Normal Approximation to Binomial Calculator

How to Use the Normal Approximation to Binomial Calculator

This calculator estimates a binomial probability using the normal distribution with continuity correction. You can use it to approximate the following probabilities:

• Exactly x successes, P(X = x)
• At most x successes, P(X ≤ x)
• Less than x successes, P(X < x)
• At least x successes, P(X ≥ x)
• More than x successes, P(X > x)
• Between a and b successes, P(a ≤ X ≤ b)

To use the calculator:

1. Enter the number of trials, n.
2. Enter the probability of success (p), as a decimal.
3. Enter the number of successes, x. If you want a probability between two values, enter the lower and upper numbers of successes.
4. Select the probability you want to find.
5. Select how many decimal places you want in the final answer.
6. Click Calculate.

The calculator instantly returns the correct probability based on the selected probability you want to find and shows you how the value was calculated, step-by-step.

Want to understand the logic behind the tool? This calculator first checks whether the binomial distribution can reasonably be approximated by a normal distribution. If the conditions for the normal approximation to the binomial are not met, the calculator tells you so and recommends using the binomial probability calculator instead.

Tip. The calculator applies the correct continuity correction, calculates the z-score or z-scores, and returns the approximate probability with a clear step-by-step solution.

What Is the Normal Approximation to the Binomial Distribution?

The normal approximation to the binomial distribution is a method of using a normal distribution to estimate binomial probabilities.

Suppose a binomial random variable X has:

• n independent trials
• two possible outcomes on each trial
• a constant probability of success, p

Instead of calculating several individual binomial probabilities, we can sometimes approximate X using a normal distribution with the same mean and standard deviation. In this case, we need to approximate the mean and standard deviation of the binomial distribution, use these values to standardize the X variable to z, and then use a normal distribution to approximate the probabilities.

Therefore, a binomial random variable X can be approximated by a normal distribution with mean μ = np and standard deviation σ = √np(1 − p).

The normal approximation to the binomial is more useful when the binomial distribution becomes sufficiently bell-shaped.

When Can You Use the Normal Approximation to a Binomial Distribution?

Before using the normal approximation, the problem should first satisfy the conditions of a binomial experiment. There should be a fixed number of trials, each trial should have two possible outcomes, the probability of success should remain constant, and the trials should be independent.

Once these conditions are satisfied, you then go ahead to check whether the binomial distribution has sufficient expected successes or failures. The rule of thumb is to use a normal approximation to the binomial when these two conditions are met:

• Expected successes are at least 5 (i.e., np ≥ 5)
• Expected failures are at least 5 (i.e., n(1 − p) ≥ 5)

Note: Some stricter textbooks and statisticians prefer a threshold of 10 instead of 5, but 5 is the most common standard rule of thumb. Thus, you should use the recommended threshold in your textbook or by your teacher.

Tip. Our normal approximation to binomial calculator checks the np ≥ 5 and n(1 − p) ≥ 5 conditions automatically. If the conditions are not satisfied, it still gives you the answer but warns you that the normal approximation may be inaccurate. Therefore, you can ignore the warning provided np ≥ 10 and n (1 – p) ≥10.

What Is a Continuity Correction?

A binomial distribution is discrete, meaning X can only take whole-number values such as 0, 1, 2, 3, and so on. However, a normal distribution is continuous and can take any value along the number line.

Because we are using a continuous distribution to approximate a discrete distribution, we adjust the binomial boundary to make it continuous by adding or subtracting 0.5 from the X variable. This adjustment is known as the continuity correction.

For example, suppose we want to estimate P(X ≤ 10). The binomial event includes all of X = 10. However, under a continuous normal curve, the region corresponding to X = 10 extends from 9.5 to 10.5. Therefore, we can estimate the binomial probability, P(X ≤ 10), using a normal distribution by finding P(Y ≤ 10.5). In this case, Y represents the normal distribution used to approximate X.

Tip: Using the continuity correction generally produces a more accurate approximation than simply replacing X with a normal random variable without the 0.5 adjustment.

Continuity Correction Rules

One of the easiest places to make a mistake when using the normal approximation is deciding whether to add or subtract 0.5. The following table provides a quick reference to help you determine whether to subtract or add 0.5, depending on possible probabilities.

Normal Approximation to Binomial Formula

After applying the continuity correction, we standardize the resulting X value to z using the z-score formula. Specifically, we use the formula:

z = (x − μ) / σ

Where:

• x is the value obtained after continuity correction
• μ = np
• σ = √[np(1 − p)]

Therefore, the formula can also be written as:

z = z = [x − np] / √[np(1 − p)]

Already have the correct value (X), the mean (μ), and standard deviation (σ)? You can use the z-score calculator to standardize the x to z. However, if you already have a z-score and only need its probability, you can use our Z-Score Probability Calculator or Standard Normal Distribution Table.

How to Calculate the Normal Approximation to a Binomial Distribution

To calculate a binomial probability using the normal approximation manually, follow these steps:

1. Identify n, p, and the probability you need to find.
2. Check the normal approximation conditions by calculating np and n(1 − p).
3. Calculate the mean and standard deviation using μ = np and σ = √(np(1 − p)).
4. Apply the continuity correction to the required value or values.
5. Convert the corrected value to a z-score
6. Use the z-score table to find the required probability.

Example 1: Normal Approximation for an At Most Probability

A US fulfillment center finds that 35% of its orders require manual address verification. Assume that orders are independent. If 80 orders are processed, use the normal approximation to estimate the probability that at most 32 orders require manual verification.

Solution

Step 1. Identify n, p, and the probability you want to find

From the question, we know that:

• Number of trials, n = 80
• Probability of success, p = 0.35
• Number of successes, x = 32

In this case, “success” means an order requires manual address verification. Therefore, we need to find P(X ≤ 32). To find this probability manually, follow these steps:

Step 2. Check the normal approximation condition

np = 80 × 0.35

= 28

Also, n(1 − p) = 80 × 0.65

= 52

Since both values are greater than 5, we can use the normal distribution to approximate the binomial probability.

Step 3. Calculate the mean and standard deviation of the binomial

Mean, μ = np

= 80 × 0.35

= 28

The standard deviation, σ = √[np(1 − p)]

= √[80 × 0.35 × 0.65]

= 4.2661

Step 4. Apply the continuity correction

We need P(X ≤ 32). Since this is a at most probability, we need to add 0.5 to the upper boundary.

Therefore, P(X ≤ 32) = P(Y ≤ 32.5)

Step 5. Convert the correct Y value to a z-score

The corrected value is y = 32.5. We can convert it to a z-score as follows:

z = (32.5 − 28) / 4.2661

= 4.5 / 4.2661

= 1.0548

Step 6. Find the probability using a z-table

Using a z-table, P(Z ≤ 1.05) ≈ 0.8531.

This implies that the probability that at most 32 of the 80 orders require manual address verification is approximately 0.8531

Common Mistakes When Using the Normal Approximation

Some of the most common errors are:

• Not checking the approximation conditions.
• Forgetting the continuity correction.
• Using the wrong continuity correction.
• Confusing variance and standard deviation. The variance is np(1 − p), but the standard deviation is √[np(1 − p)].
• Using the wrong tail of the normal distribution. “At most” requires the area to the left, while “at least” requires the area to the right.
• Treating the approximate probability as exact. A normal approximation estimates the binomial probability; it does not calculate the exact binomial probability.

Frequently Asked Questions