The normal approximation calculator above replaces a discrete binomial or Poisson probability with an area under a normal curve, applies the half-unit continuity correction, and puts the exact answer next to it so you can see how large the error actually is. It also checks the standard validity conditions and says plainly when they fail.
Arb Digital publishes it as a companion to the exact tools rather than as a substitute for them. Our binomial distribution calculator and Poisson distribution calculator compute the true probabilities directly, and our normal distribution calculator handles areas under a normal curve. This page is the bridge between them, and its purpose is to make the size of the approximation error visible instead of asking you to take it on trust.
What This Normal Approximation Calculator Does
It computes three numbers for the same question. The normal approximation with the continuity correction applied, which is the version you should use if you are using the approximation at all. The same approximation without the correction, which is what people produce when they forget it. And the exact discrete probability, computed by summing the mass function.
Putting all three on the page turns an abstract instruction into a measurable quantity. The correction is usually described as a refinement; seeing it move the answer by several percentage points on a moderate sample makes clear that it is not optional.
The validity check reports the standard conditions — np and n(1 − p) both above ten for the binomial, and λ above ten for the Poisson — and states when they are not met. It does not stop you computing anyway, because seeing a bad approximation fail is more instructive than being blocked from producing it.
How to Use It
- Pick the distribution and enter its parameters. The binomial uses n and p; the Poisson uses λ alone.
- Choose the inequality exactly as your problem states it. For a discrete variable, "at most 25" and "fewer than 25" are different questions.
- Enter k, and b as well if you selected the range option.
- Read the error in the grid. That is the number that tells you whether the approximation was worth making.
- Check the validity note in the working line before quoting any approximate figure.
The Formula and How It Is Calculated
A binomial count with n trials and success probability p has mean np and variance np(1 − p). A Poisson count with rate λ has mean λ and variance λ. The approximation replaces the discrete distribution with a normal distribution carrying the same mean and the same variance, and then reads the probability as an area.
The complication is that the count is a whole number and the normal curve is continuous, so the probability of any single value under the normal curve is zero. The continuity correction fixes this by treating the integer k as covering the interval from k − 0.5 to k + 0.5. Penn State's STAT 414 Lesson 28 on approximations for discrete distributions sets this out for both the binomial and the Poisson case, noting that once the correction is made the calculation reduces to an ordinary normal probability.
The mapping depends on the inequality, and getting it wrong is the most common error in this topic. P(X ≤ k) becomes the area below k + 0.5, because the whole of k's interval is included. P(X < k) becomes the area below k − 0.5, because k itself is excluded. P(X ≥ k) is the area above k − 0.5, and P(X > k) is the area above k + 0.5. P(X = k) is the strip between k − 0.5 and k + 0.5, which is the only way to get a non-zero answer for a point probability at all. A range from k to b runs from k − 0.5 up to b + 0.5.
How Big Is The Correction, Really
The half unit sounds negligible next to counts in the dozens, and it is not. Its effect is roughly half a standard deviation's worth of the density at the boundary, so the relative size of the correction depends on σ rather than on the count.
For the default setup — 100 binomial trials at p = 0.3, asking for P(X ≤ 25) — σ is about 4.58, so half a unit is about eleven percent of one standard deviation. The corrected approximation lands close to the exact figure while the uncorrected one is noticeably low. The gap is not a rounding artefact; it is a systematic bias in one direction, and it does not shrink as fast as the rest of the approximation error does.
The effect is largest exactly where people are most tempted to skip it: for point probabilities, where omitting the correction gives zero, and for narrow ranges, where the endpoints are most of the interval. It is smallest for tails far from the mean, where the density at the boundary is tiny.
When The Approximation Should Not Be Used
The usual rule of thumb for the binomial is that np and n(1 − p) should both exceed about ten. Some references use five, which is more permissive and correspondingly less safe. The NIST/SEMATECH e-Handbook's section on testing whether the proportion of defectives meets requirements uses the normal approximation for exactly this kind of problem and states the minimum condition that both np and n(1 − p) reach at least five.
The reason the condition involves both quantities is skew. The binomial is symmetric only when p is one half; as p moves towards either end the distribution leans, and the normal curve, which is symmetric by construction, cannot follow it. A large n does not rescue a p of 0.005, because the distribution is still crowded against zero. In that regime the Poisson approximation to the binomial is the appropriate one instead, with λ set to np.
