Normal Approximation to Binomial Distribution

Example Binomial Distribution

A Binomial Distribution is great for finding probabilities of "Yes/No" events (like coin flips).

But when the number of trials (n) gets very large, the calculations become difficult.

Fortunately, as n gets larger, a Binomial Distribution can start to look like a smooth bell shape. We can use the Normal Distribution to get a very close answer much faster:

When Can We Use It?

We shouldn't use this approximation if the data is too skewed.

A good rule of thumb is that we can use it when both:

• np ≥ 5
• n(1 − p) ≥ 5

(Where n is the number of trials and p is the probability of success)

Setting the Parameters

To use the Normal curve, we need to find the Mean (μ) and Standard Deviation (σ) from our Binomial data:

The Continuity Correction

The Binomial distribution is discrete (it has separate bars for 0, 1, 2, and so on.), but the Normal distribution is continuous (a smooth line).

Normal Distribution
Area = 1
Certain to be in there

Normal Distribution
Area = 0.5 + 0.5 = 1
50% chance of each side

Area at a value is zero
Probability of exactly that value

Area belonging to value
Probability of being "in the bin"

Example:

A Binomial value of 3 becomes a Normal Distribution area between 2.5 and 3.5

We use 0.5 because the Binomial distribution goes in steps of 1. The "bin" extends halfway to each neighbor.

Let's try a full example.

Example: Flipping a Coin

We flip a fair coin 100 times. What's the probability of getting exactly 45 heads?

• Check:
  • n=100
  • p=0.5
  • np=50
  • n(1−p)=50
  both np and n(1−p) are greater than 5, so we are good to go.
• Find μ and σ:
  • μ = 100 × 0.5 = 50
  • σ = √(100 × 0.5 × 0.5) = 5
• Apply Correction: For "Exactly 45", we look for the area between 44.5 and 45.5
• Calculate Z-scores:
  • Z₁ = (44.5 − 50) / 5 = −1.1
  • Z₂ = (45.5 − 50) / 5 = −0.9
• Look up Area:
  • Find the area between Z = −1.1 and Z = −0.9 using the Standard Normal Distribution table
  • 1.1 →0.3643, and 0.9 →0.3159
  • the area in between is 0.3643 − 0.3159 = 0.0484, which is 4.84%

What about Ranges?

When we want a range of values, like "More than 3" or "At least 3", we just have to decide which bin edges to include:

• More than 3: We want 4, 5, 6... so we start at 3.5
• At least 3: We want 3, 4, 5... so we start at 2.5 (including 3's bin)
• Less than 3: We want 0, 1, 2... so we end at 2.5
• At most 3: We want 0, 1, 2, 3... so we end at 3.5 (including 3's bin)

Summary

• For larger values of n, the Binomial Distribution can be approximated by the Normal Distribution
• Using the Normal approximation makes the calculations much easier without losing much accuracy