Normal Distribution Calculator

What the central limit theorem does not say

The normal distribution is assumed far more often than it is checked, and the justification offered is almost always the central limit theorem — usually a version of it that is not true.

The CLT is about the distribution of the sample mean. It says that if you take repeated samples of size n and compute the mean of each, those means tend toward a normal distribution as n grows, whatever the shape of the underlying data.

It says nothing about the data itself. Incomes are not normal. Waiting times are not normal. File sizes, insurance claims and city populations are not normal, and no amount of collecting more of them makes them normal. The CLT never claimed otherwise.

This matters because the two claims license completely different things. That sample means are approximately normal is what makes a t-test and a confidence interval work on skewed data. That the data is normal would license things that are not true: reading individual percentiles off a z-table, applying the empirical rule to raw observations, or treating a three-sigma event as once-in-a-thousand.

The last one has a cost. Financial returns have famously heavy tails, so events the normal model calls once-in-a-century turn up every few years. The model was not slightly wrong; it was wrong about exactly the part that mattered.

A practical version of the distinction: use the normal for inference about a mean fairly freely, and be sceptical of it as a description of individual observations unless you have looked.

The empirical rule, unrounded

68-95-99.7 is one of the most useful things to have memorised, and the exact figures are worth knowing too because one of the roundings does real work.

±1σ is 68.27%. Close enough to 68 that the rounding never matters.

±2σ is 95.45%. This is the one that misleads. It is not 95%, and the difference is not trivial: 95.45% leaves 4.55% outside, while 95% leaves 5%.

±3σ is 99.73%. Which is why three-sigma is the conventional control-chart limit — about 1 in 370 points falls outside by chance.

The 2σ gap explains something people meet constantly without explanation. A 95% confidence interval uses 1.96, not 2, because 1.96 is where exactly 2.5% is left in each tail. The “two standard errors” rule of thumb is the empirical rule’s rounding leaking into inference, and it makes intervals about 2% wider than they need to be.

The table in the tool prints both columns side by side for your own μ and σ, so the rounding is visible rather than inherited.

One caution that applies to all of it: these figures hold only for a normal distribution. On skewed or heavy-tailed data the empirical rule can be badly wrong. What holds for any distribution is Chebyshev’s bound: at least 75% within 2σ and 88.9% within 3σ. Much weaker, and true regardless.

Why this shape turns up so often

The normal distribution is not a convention. There is a reason it appears in so many unrelated places, and knowing it tells you when to expect it.

It is what you get when many small independent effects add up. Height is influenced by a great many genetic and environmental factors, each contributing a little, none dominating. Measurement error is the accumulation of many small independent perturbations. Whenever that additive structure holds, the result tends to normal.

The key words are add and independent. Break either and you get something else.

When effects multiply rather than add, you get a lognormal. Incomes, stock prices, city sizes and file lengths grow by proportional steps, and proportional steps add up on a logarithmic scale. That is why those quantities are right-skewed with a long tail, and why taking logarithms often makes them look normal.

When one effect dominates, you get its shape. A process controlled by a single failure mechanism follows that mechanism, not a bell.

When effects are correlated, the tails get heavier. This is the financial case: returns are not independent, and in a crisis they become strongly correlated, which is precisely when the normal model understates the risk.

So the useful question is not “is this normal?” but “is this the sum of many small independent things?”. If yes, expect normal. If it is a product, expect lognormal. If it is dominated by one mechanism or by correlated shocks, expect neither.

Why one table was ever enough

There are infinitely many normal distributions, one for every pair of μ and σ. Statistics textbooks printed exactly one table. The reason is a property worth appreciating.

Every normal distribution is the standard one, shifted and stretched. Subtracting μ moves the centre to zero; dividing by σ rescales the spread to one. The shape is untouched, because both operations are linear.

So P(X < a) for any normal equals Φ((a − μ)/σ) for the standard one, and a single table of Φ answers every question about every normal distribution there is.

That is a stronger property than it sounds, and it does not hold generally. Standardising a skewed distribution does not make it standard-anything — it stays skewed, as the Z-Score Calculator shows. The normal family is closed under shifting and scaling in a way most families are not.

Two more closure properties fall out of the same structure, and both are used constantly. The sum of independent normals is normal, with variances adding. And a linear combination of normals is normal, which is what makes regression coefficients normally distributed and why so much of classical statistics is tractable at all.

The tool prints the z-score alongside every answer, so you can always check it against a table.

Out where the table stops

Printed z-tables generally stop at about z = 3.49. Beyond that they say “< 0.0002” or nothing at all, and that limit has shaped what people think is knowable.