The Poisson has its own version of the same problem: it is right-skewed for small λ, and the skewness falls as one over the square root of λ, so the approximation needs λ well into double figures before it behaves. Below about ten, sum the exact mass function instead. Where the count is a wait for a fixed number of successes rather than a fixed number of trials, our negative binomial distribution calculator covers that distribution, which also becomes more normal as its own r parameter grows.
Why Anyone Still Uses It
With exact computation available in every browser and spreadsheet, it is fair to ask why the approximation survives at all. Three answers.
It is the theoretical bridge to almost everything else in inference. A normal approximation to a binomial proportion is what a z-test for a proportion is, what a Wald confidence interval for a proportion is, and what the standard error formula in most A/B testing tools rests on. Understanding that those procedures are this approximation in disguise tells you exactly when they will misbehave — small samples, extreme proportions — which is far more useful than memorising the formulas.
It is also how you reason about a problem without a computer. Knowing that a count is roughly normal with a known mean and standard deviation lets you say immediately that a result three standard deviations out is remarkable, without summing anything. Our z-score calculator handles the standardising step and our inverse normal distribution calculator goes the other way, from a probability back to a boundary.
And exact computation is not always cheap. Summing a binomial mass function over hundreds of thousands of terms is fine once and expensive inside a loop that runs millions of times, which is why simulation and optimisation code still reaches for the closed-form approximation.
Reading The Error Figure
The grid reports the absolute error of the corrected approximation against the exact value. Absolute rather than relative, deliberately: relative error explodes in the far tail, where the exact probability is minuscule and any approximation is proportionally hopeless even when the absolute difference is one in a million.
That distinction matters when you are approximating a tail. If the exact answer is 0.00004 and the approximation gives 0.00011, the absolute error is trivially small and the relative error is nearly threefold. For a p-value, the second number is the one that matters, and the approximation should not be used. For a rough sense of whether an outcome is plausible, the first one is enough. The bars show all three figures on the same scale so the comparison is immediate.
Arb Digital checks whether the normal approximation behind a significance figure is even valid before reporting it, because on low-traffic pages it frequently is not.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Omitting the continuity correction — it biases the answer in a consistent direction and matters most for point probabilities and narrow ranges.
- Adding the half unit in the wrong direction — P(X ≤ k) uses k + 0.5 while P(X < k) uses k − 0.5, and the two questions are different.
- Checking only that n is large — the condition is on np and n(1 − p) together, so an extreme p fails it however many trials you run.
- Using the approximation for a far tail — the absolute error stays small while the relative error becomes enormous, which ruins any p-value built on it.
- Approximating when the exact value is available — every browser can sum a mass function, and the exact figure has no error at all.
Related Free Tools From Arb Digital
Compute the exact figure with the binomial distribution calculator or the Poisson distribution calculator, work with the continuous curve directly using the normal distribution calculator, standardise a value with the z-score calculator, or go from a probability back to a boundary with the inverse normal distribution calculator. The free online tools hub lists every statistics tool we publish.
Frequently Asked Questions
An adjustment of half a unit that treats a whole number k as covering the interval from k minus 0.5 to k plus 0.5, so a continuous curve can approximate a discrete count without giving every single value zero probability.
The usual rule of thumb is that np and n times (1 minus p) should both exceed about ten. Some references use five, which is more permissive and less safe.
P(X at most k) uses k plus 0.5 and P(X below k) uses k minus 0.5. P(X at least k) uses k minus 0.5 and P(X above k) uses k plus 0.5. Each inequality includes or excludes k's own interval.
Because the condition depends on p as well. A very small or very large p leaves the distribution crowded against a boundary and skewed, which a symmetric normal curve cannot follow at any sample size.
When n is large and p is very small, so that np is modest. In that regime the Poisson with a rate of np fits the binomial well while the normal does not.
The Poisson skewness falls as one over the square root of lambda, so the approximation needs lambda well into double figures. Below about ten, sum the exact mass function instead.
No. The approximation exists for hand calculation, for theory and for speed inside heavy computation. Where the exact figure can be computed, it has no approximation error at all.
This page explains a standard approximation for educational purposes. The exact figures it reports are computed by direct summation, and the approximate ones are shown so their error can be seen rather than assumed.
