# The Binomial Distribution is a useful tool for Six Sigma Projects

Lean Six Sigma is fundamentally data-driven. It heavily relies on statistics to solve real-world problems. This is especially true in the Measure and Analyze phases within the DMAIC process. Practitioners who have completed Lean Six Sigma Green Belt training are faced with the problem of determining the likelihood of defective products or services resulting from a process. You may recall from Six Sigma online courses that defective products or services are products or services that are not usable. There are two possible outcomes for defectives. The binomial distribution can be used to determine the likelihood of a process producing defective products. Let’s take a closer look to the binomial distribution and its implications for Lean Six Sigma.

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The Definition

Bernoulli J. discovered the binomial distribution in 1713. It is one of the oldest known probability distributions. The binomial distribution can be used to describe a type of probability distribution that is applicable to discrete data. It is a probability distribution that summarizes how likely it is that a value will take one or more of two independent values given a set of assumptions or parameters.

Binomial distributions are associated with data that can have either pass or fail, no-go or go. It is useful for Six Sigma teams who want to know more about the frequency of an event, rather than the magnitude. Binomial distribution is used to identify defectives data. These are non-conformities that make products or services unusable. It simply refers to the percentage of items that are not defective.

Six Sigma Binomial Distribution

The following criteria must be met for binomial data:

Each item is the product of identical conditions

Each item can result in one of two outcomes (pass/fail or go/no-go).

Each item has a constant probability of success or failure

The items’ outcomes are independent

A binomial distribution is more suitable for evaluating defects than defectives. It is most useful when there are fewer than 30 observations and the probability is greater than 10%. The binomial distribution can be used by project teams to determine how difficult it is to achieve a target given past performance.

The binomial distribution can’t be used in every situation that has only two outcomes. Consider the following question: How can someone calculate the likelihood of it snowing on a given day? A binomial distribution is not applicable as snow probability is lower in winter than in summer.

Example

Flipping a coin will create a binomial distribution. Because each trial can only take one value (heads/tails), each success has equal probability. For instance, the probability that you flip a head or a tail is 0.50. The results of one trial won’t affect the results from the next.

Formula

This is the Binomial Distribution Formula. Let’s look at the components of the formula.

The letter ‘P’ stands to indicate the probability of success. The capital letter ‘P’ refers to the desired probability, while the smaller ‘p’ refers to the actual probability.

The number of success desired is indicated by a ‘r’

“n” stands for Sample size

The symbol for factorial is ‘!’ For example, For example, the ‘5 Factorial’ is equivalent to: 5 will multiplied 4 times, 3 times multiplied with 3, 3 times multiplied again by 3, 3 times multiplied 2 and 2 times multiplied 2x. The answer is 120.

‘0 Factorial’ equals 1

Example and solution

Let’s now look at an example. This is the problem:

We all know that the two outcomes of tossing a coin are head or tail.

The probability of each outcome is 0.5, and it remains fixed over the course of time

Additionally, statistically independent outcomes exist

In this example, the important question is: What’s the chance of getting 5 heads?