Probability
Normal Approximation to Binomial Calculator
Compare a continuity-corrected normal probability for a binomial count with the exact binomial sum when the sample is not too large.
Worksheet
Press Calculate to see the result.
Compare a continuity-corrected normal probability for a binomial count with the exact binomial sum when the sample is not too large.
What this calculator answers
Binomial probabilities are exact but tedious once the trial count is large, so a normal curve centered at np is often used as a stand-in. Because the binomial is a count and the normal is continuous, the approximation for P(X ≤ k) shifts the cutoff to k + 0.5. This page computes that z, reads the normal cdf, and, when n is 200 or less, places the exact binomial cdf beside it. The comparison is the point of the page. A plain binomial worksheet does not show the approximation, and a plain normal worksheet does not know about counts.
The page checks np and n(1 − p) against 10, builds the continuity-corrected z, and converts it with the standard normal cdf. The exact binomial cdf is computed only when n is at most 200. Larger n skips the exact sum and says so. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
Continuity-corrected normal approximation to P(X ≤ k) uses this relationship.
z = (k + 0.5 − np) / √(np(1 − p)), approximate probability = Φ(z)
- np. The binomial mean, used as the center of the normal curve.
- np(1 − p). The binomial variance under the square root.
- k + 0.5. The continuity correction for the event X ≤ k.
- Φ(z). The normal approximation to the binomial cdf.
A count of k occupies the continuous interval from k − 0.5 to k + 0.5. The event X ≤ k therefore reaches the right edge of that interval, which is k + 0.5. Standardizing that edge with the binomial mean and standard deviation produces z, and the standard normal cdf turns z into a probability. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The form is prefilled with a small example so every arithmetic step can be checked by hand. Change one field after you finish the walkthrough and confirm the ticket moves the way the formula predicts.
The inputs are n = 100, p = 0.4, and k = 45. np = 40 and n(1 − p) = 60. The rule of thumb asks both to be at least 10, and here that check passes. The binomial variance is 24 and the standard deviation is 4.899.
For P(X ≤ 45) the continuity correction uses the boundary 45 + 0.5 = 45.5. z = (45.5 − 40) / 4.899 = 1.1227. The normal approximation is the standard cdf of that z, which equals 0.8692.
n is 100, which is at most 200, so the exact binomial cdf is 0.8689. The approximation minus the exact cdf is 0.000304. The comparison is the point of this page.
| Approximate P(X ≤ k) | 0.8692 |
|---|---|
| Exact binomial cdf | 0.8689 |
| Continuity-corrected z | 1.1227 |
| np | 40 |
| n(1 − p) | 60 |
A second situation
A sample that fails the np rule of thumb
The inputs are n = 30, p = 0.1, and k = 4. np = 3 and n(1 − p) = 27. The rule of thumb asks both to be at least 10, and here that check fails. The binomial variance is 2.7 and the standard deviation is 1.6432.
For P(X ≤ 4) the continuity correction uses the boundary 4 + 0.5 = 4.5. z = (4.5 − 3) / 1.6432 = 0.9129. The normal approximation is the standard cdf of that z, which equals 0.8193.
n is 30, which is at most 200, so the exact binomial cdf is 0.8245. The approximation minus the exact cdf is −0.00516. The comparison is the point of this page.
| Approximate P(X ≤ k) | 0.8193 |
|---|---|
| Exact binomial cdf | 0.8245 |
| Continuity-corrected z | 0.9129 |
| np | 3 |
| n(1 − p) | 27 |
np is 3 and n(1 − p) is 27. At least one is below 10, so treat the normal approximation as rough and prefer the exact cdf when it is shown.
When np falls below 10, the bell curve is a poor stand-in for a skewed count, and the gap between the approximation and the exact cdf widens. The page still shows both numbers so the failure is visible rather than hidden behind a refusal. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
The count k moved from 45 to 50, so the continuity-corrected cutoff moved with it. The headline result went from 0.8692 to 0.984.
