Normal Distribution Calculator

A normal distribution is continuous, so it has no gaps and no steps — between any two values there is always another. That has a consequence people often find counter-intuitive: the probability of any one exact value is zero. The height of the curve above a point is a density, not a chance. Only an interval has a probability, and that probability is the area sitting above it.

This is why the entire curve is scaled to enclose an area of exactly 1, and why every answer on this page is a share of that total. It is also why the four regions offered above are the only shapes you ever need: any question about a normal distribution reduces to an area to the left of a point, to the right of it, between two points, or outside them.

Set the calculator to the mean of a sample of n and one thing changes: the spread becomes σ/√n instead of σ. That single substitution is the whole of the central limit theorem as it affects this arithmetic. Averages are more predictable than individual observations, and they get more predictable with the square root of the sample size — quadruple n, and the spread halves.

Two limits are worth stating plainly, because they are where the theorem is most often over-read. It does not make the underlying data normal. A skewed population stays skewed no matter how large the sample; it is the distribution of the mean that becomes normal, not the distribution of the values. And it does not name a sample size at which the approximation becomes safe. The familiar n ≥ 30 is a convention with no theorem behind it: a population that is already normal needs no minimum at all, and a heavily skewed one can need far more than thirty. This page flags a small n beside the answer rather than presenting a confident probability it has not earned.

The cumulative normal Φ(z) is computed from its mathematical definition in double precision: an all-positive series form of the error function near the centre, and a continued fraction evaluated by the modified Lentz algorithm in the tails. Neither branch uses a fitted coefficient table, so the accuracy of each follows from the mathematics rather than from a curve fit. The inverse — the z that cuts off a given area — is obtained by bisection on that same function, which needs no magic constants and converges to the last representable bit.

Tails are computed as tails. A small upper-tail probability is never formed as 1 − Φ(z), because subtracting a number very close to 1 from 1 discards most of the significant digits; the tail is computed directly instead. The same reasoning governs the region between two bounds, which is measured from whichever side keeps its precision. The practical effect is that a probability like 1.1 × 10⁻¹⁹ arrives with its digits intact instead of collapsing to zero.

These values are pinned by an automated test suite that runs on every change to this site. It asserts agreement with our own published z-score table, including at the two cells that sit closest to a rounding boundary and would therefore be the first to disagree if accuracy ever slipped. Where the normal model itself is the weak link — a boundary far into the tail, a sample mean at small n, or a distribution that extends below zero for a quantity that cannot be negative — the page says so beside the answer instead of presenting the figure unqualified.

All arithmetic runs in your browser. The page makes no network request to produce a result, so nothing you type is transmitted to us or stored.