Normal Approximation Calculator

The binomial distribution is exact and, for large n, exhausting: nobody sums a hundred binomial terms by hand. The normal approximation is the classical escape — the binomial’s histogram melts into the bell curve as n grows — and the continuity correction is the small act of respect (half a unit, either way) that makes the escape honest. This calculator computes your binomial probability three ways at once: exactly, by the normal approximation with the correction, and without it, so the quality of the shortcut is itself one of the outputs.

Why a Discrete Distribution Melts Into a Curve

The binomial counts successes in n independent trials; its mean is np, its spread √(np(1−p)), and its shape — this is the de Moivre–Laplace theorem, the central limit theorem’s first-born special case — approaches the normal curve with that mean and spread as n grows, provided p is not too near the edges. The approximation replaces a sum of binomial terms (exact, and for n = 100 already unfriendly) with a single lookup of the normal CDF at a standardised boundary. Everything hinges on the boundary’s placement, and that is where the continuity correction earns its name: the binomial lives on integers, its probability mass stacked in columns at 0, 1, 2, …, while the normal spreads continuously. The column at k owns the territory from k − 0.5 to k + 0.5, so P(X ≤ k) honestly corresponds to the curve’s area up to k + 0.5 — half a unit of respect paid to the discreteness — and the calculator’s three-way output lets you watch that half-unit earn its keep on your own numbers.

Worked Example: A Hundred Fair Coins

The defaults ask the classic question: in 100 fair flips, P(X ≤ 55) heads. Exact binomial answer: 0.8644. The normal curve with μ = 50, σ = 5: the corrected boundary is 55.5, z = 1.10, area 0.8643 — agreement to the fourth decimal, an approximation performing like a measurement. Without the correction the boundary is 55, z = 1.00, area 0.8413 — wrong by more than two percentage points, the entire error concentrated in that neglected half-column at the boundary. The example is the correction’s whole brief: the half-unit costs nothing (it is the same formula, shifted) and buys, typically, an order of magnitude of accuracy. Where the approximation strains is elsewhere — at the distribution’s edges, for extreme p, and in the far tails, where relative errors grow even as absolute ones shrink — and the exact column of the output stands as the permanent audit of the shortcut.

The Rules of Thumb, and Their Limits

Textbooks license the approximation when np ≥ 10 and n(1−p) ≥ 10 (older texts say 5; the stricter figure is the safer inheritance): the conditions guarantee the distribution’s bulk sits clear of both boundaries with enough columns under the curve for the melting to have occurred. The calculator’s note applies the rule to your inputs and says plainly when it is withholding its blessing. Three cautions refine the licence. Tail probabilities (the 0.1% events) are priced worst by the approximation in relative terms — an answer of 0.0011 against an exact 0.0016 is a small absolute miss and a large proportional one, and for rare-event work (the hub’s false-positive territory) the exact sum or a Poisson approximation serves better. Skewed p at modest n leaves the binomial lopsided long after the rule of thumb is met; the correction helps the near side and cannot symmetrise the far. And the exactly-k mode is the approximation’s weakest act — a single column’s area estimated from a density — printed for completeness and best replaced by the exact figure whenever it matters.

Where the Approximation Works for a Living

Its ancestral job is quality control and polling arithmetic: the sampling distribution of a proportion is a rescaled binomial, so every margin-of-error computation (the hub’s sampling-error tool prices them) is this page’s theorem in a suit. Acceptance sampling in manufacturing (is this lot’s defect count within the curve’s promise?), A/B testing’s z-tests for conversion differences, and the Wald confidence interval for a proportion are all the normal approximation drawing a salary. Its modern role has shifted from necessity to judgment: computers sum the exact binomial for any sane n (this page does, iteratively, for every query), so the approximation survives as the thinking tool — the way to see at a glance that 55 heads in 100 is unremarkable (1.1 standard deviations of the count) before any exact machinery is consulted. Fluency in the shortcut is statistical literacy; the exact column is its proofreader.

