Statistics and Parameters

 

One of my hobbies is woodworking. That involves adjusting machines to cut to a particular size. I usually take a test cut on a piece of scrap and adjust the machine accordingly. If the cut is too deep wind back a little, if it is too shallow increase a little more. Then repeat until the size is correct to the accuracy I need.

Many tasks involve some sort of adjustment, the vial filling simulation is another example. It is important to have an effective strategy, over-adjusting can make things worse.

The quality guru W. Edwards Deming used a 'funnel' experiment to illustrate the pitfalls of commonly used strategies:

The funnel is initially positioned directly over the target. A marble is dropped through the funnel, lands on the target and rolls away in a random direction. This is repeated a number of times with the rest positions of the marbles being recorded.

	To minimize the variation should you: Leave the funnel positioned directly over the target, do not adjust it Correct the funnel position after each drop, based on the rest position of the previous marble.

We have taken a sample of 20 vials to avoid relying on a single result. The idea is that this 'averages out' the vial to vial variation and gives a more reliable estimate.

This is an example of 'inferential statistics'. We have used the sample mean to estimate the process mean (also known as the population mean).

• The sample mean is called a 'statistic' and is represented by . A statistic is a hard and fast value calculated from the sample data
  
• The process mean is called a parameter and is represented by μ (by convention parameters are represented by Greek letters).

The sample mean is not a perfect estimate of the process mean. If we take a number of samples from the filling machine we will find that the mean varies from sample to sample, although the variation in the sample means is less than the individual vial to vial variation.

The larger the sample, the more reliable the estimate. We can never know the exact value of the process mean, although we can estimate it to any required accuracy by taking a sufficiently large sample.

Later in the course we will discover how to quantify the likely error in the accuracy of a sample.

Do not tamper with the process if the apparent error can be explained by the natural variation in the sampling process.