If X is a random variable that follows a binomial distribution with n trials and p probability of success on a given trial, then we can calculate the mean (μ) and standard deviation (σ) of X using the following formulas:
- μ = np
- σ = √np(1-p)
It turns out that if n is sufficiently large then we can actually use the normal distribution to approximate the probabilities related to the binomial distribution. This is known as the normal approximation to the binomial.
For n to be “sufficiently large” it needs to meet the following criteria:
- np ≥ 5
- n(1-p) ≥ 5
When both criteria are met, we can use the normal distribution to answer probability questions related to the binomial distribution.
However, the normal distribution is a continuous probability distribution while the binomial distribution is a discrete probability distribution, so we must apply a continuity correction when calculating probabilities.
In simple terms, a continuity correction is the name given to adding or subtracting 0.5 to a discrete x-value.
For example, suppose we would like to find the probability that a coin lands on heads less than or equal to 45 times during 100 flips. That is, we want to find P(X ≤ 45). To use the normal distribution to approximate the binomial distribution, we would instead find P(X ≤ 45.5).
The following table shows when you should add or subtract 0.5, based on the type of probability you’re trying to find:
| Using Binomial Distribution | Using Normal Distribution with Continuity Correction |
|---|---|
| X = 45 | 44.5 |
| X ≤ 45 | X |
| X | X |
| X ≥ 45 | X > 44.5 |
| X > 45 | X > 45.5 |
The following step-by-step example shows how to use the normal distribution to approximate the binomial distribution.
Example: Normal Approximation to the Binomial
Suppose we want to know the probability that a coin lands on heads less than or equal to 43 times during 100 flips.
In this situation we have the following values:
- n (number of trials) = 100
- X (number of successes) = 43
- p (probability of success on a given trial) = 0.50
To calculate the probability of the coin landing on heads less than or equal to 43 times, we can use the following steps:
Step 1: Verify that the sample size is large enough to use the normal approximation.
First, we must verify that the following criteria are met:
- np ≥ 5
- n(1-p) ≥ 5
In this case, we have:
- np = 100*0.5 = 50
- n(1-p) = 100*(1 – 0.5) = 100*0.5 = 50
Both numbers are greater than 5, so we’re safe to use the normal approximation.
Step 2: Determine the continuity correction to apply.
Referring to the table above, we see that we should add 0.5 when we’re working with a probability in the form of X ≤ 43. Thus, we will be finding P(X
Step 3: Find the mean (μ) and standard deviation (σ) of the binomial distribution.
μ = n*p = 100*0.5 = 50
σ = √n*p*(1-p) = √100*.5*(1-.5) = √25 = 5
Step 4: Find the z-score using the mean and standard deviation found in the previous step.
z = (x – μ) / σ = (43.5 – 50) / 5 = -6.5 / 5 = -1.3.
Step 5: Find the probability associated with the z-score.
We can use the Normal CDF Calculator to find that the area under the standard normal curve to the left of -1.3 is .0968.
Thus, the probability that a coin lands on heads less than or equal to 43 times during 100 flips is .0968.
This example illustrated the following:
- We had a situation where a random variable followed a binomial distribution.
- We wanted to find the probability of obtaining a certain value for this random variable.
- Since the sample size (n = 100 trials) was sufficiently large, we were able to use the normal distribution to approximate the binomial distribution.
This is a complete example of how to use the normal approximation to find probabilities related to the binomial distribution.