Normal Distribution Calculator

The standard normal curve is the one every statistics table is printed from, and this page is the table in both directions. Give it a z-score and it returns the share of the curve below, above, outside and between; give it a share and it returns the z-score that cuts it off. The second direction is the one printed tables handle badly — they go from score to area, so getting a critical value back out means reading the table backwards and interpolating between rows.

How it is calculated

Φ(z) = ∫ from −∞ to z of (1/√(2π)) e^(−t²/2) dt · z = Φ⁻¹(p)

There is no closed form for that integral, so the curve has to be summed. This page uses the Taylor series published by George Marsaglia in the Journal of Statistical Software in 2004, which the article reports as accurate to an absolute error below 8 × 10⁻¹⁶, with the tail version used here holding 14 to 16 correct digits. Going the other way has no formula at all: the quantile is found by bisection on Φ, and the bracket is halved until it is narrower than one part in 2⁵² of its own upper end — the smallest gap a double can still hold there — or until 200 halvings have gone by, which is roughly three times what that criterion ever needs. Past a z of about 17 the tail is reported as zero, because by then it is smaller than 10⁻⁶⁰ and no longer means anything.

Source: Marsaglia, G. (2004), "Evaluating the Normal Distribution", Journal of Statistical Software 11(4) — true values printed for cPhi(x) = 1 − Φ(x), computed in Maple to 30 digits; the article gives its own C function an absolute error below 8 × 10⁻¹⁶ and its tail version 14 to 16 correct digits

Questions people ask

Why does a printed table give the area from 0 to z instead of the area below z?

To halve the size of the table. The curve is symmetric, so the area below −z and the area above z are the same number, and tabulating only the right half from the centre outward covers both. The cost is that every lookup needs an extra step: add 0.5 for a positive z, or subtract the result from 0.5 for a negative one. The handbook works exactly that arithmetic in its own examples, and it is where most table-reading mistakes come from.

What is the difference between the area outside ±z and the area above z?

A factor of two, and a decision you should have made before seeing the data. The area above z is the one-tailed figure — the share of the curve beyond your value in the direction it happens to lie. The area outside ±z counts both directions, and is the right one if a surprise either way would have interested you. Picking the tail after seeing which gives the smaller number is not a calculation, it is a choice, and it halves your p-value for nothing.

When can I use the normal distribution rather than t?

When you know the population standard deviation rather than estimating it from the same sample. That is rarer than it sounds: standardised tests, calibrated instruments and long-running processes qualify, a fresh sample of thirty measurements does not. With an estimated standard deviation the t distribution accounts for the extra uncertainty, and its tails are fatter for it — noticeably so below about thirty degrees of freedom, negligibly above.

My data is not normal. Does this still apply?

To the data, no. To the mean of a reasonably large sample, usually yes, which is the distinction that matters. The central limit theorem is about the sampling distribution of the mean, not about the observations, so a normal-based confidence interval for a mean can be perfectly sound on skewed data while a normal-based percentile for an individual observation is badly wrong. Income is the standard example: two standard deviations above mean income is nowhere near the 97.7th percentile.

How accurate are these numbers in the tails?

Better than any printed table. The values here are checked against the true tail probabilities Marsaglia computed to thirty digits, including at z = 5.6 where the one-tailed area is about 1.07 × 10⁻⁸ — a region where three-decimal tables simply print zero. The floor is a hard one though: past about z = 17 the answer is reported as zero rather than as a number smaller than the model behind it could justify.

Why are the critical values 1.96 and 1.645 everywhere?

Because they cut off five percent of the curve in the two ways people usually want. 1.96 leaves 2.5 percent in each tail, so ±1.96 encloses 95 percent — that is where the 95 percent confidence interval comes from. 1.645 leaves 5 percent in one tail. Both are in the summary the handbook prints above its table, and you can get either one from this page by asking for the area rather than the score.

Sources

The documents this page reads its numbers out of, linked so you can check them yourself.

1. Marsaglia, G. (2004), "Evaluating the Normal Distribution", Journal of Statistical Software 11(4) — true values printed for cPhi(x) = 1 − Φ(x), computed in Maple to 30 digits; the article gives its own C function an absolute error below 8 × 10⁻¹⁶ and its tail version 14 to 16 correct digits
2. NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.7.1, Cumulative Distribution Function of the Standard Normal Distribution — the handbook works three examples off this table: "Look for 1.5 in the X column, go right to the 0.03 column to find the value 0.43699. Now add 0.5 ... to obtain the final result of 0.93699", the mirrored 1 − 0.93699 = 0.06301 for −1.53, and 0.69146 − 0.15866 = 0.5328 for the area between −1 and 0.5

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