What Is Normal Approximation to the Binomial?
When n is large and p is not too close to 0 or 1, the binomial distribution looks like a normal distribution with mean np and standard deviation √(np(1 − p)). The normal approximation to the binomial uses that curve to estimate probabilities that would otherwise need many terms.
Because a binomial count is discrete and the normal curve is continuous, the calculator applies a continuity correction of 0.5 by default. It shows the approximate probability, the exact binomial probability, the error, and whether np and n(1 − p) are both at least 10.
Below you will find the formulas, variables, a worked example and answers about the continuity correction and when to use the exact binomial instead.
How to Use the Normal Approximation to the Binomial Calculator
- Enter n and p.
- Choose the probability type and enter k (and k₂).
- Keep continuity correction on unless your course says otherwise.
- Compare the approximation with the exact value and check the validity rule.
Normal Approximation to the Binomial Formula
Normal parameters
With continuity correction
Normal Approximation to the Binomial Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| n | Number of trials | |
| p | Success probability per trial | |
| μ = np | Mean of the approximating normal | |
| σ | Standard deviation | √(np(1 − p)) |
| k | Number of successes of interest |
Normal Approximation to the Binomial Worked Example
For n = 100 and p = 0.4, μ = 40 and σ = √(100 × 0.4 × 0.6) = 4.899. To approximate P(X ≤ 45) use 45.5: z = (45.5 − 40) ÷ 4.899 = 1.123, so Φ(1.123) ≈ 0.869. The exact binomial value is about 0.869, so the approximation is excellent here.
With the example values loaded in the calculator above, the steps are:
- Mean μ = np = 100 × 0.4 = 40; σ = √(np(1 − p)) = 4.89898.
- Continuity correction ON: use half-unit boundaries such as k ± 0.5.
- Normal approximation of P(X ≤ 45) = 0.86921388; exact binomial = 0.86890955.
Result: P(X ≤ 45), normal approximation = 0.86921388; P(X ≤ 45), exact binomial = 0.86890955.
How to Read the Approximation
For n = 100 and p = 0.4, the mean is 40 and the SD is √(100 × 0.4 × 0.6) = 4.899. To approximate P(X ≤ 45), use 45.5 with the continuity correction: z = 5.5 ÷ 4.899 = 1.123, so the normal gives about 0.869. The exact binomial value is 0.8689, so the approximation is very close.
The approximation improves as n grows and as p moves toward 0.5. The calculator shows the approximation error against the exact value, and np and n(1 − p), which should both be reasonably large before you rely on it.
Where Normal Approximation to the Binomial Is Used
Large n
Estimate binomial probabilities when summing many terms is tedious.
Continuity correction
See how the correction improves the approximation.
Checking accuracy
Compare with the exact binomial.
Exams
Practise the standard approximation steps.
Common Normal Approximation to the Binomial Mistakes to Avoid
- Check np and n(1 − p) are both large enough, commonly at least 10.
- Apply the continuity correction by adding or subtracting 0.5.
- The approximation is weaker when p is near 0 or 1.
- Modern software can compute exact binomial values for large n, so use the exact value when you can.
Normal Approximation to the Binomial Terms and Definitions
- Continuity correction
- Adjusting by 0.5 so a discrete count is matched with a continuous normal area.
- Central limit theorem
- The reason a sum of many independent trials is close to normal.
Normal Approximation to the Binomial FAQs
When can I use the normal approximation to the binomial?
When n is large and p is not too close to 0 or 1, usually when both np and n(1 − p) are at least about 10.
What is the continuity correction?
It is the adjustment of 0.5 made because a discrete count is approximated by a continuous curve. For P(X ≤ 45) you use 45.5.
What are the mean and standard deviation?
The mean is np and the standard deviation is √(np(1 − p)). For n = 100 and p = 0.4 they are 40 and 4.899.
Why not always use the exact binomial?
You can, with software. The approximation matters for hand calculation and for understanding, and it gives results that are close for large n.