The probabilities out there are perfectly well defined and often the ones that matter. The upper tail at z = 5 is 2.87 × 10⁻⁷ — the “five sigma” threshold particle physics uses to announce a discovery. At z = 6 it is 9.87 × 10⁻¹⁰, the origin of the name Six Sigma.

Computing them needs some care. The obvious approach — take 1 − Φ(z) — fails, because past about z = 8 the CDF rounds to exactly 1.0 in double precision and the subtraction returns a hard zero. A calculator built that way reports “probability 0” for events that are rare rather than impossible.

This page computes upper tails directly instead, so P(X > a) stays accurate however far out you go. The suite asserts both halves of that: it checks that 1 − Φ(10) really is exactly zero, and that the direct computation gives 7.62 × 10⁻²⁴.

The caveat is the one from the CLT section, and it is the important one. A five-sigma probability under a normal model is only as good as the model. For genuinely normal data it is exact; for financial returns or anything with heavy tails, the real probability of a five-sigma move is far higher — which is why those events keep happening.

How to check whether your data is normal

Given how much rests on the assumption, it is worth knowing how to test it — and which tests to distrust.

Look at it first. A histogram catches gross departures immediately: two peaks, a hard boundary at zero, a long right tail. Most bad normality assumptions are visible in one chart.

A Q-Q plot is the sharper tool. Plot your sorted data against the quantiles a normal would give. Normal data lands on a straight line; curvature at the ends is skew, and an S-shape is heavy or light tails. It shows where the departure is, which a test statistic never does.

Compare the mean and the median. A normal distribution has them equal. A large gap is skew, and the Mean, Median and Mode Calculator reads the direction off it directly.

Be careful with formal normality tests. Shapiro-Wilk and Anderson-Darling have a perverse property: on small samples they lack the power to detect real departures, and on large samples they reject for departures far too small to matter. A test on 5,000 observations will find non-normality in almost anything, which tells you nothing about whether it affects your analysis.

The practical standard: for inference about a mean, mild non-normality is usually harmless because the CLT is doing the work. For anything that reads probabilities off individual observations — percentiles, tail risks, specification limits — the assumption has to be right, and that is where a Q-Q plot is worth the minute it takes.

Where this actually gets used

Four settings, with the caveat that applies to each.

Standardised testing. IQ scales and similar are constructed to be normal — scores are mapped onto a normal with a chosen mean and spread. The percentile is exact here precisely because normality was designed in rather than assumed.

Process control. Manufacturing tolerance and control limits assume a stable normal process. The assumption is usually reasonable and always worth checking, because a shifted or multi-modal process breaks the limits rather than merely straining them.

Measurement error. The classic case, and the one the distribution was developed for: repeated readings of the same quantity, where many small independent perturbations add. This is where the normal model is on its strongest ground.

Inference. Confidence intervals, t-tests and regression all rest on the normality of sampling distributions, not of the data. This is the CLT case, and it is the reason those methods work as well as they do on data that is nothing like normal.

The pattern across all four: normality is safest where it was engineered in or where the quantity is a sum of many small effects, and least safe where individual extreme values carry the consequence.

Sources and methodology

References for the exact values and the assumption-checking advice.

• NIST/SEMATECH e-Handbook — the normal distribution and its tabulated values
• NIST/SEMATECH e-Handbook — the normal probability plot, for checking the assumption rather than assuming it
• NIST/SEMATECH e-Handbook — Anderson-Darling and the limits of formal normality tests
• Z-Score Calculator — standardising, and why it does not make data normal
• Confidence Interval Calculator — where the 1.96 rather than 2 comes from
• How we calculate — methodology

Method. Probabilities come from the regularised incomplete gamma function rather than a polynomial approximation to the error function, which is what makes the figures exact to about 1e-15 instead of the ~1.5e-7 of the Abramowitz & Stegun form still used elsewhere on this site. Upper tails are computed directly rather than as 1 − Φ: past about z = 8 that subtraction underflows to a hard zero, and the suite asserts both that 1 − Φ(10) really is zero and that the direct value is 7.62 × 10⁻²⁴. The empirical-rule figures printed by the tool are computed live from the same engine, not hardcoded, so the 68.27 / 95.45 / 99.73 column cannot drift from the curve beside it. The five-sigma figure quoted above is asserted against the published 2.8665 × 10⁻⁷. That engine is verified on every change against 88 assertions, including the identity that the square of a standard normal is chi-square with one degree of freedom. The count and the per-case breakdown are published on the formula verification page.

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