Control Chart - Part 1

I have been involved in Quality Assurance. Long ago I plan to write some article on Quality Assurance to gave a greater understand to everyone. The following few parts will be about Control Chart will be published in a few days.

What is Control Chart?

Control chart is the primary method for evaluation of a process by a graphic comparison of process data to calculated Control Limits. It was first introduced by Walter A. Shewhart, an engineer at Western Electric & Bell Telephone therefore also known as Shewhart Control Chart.

Shewhart Concept of Control Chart

The fundamental objective of Shewhart is to achieving economic operation of a process. During normal operation, process behavior falls within certain predictable limits of variation. This is called “controlled variation”. Otherwise, performance deviation outside these limits signals the presence of problems that are jeopardizing the economic success of the process. This is “uncontrolled variation”. Control chart is utilizes to identify the difference of “controlled variation” and “uncontrolled variation”. The cause to uncontrolled variation is special cause whereas in a controlled variation is common cause.

Determining k for control limit

How do we determine the control limit? What is the value of the k should be considers?

Normal Distribution process
Assuming the process we are controlling is normally distributed the probability that an observed value of a statistic falling outside the control limits is 0.0027 or 0.27% when k is 3.

It is a common practice that k is 3 for normal distribution. It is very rare that a stable process that the average of its samples to falls out of ± 3 standard deviation (σ). The chance is only 0.27%, but not impossible.

Non-normal distribution process
Even if the original characteristic of process does not follow a normal distribution, the value of k can generally fixed as 3. This is proven by the “Central Limit Theorem”. 

So the control limit for any control chart basically are considering k = 3 (3-sigma).

Shewhart propose k to be 3 because is it is acceptable economical value and it is selected based on empirical evidence that it works. It is not that it normal distribution process or due to Central Limit Theorem.

Central Limit Theorem

The theorem states that the sum of a large number of independent observations from the same distribution under certain general conditions is an approximate to normal distribution

Experiment:

To illustrate Central Limit Theorem better, we conduct a simple experiment with dice.

A dice has six surfaces. For an unbiased dice, we know the probability for each surface to occurs is same (1/6 chance). The distribution for that dice will be uniform. But we take average of three throws the distribution will not be uniform (round them up to closes integer).

The more trials, you will notice the distribution of the average will slowly become approximate to normal distribution.

The diagram on the below shows that a distribution that is not normally distributed. When we takes sampling data of n=5 and average data of the five samples, the more trials we do the distribution tends to becomes close to normal distribution.

Try the following link for better understand of it, it has an applet to demonstrate this.