The Binomial Distribution And How It’s Used In Six Sigma Projects
Data is at the heart of Lean Six Sigma methodology. In particular, the Measure and Analyze phases of “Define, Measure, Analyze, Improve, and Control” significantly rely on statistics in order to solve real-life problems. Determining the likelihood of a process resulting in defects is a big challenge that many practitioners that completed their Lean SS Green Belt training confront.
If you remember from your Six Sigma classes, defectives are useless goods or services. There are two possible results in regards to defects: defective and non-defective. The binomial distribution may be used to calculate the likelihood that a procedure would produce faulty items. Let’s examine the binomial distribution in more detail and discuss what its implications for Lean Six Sigma are.
What Is The Binomial Distribution
Discovered in 1714 by Bernoulli J., the binomial distribution is amongst the first probability distributions discovered. The binomial distribution is a type of probability distribution used for discrete information/data. It encapsulates the chances of a value taking 1 of 2 possible outcomes given a certain set of constraints or presumptions.
Any type of data that is constricted to only having 1 of 2 values, such as passing or failing, going or not going, etc., are related to the binomial distribution. If a Six Sigma team wants to find out more details about the frequency of an occurrence rather than its size, it is highly helpful. Defective data, which refers to non-conformities related to goods, services or products that make them useless, are characterized by the binomial distribution. Simply said, it defines the proportion of things without defects.
Six Sigma And The Binomial Distribution
The following requirements must be met by binomial data:
- Every item must be the outcome of the same circumstances.
- There are two potential results for each item
- For every item, the likelihood of failure or success is fixed.
- The results of the tests are distinct.
A binomial distribution is therefore more appropriate for judging flaws than defectives. It works best when there are fewer than thirty observations and a likelihood of at least 10 percent. The binomial distribution may be used by project team members to determine how challenging it will be to meet a specific goal in light of prior performance.
For any case with only two possible outcomes, the binomial distribution shouldn’t be applied. Think about the scenario when someone wishes to estimate the likelihood of snowfall on a specific day. Since there is less change in the weather in the summertime than there is during the winter, a binomial distribution cannot be used.
An Example Of Binomial Distribution
A binomial distribution can result from tossing a coin. This is due to the fact that each attempt can only end in 1 of 2 outcomes (tails or heads), each result will have the same chance (flipping a coin to determine whether it is tails or heads has a likelihood of 0.50), and thus the outcomes of one test will not affect those of another.
The Binomial Distribution Formula
The formula for the Binomial Distribution is as follow:
P(r) = (n! / r! (n – r)!) pr (1 – p) n – r
P = Desired PRobability
p = Actual Probability of Success
r= Number of successes desired
N = Sample size
This is what each element represents:
- Chances of success are represented by (P.) The intended probability is represented by an uppercase (P) whereas the actual likelihood is represented by a tiny (p).
- (r) represents the required amount of victories.
- (n) denotes the sample size.
- The factorial sign is (!). For instance, the expression (5 Factorial) means that five will be multiplied by four, four will be multiplied by three, three will be multiplied by two, and two will be multiplied by one, etc. 120 is the answer.
- The value of (0 Factorial) is always one.
The Solution To This:
Let’s go through another example and image. This is the next problem:
- We are aware that there are only two possible results when flipping a coin: tails or heads.
- Each result has a set likelihood of 0.5 fixed during time.
- Furthermore, results are random, and independently variable.
In this case, the crucial query is: What really is the likelihood of receiving five heads after tossing the coin eight times?
Using the binomial distribution formula can help us find the correct answer.
- The likelihood of receiving five heads must be determined. As a result, (r) will have a fixed value of five.
- We’ll flip the coin eight times. We shall thus use an 8-person sample size.
- There is a 0.5 percent chance of receiving heads.
To make it easier to understand the result immediately, the full information is stacked in the image below. Examine the (eight factorial), (five factorial), and (three factorial) values. Regarding the 0.5 part of the formula, note the “superscript five” and “superscript eight minus five”. The result will be 0.2187 after all of the values located within the formula have been substituted. If we flip the coin eight times and we use the binomial distribution formula, that would be the likelihood of receiving five heads.
P(r) = (n! / r! (n – r)!) pr (1 – p) n – r
P(5) = (8 / 5 (8 – 5)!) 0.55 (1 – 0.5) 8 – 5
P(5) = (8*7*6*5*4*3*2*1 / 5*4*3*2*1(3*2*1)) 0.5*0.5*0.5*0.5*0.5 (0.5*0.5*0.5)
The answer is:
Other Ways You Can Calculate Probability Using The Binomial Distribution
Using a binomial table would be another method for calculating probability using the binomial distribution. You must be aware of the exact value of (x) in order to employ this effective strategy (the wanted number of successes). (X) values can be typically listed in a binomial table’s left column. The column located at the top part of the binomial table contains the (p) values, which represent the likelihood that any given experiment would be successful. The likelihood for your scenario is determined by finding the intersection of the (x) value and the (p) value.
Using a statistical software program can be, of course, one of the simplest approaches to compute probability using a binomial distribution. Although there are plenty of other software programs available, Minitab is frequently used by many Six Sigma experts to compute probability determined by a number of distributions.
Any competent Six Sigma expert would be aware of the advantages of using the binomial distribution to estimate the likelihood of producing faulty items. Knowing the likelihood that a process may result in subpar products or services provides a data-driven understanding of a process’ performance and raises understanding of the value of upgrading the process. It eliminates the element of guessing in process development. You can apply the binomial distribution towards your own dataset now that you are aware of why it can be so helpful in Six Sigma initiatives.
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