The **normal approximation to the binomial **is when you use a continuous distribution (the normal distribution) to approximate a discrete distribution (the binomial distribution). According to the Central Limit Theorem, the sampling distribution of the sample means becomes approximately normal if the sample size is large enough.

## Normal Approximation to the Binomial: n * p and n * q Explained

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The first step into using the normal approximation to the binomial is making sure you have a “large enough sample”. How large is “large enough”? You figure this out with two calculations: n * p and n * q .

*n*is your sample size,*p*is your given probability.*q*is just 1 – p. For example, let’s say your probability p is .6. You would find q by subtracting this probability from 1: q = 1 – .6 = .4. Percentages (instead of decimals) can make this a little more understandable; if you have a 60% chance of it raining (p) then there’s a 40% probability it*won’t*rain (q).

When n * p and n * q are greater than 5, you can use the normal approximation to the binomial to solve a problem.

## Normal Approximation: Example #1 (Video)

Watch the video for an example:

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## Normal Approximation: Example #2

Sixty two percent of 12th graders attend school in a particular urban school district. If a sample of 500 12th grade children are selected, find the probability that at least 290 are actually enrolled in school.

## Part 1: Making the Calculations

**Step 1:** *Find p,q, and n:*

- The probability
*p*is given in the question as 62%, or 0.62 - To find
*q*, subtract p from 1: 1 – 0.62 = 0.38 - The sample size
*n*is given in the question as 500

**Step 2:*** Figure out if you can use the normal approximation to the binomial.* If n * p and n * q are greater than 5, then you can use the approximation:

**n * p = 310 and n * q = 190. **

These are both larger than 5, so you can use the normal approximation to the binomial for this question.

**Step 3:** *Find the mean, μ by multiplying n and p*:

n * p = 310

(You actually figured that out in Step 2!).

**Step 4:** *Multiply step 3 by q *:

310 * 0.38 = 117.8.

**Step 5:** *Take the square root of step 4 to get the standard deviation, σ*:

√(117.8)=10.85

**Note**: The formula for the standard deviation for a binomial is √(n*p*q).

## Part 2: Using the Continuity Correction Factor

**Step 6:** *Write the problem using correct notation*. The question stated that we need to “find the probability that at least 290 are actually enrolled in school”. So:

P(X ≥ 290)

**Step 7:**

*Rewrite the problem using the continuity correction factor*:

P (X ≥ 290-0.5) = P (X ≥ 289.5)

**Note:**The CCF table is listed in the above image, but if you haven’t used it before, you may want to view the video in the continuity correction factor article.

**Step 8:** *Draw a diagram with the mean in the center*. Shade the area that corresponds to the probability you are looking for. We’re looking for X ≥ 289.5, so:

**Step 9:*** Find the z-score*.

You can find this by subtracting the mean (μ) from the probability you found in step 7, then dividing by the standard deviation (σ):

**(289.5 – 310) / 10.85 = -1.89**

**Step 10:** *Look up the z-value in the z-table: *

The area for -1.89 is 0.4706.

**Step 11:** *Add .5 to your answer in step 10 to find the total area pictured:*

0.4706 + 0.5 = 0.9706.

That’s it! The probability is .9706, or 97.06%.

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## References

Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences, Wiley.

Everitt, B. S.; Skrondal, A. (2010), The Cambridge Dictionary of Statistics, Cambridge University Press.

Vogt, W.P. (2005). Dictionary of Statistics & Methodology: A Nontechnical Guide for the Social Sciences. SAGE.

Lindstrom, D. (2010). Schaum’s Easy Outline of Statistics, Second Edition (Schaum’s Easy Outlines) 2nd Edition. McGraw-Hill Education