Q. 171  The most common distribution for Defectives type of data is Binomial Distribution. What are the key features of Binomial Distribution? Explain with examples. 

 

Note for website visitors - Two questions are asked every week on this platform. One on Tuesday and the other on Friday.

• All questions so far can be seen here - https://www.benchmarksixsigma.com/forum/lean-six-sigma-business-excellence-questions/
• Please visit the forum home page at https://www.benchmarksixsigma.com/forum/ to respond to the latest question open till the next Tuesday/ Friday evening as per Indian Standard Time.  
• The best answer is always shown at the top among responses and the author finds honorable mention in our Business Excellence dictionary at https://www.benchmarksixsigma.com/forum/business-excellence-dictionary-glossary/ along with the related term.

 

Benchmark Six Sigma Expert View by Venugopal R

 

In General, data can be categorized as Continuous data and Attribute data. In attribute type of data, we have further classifications viz. Count, Yes / No, Ordered etc. The principle of Binomial distribution applies for situations where we can have two outcomes, like the “Yes / No” type of data. Other requirements for being eligible for application of Binomial distribution are to have a fixed number of independent trials and the probability of the outcomes remain same throughout the exercise.

 

One of the most popular example used to illustrate Binomial distribution is the outcomes relating to throwing of a dice, which has 6 faces. Let us define the outcome as obtaining ‘2’, when the dice is thrown 5 times.

1.       The ‘success’ is defined as say, ‘Obtaining the number 2’.

2.       The number of trials is 5

3.       The probability of success (obtaining the number 2) for each throw (trial) is 1/6 – and this probability remains the same

 

The  overall probability of success can be calculated by the formula _nC_r P^r (1-P)^n-r

Where n is the number of trials, r is the number of successes, P is the probability of success for one trial.

 

Where historic probabilities are of outcomes are available, the Binomial calculations can help in estimating the expected nature of outcomes for a given number of occurrences.

 

The principle of Binomial distribution is applied to develop sampling plans for attribute data pertaining to ‘defectives’. A ‘defective’ is an item that contains one or more ‘defect’ and hence, if I have a sample of 10 items, the number of ‘defectives’ in the sample can possibly vary from 0 to 10. Here the number of samples becomes the ‘n’ value and the outcome is either the item is defective / or not. Using the sample observation, the plans are used to estimate the proportion defective in the lot and decide whether the lot can be accepted or not. Another application is on the Attribute control chart, ‘p-charts’ that are used for plotting the proportion defective data.

 

The probability density distribution for Binomial will appear symmetric, but it is a discrete distribution and different from normal distribution, which is continuous.

In day to day life there are number of examples of binomial distribution like

1. exam result pass or fail

2. application reject or not reject

3.bus will come or not come

4.product defective or not defective

 

Binomial distribution has following characteristics

1.This distribution has only two possible outcomes for n number of trials like yes or no,go not go ,pass or fail,

More familiar example  for binomial distribution is coin toss where only two possible outcomes are possible either Head or Tail for n number of trials

2. Trails are fixed say “n “ numbers and all are identical.

3.Probability of success (p) remains constant  from trial to trial

4. Result of each trial is independent of other trial.

Example:Suppose in exam there are 10 questions and each question has four possible answers out of which only one is correct then probability of any answer being correct is 1/4=0.25 - This probability of correct answer will be same for each question.

 

To calculate the probability of all answers being correct:

probability of success on single trial = 0.25

Trials = 10

Successes = 10

We all know that Binomial distribution is used when we are looking at number of defectives rather than the defect itself. For example, Inspection of a part can have outcome of Pass or Fail, here there are 2 outcomes and we can count the numbers of passed or failed items based on our interest.

 

>This is one of the requirements to have only Two outcomes for a trial for the defectives to follow a Binomial distribution 

In statistics we collect samples to study the population parameter, say for example a bank manager wants to know how many customers are interested in vehicle loan and out of the 100 customers 40 are not interested. So the answers from each customer was independent (Either they say Yes or No) 

> The number of such trials needs to be fixed in order to have the experiment follow Binomial distribution. 

> Because we have independent trials the probability of each outcome is constant. 

The best answer is that of Nilesh Akre for providing a well structured answer, mentioning all properties and clarifying with an example. For Benchmark Expert View, please go through Venugopal's answer.