Normal Approximation Calculator
Approximate binomial distribution probabilities using normal distribution with continuity correction
Normal Approximation to Binomial
Total number of independent trials or occurrences
Probability of success on a single trial (0 ≤ p ≤ 1)
Number of successful outcomes to calculate probability for
Select the type of probability to calculate
Normal Approximation Validity
Normal approximation is valid for this problem.
Approximation Results
Problem Statement
P(X = 40)
Continuity Correction
P(39.5 < X < 40.5)
Z-Score Calculation
Z₁ = (39.5 - 50.0) / 5.0 = -2.1000
Z₂ = (40.5 - 50.0) / 5.0 = -1.9000
Final Probability
Step-by-Step Example
Coin Flip Problem
Problem: Fair coin flipped 100 times. What's P(X ≤ 40)?
Given: N = 100, p = 0.5, n = 40
Validity Check: N×p = 50 ≥ 5 ✓, N×(1-p) = 50 ≥ 5 ✓
Solution Steps
1. μ = N×p = 100×0.5 = 50
2. σ² = N×p×(1-p) = 100×0.5×0.5 = 25
3. σ = √25 = 5
4. Continuity correction: P(X ≤ 40) → P(X < 40.5)
5. Z = (40.5 - 50) / 5 = -1.9
6. P(Z ≤ -1.9) = 0.0287 or 2.87%
Continuity Correction Table
Key Formulas
Binomial Parameters
Mean: μ = N × p
Variance: σ² = N × p × (1-p)
Std Dev: σ = √(N × p × (1-p))
Z-Score
Z = (x - μ) / σ
Validity Rules
N × p ≥ 5
N × (1-p) ≥ 5
Quick Tips
Always check validity conditions before approximation
Use continuity correction for better accuracy
Normal approximation works best when p ≈ 0.5
Larger sample sizes give better approximations
Understanding Normal Approximation to Binomial
What is Normal Approximation?
Normal approximation to the binomial distribution is a method of estimating binomial probabilities using the normal distribution. This is particularly useful when calculating exact binomial probabilities becomes computationally intensive for large values of N.
When to Use It?
- •When N is large (typically N ≥ 30)
- •Both N×p ≥ 5 and N×(1-p) ≥ 5
- •Exact binomial calculations are impractical
- •Quick approximation is needed
Why Continuity Correction?
The continuity correction compensates for the fact that we're approximating a discrete distribution (binomial) with a continuous one (normal). Without it, the approximation can be quite inaccurate, especially for smaller sample sizes.
Steps for Calculation
- 1.Check validity conditions
- 2.Calculate μ and σ
- 3.Apply continuity correction
- 4.Calculate Z-score
- 5.Find probability from Z-table