Continuity Correction Calculator
Apply continuity correction for normal approximation to binomial distribution
Calculate Continuity Correction
Total number of independent trials
Number of successful outcomes
Probability of success (0 to 1)
Approximation Validity Check
np = 100 × 0.5 = 50.00
✅ ≥ 5 (Good)n(1-p) = 100 × 0.500 = 50.00
✅ ≥ 5 (Good)Continuity Correction Results
Original Problem
With Continuity Correction
Normal Distribution Parameters: μ = 50.00, σ = 5.000
Variance: σ² = np(1-p) = 25.000
Continuity Correction Rules
| Discrete (Binomial) | Continuous (Normal) with Correction |
|---|---|
| P(X = n) | P(n - 0.5 < X < n + 0.5) |
| P(X < n) | P(X < n - 0.5) |
| P(X ≤ n) | P(X < n + 0.5) |
| P(X > n) | P(X > n + 0.5) |
| P(X ≥ n) | P(X > n - 0.5) |
Example Calculation
Problem Setup
Scenario: A coin is flipped 100 times (N = 100)
Probability of heads (success): p = 0.5
Question: What is P(X = 60)?
Validity check: np = 50 ≥ 5 ✓, n(1-p) = 50 ≥ 5 ✓
Solution with Continuity Correction
1. Normal parameters: μ = np = 50, σ = √[np(1-p)] = √25 = 5
2. Apply continuity correction: P(X = 60) → P(59.5 < X < 60.5)
3. Standardize: Z₁ = (59.5 - 50)/5 = 1.9, Z₂ = (60.5 - 50)/5 = 2.1
4. Result: P(59.5 < X < 60.5) = Φ(2.1) - Φ(1.9) ≈ 0.0108
When to Use Continuity Correction
Binomial → Normal
Approximating discrete binomial with continuous normal distribution
Large Sample Size
Both np ≥ 5 and n(1-p) ≥ 5 for valid approximation
Discrete → Continuous
Any discrete distribution approximated by continuous one
Key Concepts
Correction factor is always ±0.5
Bridges discrete and continuous distributions
Improves approximation accuracy
Based on Central Limit Theorem
Essential for statistical inference
Understanding Continuity Correction
What is Continuity Correction?
Continuity correction is a statistical technique used when approximating a discrete probability distribution (like binomial) with a continuous distribution (like normal). The correction factor of ±0.5 accounts for the difference between discrete and continuous probability models.
Why is it Needed?
- •Discrete distributions have gaps between values
- •Continuous distributions have no gaps
- •Correction bridges this conceptual difference
- •Improves accuracy of normal approximation
How to Apply
Step 1: Check validity (np ≥ 5 and n(1-p) ≥ 5)
Step 2: Identify the problem type (=, <, ≤, >, ≥)
Step 3: Apply appropriate correction (±0.5)
Step 4: Calculate using normal distribution
Central Limit Theorem Connection
The Central Limit Theorem states that sample means approach a normal distribution as sample size increases. Continuity correction enhances this approximation by accounting for the discrete nature of the original distribution.