Continuity Correction Calculator

Improve binomial probability approximations by applying continuity correction to normal distribution.

Last updated: March 2026

Calculate Corrected Probability

Number of Trials (n)

Success Probability (p)

Number of Successes (x)

Probability Type

Mean (μ = np)

50.0000

Std Dev (σ)

5.0000

With Correction

0.184060

Without Correction

0.158655

What is Continuity Correction?

Continuity correction bridges the gap between discrete and continuous distributions. The binomial distribution is discrete (only whole number outcomes like 0, 1, 2...), while the normal distribution is continuous. When approximating binomial probabilities with the normal distribution, we need to adjust for this difference.

Instead of finding P(X = 5), we calculate P(4.5 < X < 5.5) using the normal distribution. This ±0.5 adjustment improves accuracy, especially for smaller sample sizes or when probabilities are extreme (close to 0 or 1).

Without continuity correction, normal approximations can deviate significantly from true binomial probabilities. With correction, even modest sample sizes (n ≥ 30) give reliable approximations. This is why statistical software often includes continuity correction as an option.

How to Apply Continuity Correction

Adjustment Rules

P(X = x):

Calculate P(x - 0.5 < X < x + 0.5)

P(X ≤ x):

Calculate P(X < x + 0.5)

P(X ≥ x):

Calculate P(X > x - 0.5)

P(X < x):

Calculate P(X < x - 0.5)

P(X > x):

Calculate P(X > x + 0.5)

Formula Steps

1. Calculate μ = np and σ = √(np(1-p))

2. Adjust x value by ±0.5 based on inequality type

3. Standardize: z = (adjusted x - μ) / σ

4. Use normal CDF: Φ(z) to get probability

When to Use

✓ Use when approximating binomial with normal instead of exact binomial calculation. ✓ More important when np or n(1-p) is small. ✓ Especially helpful when np or n(1-p) is between 5-10. ✗ Skip when n is very large (n > 1000) — correction becomes negligible.

Example: Coin Flip Approximation

Flip a fair coin 100 times. What's the probability of getting at least 55 heads?

Given:

n = 100, p = 0.5, x = 55, find P(X ≥ 55)

Step 1:

μ = 100 × 0.5 = 50, σ = √(100 × 0.5 × 0.5) = 5

Step 2 (No correction):

z = (55 - 50) / 5 = 1.0, P(X ≥ 55) ≈ 15.87%

Step 2 (With correction):

z = (55 - 0.5 - 50) / 5 = 0.9, P(X ≥ 55) ≈ 18.41% (more accurate)

Insight:

The correction adjusted the boundary from 55 to 54.5, giving a 2.54% improvement in accuracy over the uncorrected normal approximation.

Frequently Asked Questions

When does continuity correction matter most?

When np or n(1-p) is small (5-20 range). For very large n or probabilities near 0.5, correction has minimal impact. Always use it to be safe.

Why ±0.5 specifically?

Each integer x in a discrete distribution can be thought of as occupying the interval [x-0.5, x+0.5). This represents one unit width centered on the integer.

Does correction always improve accuracy?

Yes. It reduces bias from the discrete-to-continuous approximation. Effect is smallest for large n, but it never hurts.

Can I use correction with other distributions?

Yes, continuity correction applies whenever approximating discrete with continuous distributions (e.g., Poisson with normal).

What if I forget which direction to adjust?

Remember: P(X ≤ x) gets +0.5 adjustment to upper bound (becomes x+0.5), P(X ≥ x) gets -0.5 to lower bound (becomes x-0.5).

Is correction used in software automatically?

Depends on the software. Some packages (R, Python) may require specifying it. Always check documentation. Many require explicit parameter.