What this calculator does
The normal approximation to the binomial distribution replaces an exact binomial calculation with a normal (bell-curve) one that shares the same mean and standard deviation. It works well once n is reasonably large and p is not too close to 0 or 1, and it is far easier to work with than the exact binomial formula when n runs into the hundreds or thousands.
The commonly cited rule of thumb for when the approximation is trustworthy is that both np and n(1−p) should be at least 5. Below that, the binomial distribution is too lopsided for a symmetric bell curve to describe it well, and the approximation can be noticeably off, especially out in the tails.
The formula
The mean of the approximating normal distribution is μ = np, and its standard deviation is σ = √(np(1−p)), using the same n and p as the underlying binomial. This calculator checks np ≥ 5 and n(1−p) ≥ 5 automatically. If you also supply a value of x, it estimates P(X ≤ x) or P(X ≥ x) using the continuity correction, which shifts x by 0.5 to account for the binomial being a discrete distribution being approximated by a continuous one.
| Term | Meaning |
|---|---|
| n | The number of independent trials in the underlying binomial distribution. |
| p | The probability of success on each individual trial. |
| μ (mu) | The mean of the approximating normal distribution: μ = np. |
| σ (sigma) | The standard deviation of the approximating normal distribution: σ = √(np(1−p)). |
| Continuity correction | Adjusting a discrete value of x by 0.5 before applying a continuous normal approximation, which noticeably improves accuracy near the boundary. |
The inputs explained
| Field | What to enter |
|---|---|
| Number of trials (n) | The number of trials in the binomial distribution being approximated. |
| Probability of success (p) (%) | The probability of success on a single trial, as a percentage. |
| Also estimate a probability? | Leave as "No" for just the mean and standard deviation, or choose a direction to also estimate a probability for a specific value of x. |
| Value of x (only used if estimating a probability) | The value of x used only when estimating P(X ≤ x) or P(X ≥ x). Ignored otherwise. |
When to use it
Approximating a large binomial calculation
Computing exact binomial probabilities by hand becomes impractical once n reaches the hundreds, since it involves large factorials. The normal approximation turns that into a quick lookup against the standard normal distribution instead.
Checking whether the approximation is even appropriate
Before relying on a normal approximation in coursework or a quick estimate, the np ≥ 5 and n(1−p) ≥ 5 check flags cases where p is so extreme, or n so small, that a bell curve is a poor stand-in for the real distribution.
Estimating a tail probability quickly
Questions like "what is the chance of at least 60 successes out of 100 trials at a 50% success rate" can be estimated in seconds with the continuity-corrected normal approximation, without touching the exact binomial formula.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How the np ≥ 5 rule holds up as p becomes more extreme
A fixed 100 trials, across a range of success probabilities.
| Probability of success (p) | Mean (μ = np) | Standard deviation (σ) | Rule-of-thumb check (np ≥ 5 and n(1−p) ≥ 5) |
|---|---|---|---|
| 1% | 1.000 | 0.9950 | Not satisfied: the normal approximation may be unreliable here, especially in the tails |
| 3% | 3.000 | 1.706 | Not satisfied: the normal approximation may be unreliable here, especially in the tails |
| 5% | 5.000 | 2.179 | Satisfied: the normal approximation should be reasonably accurate |
| 10% | 10.000 | 3.000 | Satisfied: the normal approximation should be reasonably accurate |
| 50% | 50.000 | 5.000 | Satisfied: the normal approximation should be reasonably accurate |
| 97% | 97.000 | 1.706 | Not satisfied: the normal approximation may be unreliable here, especially in the tails |
How the estimated probability changes across values of x
A fixed n = 100 and p = 50%, across a range of x values.
| Value of x | Estimated P(X ≤ x) | z used (continuity-corrected) |
|---|---|---|
| 40 | 2.87% | -1.900 |
| 45 | 18.4% | -0.900 |
| 50 | 54.0% | 0.100 |
| 55 | 86.4% | 1.100 |
| 60 | 98.2% | 2.100 |
Questions
Why use a normal approximation instead of the exact binomial formula?
The exact binomial formula involves computing large factorials or repeated products, which becomes slow and numerically awkward once n is in the hundreds or thousands. The normal approximation reduces the same question to a lookup against the standard normal distribution.
What does the continuity correction actually do?
The binomial distribution only takes whole-number values, while the normal distribution is continuous. Shifting the boundary value by 0.5 before converting to a z-score accounts for that mismatch and noticeably improves the accuracy of the estimate, especially for smaller n.
What happens if the np ≥ 5 rule is not satisfied?
The approximation can still be calculated, but it may be inaccurate, particularly in the tails of the distribution. In that situation the exact binomial calculation, or the Poisson approximation for rare events, is usually more reliable.
Is this the same as the binomial distribution calculator on this site?
No. The binomial distribution calculator and the binomial probability calculator compute exact probabilities directly from n, p and k. This calculator instead works out the normal curve that approximates that same distribution, which is a different, faster method for large n.
For the exact probability of a specific outcome rather than an approximation, use the binomial probability calculator or the full binomial distribution calculator.