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.
Where Normal Approximation to the Binomial Is Used
Large-sample probabilities
Estimating P(X ≤ k) when n is in the hundreds or thousands.
Polling and surveys
Estimating the chance a sample count falls in a range.
Quality control
Estimating the chance of more than k defects in a large batch.
Checking the approximation
Seeing how close the normal estimate is to the exact answer.
Common Normal Approximation to the Binomial Mistakes to Avoid
- Check np ≥ 10 and n(1 − p) ≥ 10 first.
- For P(X < k) use k − 0.5 as the upper boundary.
- For exact answers with small n, use the binomial formula directly.
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 both np and n(1 − p) are at least 10 (some texts use 5).
What is the continuity correction?
A 0.5 adjustment: P(X ≤ k) uses k + 0.5 and P(X ≥ k) uses k − 0.5.
What are the mean and standard deviation?
μ = np and σ = √(np(1 − p)).
Why not always use the exact binomial?
You can with software. The approximation is handy by hand and for very large n.