Binomial Distributions

Elizabeth Jamischak • Apr 12, 2023

Reviewing your toolbox

Welcome to the first of our quarterly newsletters for 2023!


You may recall, the blog posts in 2022 followed the Six Sigma DMAIC (Define, Measure, Analyze, Improve, Control) methodology therefore for 2023, we wanted to dive a bit deeper into some select tools from both the Six Sigma and Lean toolboxes. If there are any specific tools you would like highlighted in a post this year please email lstuart@ezsigmagroup.com and we will be sure to include it.


Without further ado, we shale begin our discussion with Binomial Distributions.


In normal distributions, we could calculate probabilities using the table of “areas under the curve”. The normal distribution, however, is a representation of continuous type data. The binomial distribution results from trials where there are only two possible outcomes.


Since these types of trials characterize discrete (attribute) data, such as “defect versus no defect”, we can apply binomial probability. If the binomial distribution approximates the (bell-shape) normal distribution, we can then calculate probabilities using the table of areas under the normal curve.


What does that mean?


As suggested by the prefix “bi” which signifies “two”, the binomial distribution describes the possible number of times a particular event or outcome will occur in a sequence of observations or trials.


This type of event is binary, in that it may or may not occur. Common examples of this would be the tossing of a coin to see how many times it lands “heads-up” or rolling a die to see how often the “number 4” appears. These are success\failure, yes\no, pass\fail responses. In our work, either a defect is present, or it is not.


Now, this may seem quite simple to you right now... I flip a coin once and I can know that either heads will turn up, or it won’t... it has a 50% chance... or does it?


What if you flipped the coin ten times? Would you always see 5 heads out of each set of 10 tries? Conduct you own test right now by repeating this experiment 4 times, each trial consisting of 10 tosses of the coin. Each occurrence of “heads” can be represented by the value 1.


What if you were to repeat a trial, but this time flip the coin 100 times. How would the results change? How does probability figure in something as simple as the toss of a coin, and what should we understand about this concept when conducting our six sigma projects, (or when you are attending a casino).


What is important is to understand how probability is reflected in the binomial distribution therefore, our focus is not on memorizing the math, but understanding the concept and application.


The “Heart” of the Calculation

When a coin is tossed, a card drawn from a deck, (or a student guesses on a multiple-choice test), they all have something in common.


In each case, the events;

  1. can only have one of two possible results (we could say “dichotomous”, but that would be pretentious!).
  2. are independent (a preceding event does not influence the subsequent event(s).
  3. are randomly selected (there is no bias in the outcome or selection)
  4. are mutually exclusive (the occurrence of one event excludes the possibility of the other event – can’t be both)


With this in mind, we can now have a quick look at the binomial probability of observing r outcomes in n number of trials:


Where p(r) is the probability of exactly r successes or outcomes, n is the number of events, and p is the probability of success for any one trial.


Don’t be intimidated by this expression. You do not have to memorize it, only understand how it works.


Let’s take our example of the coin toss. What would be the probability of observing exactly 3 heads if a fair coin is flipped 6 times?

  • A fair coin (or test) has no bias – that is, either result has equal chance of being selected. There is no influence on the outcome.



Let’s do an example with the following number from a coin toss…


n = 6 (number of events)

r = 3 (the exact number of successful outcomes we want to see)

p = .5 (the coin is fair, so heads has a 50% chance of being observed)


Now, let’s complete the formula…


We’ve seen this formula before except for the exclamation marks – what do they mean?


The exclamation mark indicates factorial notation. 6! means 6 x 5 x 4 x 3 x 2 x 1. Your calculator should have a factorial notation button that will make this easier. 6! = 720.



So, let’s simplify this one more time...


The Binomial Distribution



The results can be illustrated in graphical form, presenting a distribution curve whose shape may not be that unfamiliar to you.


The first distribution represents that same trial we have just calculated the probability on: p = .5, n = 6, r = 3


Notice that the probability indicated for observing three heads out of six tosses of the coin is .3125. In addition, you may be thinking that this looks a lot like the normal distribution. We’ll address that further on in this post.


What if we were to change the probability estimate from .5 to something else. The results of this change are shown below.


Notice how the distribution is skewed positively to the right, and that the probability of observing 3 heads has been reduced from .3125 to .18522. The highest probability for the occurrence of “heads” is now 2 out of six, since the probability for 2 is .32413.


The distribution no longer resembles the normal curve.


In real life, we may need to make probability estimates using large numbers. While not always possible, it is advantageous to use the normal distribution to approximate a binomial distribution.


Applications of the Binomial Distribution


In this post we have mentioned coin tosses and students guessing at multiple-choice tests, but the binomial distribution has practical applications. It is commonly used in quality control (estimating defects), public opinion surveys, medical research, and estimating insurance casualty claims.


Another situation you may relate to is the reservation process for the airline industry. If, on average, 7% of reservations are “no-shows”, you may want to take more reservations than there are available seats. That way, after netting out the “no-shows”, you will still have a full flight. If you are willing to take a 5% risk of not having enough seats for those who show up, how many reservations can you take in excess of the 200 seats to stay within that allowable risk?


