If you enjoy statistics this is the perfect article for you! But even if hearing the words hypergeometric distribution makes you want to yack all over the floor at least give this article a quick skim as it could prove useful for you someday… especially if you ship a product or service to a customer!
To help me with this article I’m going to make up a completely fabricated story.
Let’s say a producer of widgets (replace widgets with the product your company produces if you wish) recently shipped 1,000 units to their customer.
Let’s also say that, prior to shipping these units, the producer randomly inspected 100 widgets.
After inspection, the producer ships the 1,000 widgets to their customer, collects payment, and moves on to the next order.
The Angry Customer
A few weeks later the producer gets a phone call from their customer – who is less than pleased to say the least – explaining that they’ve discovered 4 defective widgets and they hadn’t even looked at all the widgets!
The customer goes on to say that they are sending all 1,000 widgets back and they are giving the producer 10 days to make things right.
Calling All Inspectors
Once the widgets arrive back in the plant a team quickly inspects all 1,000 widgets. Once they are complete they realize that 36 of the widgets were actually defective!
Identified Root Cause
Luckily, the root cause of the defective units was quickly identified and an immediate countermeasure was put into place ensuring this problem would never happen again.
How Did This Order Ever Ship?
The next question that was asked was how this order ever shipped. It was hypothesized that since 100 widgets were 100% inspected at least 1 of the 100 widgets should have been found to defective which would have forced a more thorough inspection of the entire lot.
Bring in the Black Belt
Not sure how to solve this problem the team asked their local Six Sigma Black Belt to help them determine the probability of catching at least 1 defective product while inspecting 100 units out of a lot of 1,000.
The Hypergeometric Distribution
After learning more about the situation the black belt decides to use the Hypergeometric Distribution, which Wikipedia so eloquently defines as a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement.
Here is how the black belt set things up in Minitab, which is the statistical software package they use.
In this example we are dealing with a finite population of 1,000 widgets so the “Population size” is 1,000.
Since we know, in this example, 36 widgets were defective and being defective is the “Event” in this case, the “Event count in the population” is 36.
Since the company sampled 100 widgets before shipping the lot, the “Sample size” is 100.
And since, for the Hypergeometric Distribution, X is the number of defective widgets we found in the sample of 100 widgets… X can range from 0 (we found no defective widgets) up to 36 (we found all defective widgets).
Now then, since the Black Belt wants to know the probability of finding at least 1 defective widget, they want to find the probability that X is greater than 0.
So, by using an “Input constant” of 0, they learn that the probability of having no defective widgets in this lot of 1,000 widgets is ~ 2% as shown above.
In other words, there is a 98% chance that at least 1 of the 36 defective widgets should have been detected during the inspection process.
With this information, the team came to the conclusion that their inspection process was broken. There were three theories that needed to be investigated.
- The measurement system is not repeatable and/or reproducible
- The units were not inspected at all
- Defective units were allowed to pass
So, as is the case with many things related to continuous improvement, answering one question has created several more that need to be addressed.
What do you think?
What do you think of this analysis? Would you have done it another way? What about the conclusions of the team… can you think of another theory of why these widgets were sent to the customer?