) by construction. This is not trivia — it is the licence for the commonest move in the subject, computing a right slice as 1 minus a left slice. It is also the answer to a question that appears verbatim in exams: "what is the total area under a normal curve?" — 1, for every normal curve regardless of mean or spread, because changing μ moves the curve and changing σ reshapes it, but neither can change what it totals. The area under the normal curve page explores the consequences, including why single points have probability zero.
Worked slices
- Left of −0.75: straight from the negative table — 0.2266. Note the pleasing check: the area left of −0.75 equals the area right of +0.75, by symmetry.
- Between 0.25 and 1.75: Φ(1.75) − Φ(0.25) = 0.9599 − 0.5987 = 0.3612. Both boundaries on the same side is the variation students least expect; the subtraction does not care.
- Right of 2.05: 1 − Φ(2.05) = 1 − 0.9798 = 0.0202 — about 2% of the distribution, the neighbourhood where "unusual" begins.
- Between the mean and 1.25: 0.8944 − 0.5000 = 0.3944. This body area has its own column in the unit normal table, where the post on the body of the table explains the terminology.
From raw values to areas
Real questions name raw boundaries — heights, scores, times — so the first step is always standardisation, z = (X − μ) / σ, applied to each boundary separately; the slices are then read in z units. A region "between 165 cm and 180 cm" in a population with μ = 170, σ = 6 becomes "between z = −0.83 and z = 1.67": Φ(1.67) − Φ(−0.83) = 0.9525 − 0.2033 = 0.7492. The area did not change when the units did — that invariance is precisely what makes the standard table universal, and the standard normal distribution table page is devoted to it.
Also asked as: what is the total area under a normal curve · what is the total area under the normal curve · how to find the area under a standard normal curve · how to find area under normal curve.
Related questions
Frequently asked questions
Can an area under the curve be greater than 1?
No. The total is 1, so every slice is between 0 and 1. A computed "area" above 1 (or below 0) means the wrong table column was used or a subtraction ran backwards — the two classic errors, both easy to screen for.
Is the area the same as the height of the curve?
No. Height is density — the standard normal peaks at about 0.3989, which is not a probability. Only areas (width × height accumulated over a range) are probabilities. This is why the probability of one exact value is zero: a point has height but no width, hence no area.