Sampling Without Replacement


Before concluding this post, we will touch briefly on the issue of sampling with or without replacement. As suggested at the start of this post, the binomial distribution requires that you sample with replacement.


To sample without replacement is handled by what is referred to as the hypergeometric distribution. However, as long as the population size is much larger (20 x or more) than the sample size, we can use the binomial distribution. In this instance, we would be approximating the hypergeometric distribution with the binomial distribution.


Summary


We have discussed that the binomial distribution describes the possible number of times that a given event will occur over a series of observations. It uses discrete (attribute) data where there can only be two possible outcomes, yes\no, pass\fail, good\bad, etc.


Our review identified the fact that for a binomial distribution to be applied, the events must also be independent, randomly selected, and mutually exclusive.


We understand that the shape of the binomial distribution is described by two parameters, the number of values for N and the probability assigned to a specific outcome being observed (p).


We can now use this knowledge to estimate the probability of a specific number of events occurring, but to determine the cumulative probability of a range of events, we can use the normal distribution to approximate the binomial distribution, and using the resultant Z value and assign a probability estimate based on the area under the normal curve.

 

For further information on Binomial Distribution or other Lean Six Sigma tools feel free to reach out to info@ezsigmagroup.com

" I believe that the Binomial Theorem and a Bach Fugue are, in the long run, more important than all the battles in history" - James Hilton (English novelist 1900-1954)







"Life is good for only two things, discovering mathematics and teaching mathematics" - Simeon-Denis Poisson (French scientist 1781-1840)

By Elizabeth Jamischak 21 Dec, 2022
This year we have been posting about the DMAIC (Define, Measure, Analyze, Improve, Control) methodology which is the hallmark of Six Sigma theory and execution. Since this is the final blog for 2022 we decided to include both ‘Improve’ and ‘Control’ within this post. So far we have discussed the importance of defining the problem that we want to fix or area we want to improve and the value of spending time on developing a robust Project Charter, how to protect our project from scope creep by using tools such as a Data Collection Plan and how to uncover the actual Root Cause of the problem we are wanting to resolve or area we are planning to improve. In this post we will list common tools which will aid in collating all the information we have gathered thus far in our project and interpreting that information into potential solutions. However, our primary focus will be on how to actually implement change and then how to monitor / control those changes. To jump right in, some of the most common tools within the Improve phase to identify potential solutions include: Failure Mode and Effects Analysis Stakeholder Analysis Kaizen Event Poka-Yoke Cost-Benefit Analysis Matrix Diagram Once you have a list of possible solutions, selecting the best one AND implementing often brings with it stress and anxiety therefore putting a plan in place is essential for successful change. There are many templates and resources available online for developing an Implementation Plan. We recommend including the following elements: Strategy Process Action Each of these elements can be broken down by applying the 5W2H Method. For example, Strategy Who is managing the implementation of the change(s)? What is our ultimate goal with this project? When does the change(s) need to be completed? Where does this project tie into our mission & vision? Why have we selected ______ solution instead of _______ solution? How will the changes be monitored? How much will it cost to implement the changes? Process Who will be involved in implementing the changes? What is the process and timelines for implementing this change? When do we need to set our milestones to ensure success? Where does the majority of our focus need to be to successfully implement the changes? How will we communicate the change(s) to the staff / department / organization? How much effort is required from the team to implement this change(s)? Action Who is doing _______ [list specific role]? What do we need to accomplish this week / month / quarter to hit our next milestone? When do we need to test the process to verify our changes have been fully adopted? Where will we keep the documentation for implementing the change(s)? How will daily activities for ____ need to be adjusted due to these change(s)? How much time will __________ require to work on implementing the change(s)? By answering these questions, you and your team will have a clear picture of the time, human resources, financial resources and most importantly, the communication which will be required for a successful implementation of the improvements / solution you have selected for your project. The final element in a Six Sigma DMAIC project is ‘Control’. This is the phase that makes all the others worth the time and energy because in the Control phase we focus on maintaining the change(s) / improvement(s) that have been implemented. Oftentimes, the greatest challenge at this point is maintaining the cultural / psychological change that occurred therefore it is important to review your organizations infrastructure (is it technically possible to maintain the change?), the policies and procedures (do these support the change(s)?) and the culture of your organization (is everyone on board with the change(s) / improvement(s) and intentionally incorporating the change(s) / improvement(s) into their daily activities and conversations?) Some tools and activities which typically occur within the Control Phase include… Control Plan Statistical Process Control Control Charts (ex. Xbar, run chart, X-MR chart, p chart, np chart, c chart, u chart) Cost Benefit Analysis Formal Project Closure Celebration of success and team recognition As Six Sigma practitioners it is easy to remain focused on the technical aspects of a project such as collecting and analyzing the data. Therefore, in this series we wanted to highlight some key activities and soft skills which can have a dramatic impact on the success of your project. We trust you will continue to thrive and learn from each experience and that you will allocate additional time and energy to developing your plans at each stage of the DMAIC process and that you will focus on how your project will and is, impacting the culture of your organization. Thank you for joining us on this journey and we look forward to connecting in 2023!
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