The nudged inputs
The inputs are n = 100, p = 0.4, and k = 50. np = 40 and n(1 − p) = 60. The rule of thumb asks both to be at least 10, and here that check passes. The binomial variance is 24 and the standard deviation is 4.899.
For P(X ≤ 50) the continuity correction uses the boundary 50 + 0.5 = 50.5. z = (50.5 − 40) / 4.899 = 2.1433. The normal approximation is the standard cdf of that z, which equals 0.984.
n is 100, which is at most 200, so the exact binomial cdf is 0.9832. The approximation minus the exact cdf is 0.000717. The comparison is the point of this page.
| Approximate P(X ≤ k) | 0.984 |
|---|---|
| Exact binomial cdf | 0.9832 |
| Continuity-corrected z | 2.1433 |
| np | 40 |
| n(1 − p) | 60 |
How to read the result
Read the headline as an approximation, then look at the exact cdf on the next line when it is present. A difference of a few ten-thousandths means the rule of thumb was comfortable. A difference of several hundredths means the normal stand-in is strained. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Write the result with its unit, the inputs that produced it, and the name of the formula. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- The trials are binomial: fixed n, independent, constant p.
- The event of interest is X ≤ k, so the correction is +0.5 rather than −0.5.
- np and n(1 − p) are both at least 10 if you want the usual accuracy.
- The exact comparison is available only through n = 200 on this page.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Using k instead of k + 0.5 and calling the result continuity-corrected.
- Applying the +0.5 correction to P(X ≥ k), which needs k − 0.5.
- Trusting the approximation when np is 3 because the page still printed a number.
- Pooling this page with a two-proportion z test. Nothing here is pooled, and nothing here is a hypothesis test.
Where a different calculator is the honest choice
Use the binomial distribution calculator when you want only the exact mass and cdf. Use the normal distribution calculator when the variable was continuous to begin with. Use the Poisson calculator when successes are rare and the mean, not n and p separately, is the natural summary. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: Binomial Distribution Calculator, Normal Distribution Calculator, Poisson Distribution Calculator.
Check yourself
For n = 100 and p = 0.4, what is the continuity-corrected normal approximation to P(X ≤ 45), and how close is the exact cdf?
np is 40 and n(1 − p) is 60, both above 10. z is 1.1227, the approximation is 0.8692, and the exact binomial cdf is 0.8689.
Using the result in a sentence
Survey counts, defect totals, and textbook checks of the normal approximation are the settings where seeing both numbers matters. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
The rule of thumb is a warning, not a theorem. Even when both products exceed 10, the approximation is not exact, which is why the exact cdf stays on the ticket whenever n allows it. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the Normal Approximation to Binomial Calculator commit to?
This page uses Continuity-corrected normal approximation to P(X ≤ k): z = (k + 0.5 − np) / √(np(1 − p)), approximate probability = Φ(z). A count of k occupies the continuous interval from k − 0.5 to k + 0.5. The event X ≤ k therefore reaches the right edge of that interval, which is k + 0.5. Standardizing that edge with the binomial mean and standard deviation produces z, and the standard normal cdf turns z into a probability. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for Trials n, Success probability p, Successes k, and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the approximate p(x ≤ k) is 0.8692. Read the headline as an approximation, then look at the exact cdf on the next line when it is present. A difference of a few ten-thousandths means the rule of thumb was comfortable. A difference of several hundredths means the normal stand-in is strained. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Using k instead of k + 0.5 and calling the result continuity-corrected. The rule of thumb is a warning, not a theorem. Even when both products exceed 10, the approximation is not exact, which is why the exact cdf stays on the ticket whenever n allows it. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
Use the binomial distribution calculator when you want only the exact mass and cdf. Use the normal distribution calculator when the variable was continuous to begin with. Use the Poisson calculator when successes are rare and the mean, not n and p separately, is the natural summary. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
np is 40 and n(1 − p) is 60, both above 10. z is 1.1227, the approximation is 0.8692, and the exact binomial cdf is 0.8689. Name the formula, the sample size or the inputs, and the unit before you compare the figure with someone else's. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.