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

N × p = 50.0 ≥ 5
N × (1-p) = 50.0 ≥ 5

Normal approximation is valid for this problem.

Approximation Results

50.0
Mean (μ)
25.0
Variance (σ²)
5.0
Std Dev (σ)

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

1.0852%
Decimal: 0.010852

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

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)

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. 1.Check validity conditions
  2. 2.Calculate μ and σ
  3. 3.Apply continuity correction
  4. 4.Calculate Z-score
  5. 5.Find probability from Z-table