Hong Life CalculatorsUnderstanding the continuity correction calculator — concept and principle
This calculator approximates a binomial probability with the normal distribution, using the continuity correction, and compares it with the exact binomial answer and with the uncorrected approximation. It handles P(X = a), ≤, <, ≥, > and a range a ≤ X ≤ b. Use it for statistics homework and to see why the correction matters.
How it is calculated
μ = np; σ = √(np(1 − p))
Each whole value k is treated as the interval k − 0.5 to k + 0.5:
- P(X = a) ≈ P(a − 0.5 < Y < a + 0.5); P(X ≤ a) ≈ P(Y < a + 0.5); P(X < a) ≈ P(Y < a − 0.5)
- P(X ≥ a) ≈ P(Y > a − 0.5); P(X > a) ≈ P(Y > a + 0.5); P(a ≤ X ≤ b) ≈ P(a − 0.5 < Y < b + 0.5)
Y is normal with mean μ and standard deviation σ. The exact binomial sum is shown for comparison.
- A common rule asks for np and n(1 − p) both at least 5 (stricter texts use 10).
The principle
A binomial count can only be a whole number, so its probability sits in bars of width 1 centred on each integer. The normal curve is continuous. To cover the bar for k fully, the normal area must run from k − 0.5 to k + 0.5. Without the half-unit adjustment, the approximation cuts the end bars in half and misses part of the probability. The correction matters most when n is moderate and the interval is narrow.
Worked example
Default inputs: n = 60, p = 0.4, P(20 ≤ X ≤ 28).
- μ = 60 × 0.4 = 24; σ = √(60 × 0.4 × 0.6) = 3.79473; np and n(1 − p) = 24 · 36.
- Corrected interval: P(19.5 < Y < 28.5), z = ±1.1859.
- Normal approximation with correction = 0.7643.
- Exact binomial probability = 0.7647; error with correction = −0.00034.
- Without correction, P(20 < Y < 28) = 0.7082, an error of −0.0565.
| Number of trials n | Normal approximation with continuity correction |
|---|---|
| 15 | 5.591e-13 |
| 30 | 0.002594 |
| 45 | 0.3233 |
| 60 | 0.7643 |
| 75 | 0.3552 |
| 90 | 0.0531 |
| 120 | 0.0001397 |
How to read the result
Here the corrected approximation is within 0.0004 of the exact value, while the uncorrected one is off by more than 5 percentage points. Today the exact binomial value is easy to compute, so the normal approximation is mainly a teaching tool and a quick estimate; the correction should always be used with it.
Common mistakes
- Adding 0.5 in the wrong direction, especially for < and >.
- Forgetting to change P(X < a) into P(X ≤ a − 1) first.
- Using the approximation when np or n(1 − p) is small.
- Using σ = np(1 − p) without the square root.
Frequently asked questions
Why add or subtract 0.5?
Each whole number k stands for the interval from k − 0.5 to k + 0.5 under the continuous normal curve.
When can I use the normal approximation to the binomial?
A common rule is np ≥ 5 and n(1 − p) ≥ 5; some texts require 10.
Is the corrected approximation exact?
No, but it is usually much closer to the exact value than the uncorrected one.
Sources: Standard continuity-correction rules for the normal approximation to the binomial distribution