The Majority Question, Priced

Put the machinery to the polling use it was born for: a fair coin electorate (p = 0.5) sampled 100 strong will show 55 or more for one side with probability — switching the calculator to its ≥ mode — of about 0.184: nearly one sample in five manufactures a “55–45 lead” from a dead heat. That single number explains the lived experience of election seasons better than any commentary: samples of a hundred are weather vanes in a breeze, the breeze is binomial, and the correction-corrected curve prices it to three decimals. Scale cures it on schedule (at n = 10,000, a 55% sample share is fifty standard errors of the count from fairness — evidence no approximation needs to bless), and the distance between those two regimes is the entire education this page offers.

Two Boundaries: The Interval Question

Most real questions are intervals, and the correction applies at both fences: P(40 ≤ X ≤ 60) for the defaults’ coin becomes the curve’s area between 39.5 and 60.5 — z-values of −2.1 and +2.1, an area of 0.9643 — against the exact binomial sum’s 0.9648, agreement within five ten-thousandths across a hundred columns of discrete mass. The two-sided shape is where the approximation feels most like the theorem it is: errors at the two boundaries, each already small, partially cancel, and the bell curve’s portrait of the binomial becomes difficult to distinguish from the sitter. Interval questions about counts — control limits, capacity bands, confidence statements — are this computation, and they inherit its accuracy.

Related Tools

More from the Statistics Calculators hub: the Sampling Error Calculator for the proportion’s margin, the t-statistic Calculator for the other great standardisation, and the live False Positive Paradox Calculator for tail probabilities with consequences.

Frequently Asked Questions

What exactly is the continuity correction?

An adjustment of half a unit at the boundary between the discrete and continuous worlds. Because the integer k’s probability column spans k − 0.5 to k + 0.5 under the curve, cumulative questions shift their boundary outward by 0.5 (≤ k becomes < k + 0.5; ≥ k becomes > k − 0.5), and a point question becomes the area of the column itself. It is not a fudge factor but a change of coordinates, and the worked example’s fourth-decimal agreement is its character reference.

When does the approximation break down?

When the binomial is too skewed or too boundary-pressed for the bell to fit: small n, p near 0 or 1, or questions about the extreme tails. The calculator’s rule-of-thumb check flags the first two; the third is a matter of purpose — for tail-critical decisions (safety, significance at extreme levels), quote the exact figure the page prints first, and treat the approximation as the intuition pump it has gracefully become.

Why does the page compute the exact value at all, if the topic is the approximation?

Because an approximation unaudited is a rumour. Printing the exact answer beside the approximation converts every query into a calibration experiment for your own case — your n, your p, your boundary — which teaches more than any rule of thumb. It is also simply honest: the modern reader can have the exact number, so the tool’s subject is the relationship between the two, and relationships need both parties present.

Is this the same as the central limit theorem?

It is the theorem’s oldest special case: de Moivre proved the binomial-to-normal convergence in 1738, a century before the general CLT was framed. The sample proportion is a mean of Bernoulli variables, so its normalisation is literally the CLT applied — the approximation on this page, the sampling distribution of proportions, and the behaviour of sample means generally are one phenomenon wearing three coats, which is why the same σ/√n keeps appearing across this hub.

What about the Poisson approximation — when is that better?

When p is small and n large with np modest (rare events in many trials: defects per thousand, goals per match, mutations per genome), the Poisson distribution with λ = np hugs the binomial more closely than the normal does, especially in the tails. The normal approximation wins in the central, symmetric regime; the Poisson owns the rare-event corner. Choosing between them is reading the regime off your np and n(1−p) — large and balanced argues normal, small np argues Poisson.

Can I use this for a sample proportion?

Directly: a count X in n trials with X/n = p̂ means probability questions about the proportion are probability questions about the count, rescaled. The margin-of-error and confidence-interval machinery for proportions is this approximation standardised and dressed (z × √(p̂(1−p̂)/n)), and the hub’s sampling-error page performs that dressing; come here when the question is a probability about the count itself.