Control Chart - Part 4

X-S Control Chart
Basically, we could plot either the R chart or the S chart to accompany the X chart to monitor the process control. R chart and S chart shows the spread of the data we collected.

But most books do not touch much on S chart. In the older days it is quite difficult to calculate S (S stand for Standard Deviation), therefore S chart is less popular.

We use the same example as in Part 3, in state of the calculating the average and range, now we calculate the average and standard deviation.

STEP 1: Subgroup Average
This step is same as what we have shown earlier in X-R Control Chart (Part 3).

STEP 2: Subgroup S
In state of R now we calculate the S value for each subgroup as follows,

And subsequently second subgroup will be,

STEP 3: Construct the table for X-S
To calculate the S value one by one using calculator it will take ages to do it. But if use uses Excel or other spreadsheet we can easily calculate the standard deviation of sample by formulas.

Above Table, I use formulas in Excel to construct. For average, simply =AVERAGE(B4:F4) and for standard deviation simply =STDEV(B4:F4) , if first subgroup data 1 to 5 is in cells B4:B5.

STEP 4: Calculate Control Limit
For X chart,

Hence,

In this case A3 is 1.427 and average of all subgroup is 3.10

Notice the result is almost same as what we get using average of range in Part 3.

For S chart,

Coefficient for control chart can be obtained from APPENDIX 1. In this case, B4 and B3 are 2.089 and 0 respectively.

Note:
The data use to calculate the control limit normal should not be the same set of data that we use to plot the X-S chart. But for education purpose, I show how we do it using the same set of data.

STEP 5: Plot the X Chart and S Chart

First, plot X chart with data of each X (average) against the subgroup with control limit line for X chart as shown below.

Both X charts in Part 3 and this are identical, although the control limits calculation is different.

Next, plot S chart with data of each S against the subgroup with control limit line for S chart as shown below

Now compare this chart with the R chart (Part 3), its point up and down is the same. The difference is only the y-axis scales and the value of the UCL, LCL and CL.

What this tell us is that the R chart and S chart actually shows us the same information for those set of data. The spread of the subgroup data.

Normally when the sample size is relatively small i.e. less than 10, range is used instead standard deviation. A typical choice of n for X-R chart is either 4 or 5.

Control Chart - Part 3

X-R Control Chart

We start by explaining the more commonly use variable control chart that is X-R Control Chart. Also known as X-bar & R chart. X or X-bar stand for average whereas R stands for Range

Below table is 1 month data of rod diameter measure 5 pieces daily using vernier caliper.

To plot a X-R chart we must first divide this data into subgroup A very simple consideration could be each day is consider a subgroup. This means we have a total of 30 subgroups for that month.

STEP 1: Subgroup - Average

Next for each subgroup, we calculate the average of the subgroup.

STEP 2: Subgroup - Range

Next for each subgroup, we calculate the range of the subgroup (maximum – minimum).

STEP 3: Construct the table for X-R

After calculate each subgroup average and range data we can construct a table as below.

STEP 4 Calculate Control Limit

Before plot the control chart we need to calculate the control limit for the X chart and R chart.

For X Chart,

 From above table, we can calculate

Coefficient for control chart can be obtained from appendix 1. In this case it is 0.577.

For R Chart,

Coefficient for control chart can be obtained from appendix 1. In this case it is 2.114 & 0 respectively.

Note: The data use to calculate the control limit normal should not be the same set of data that we use to plot the X-R chart. But for education purpose, I show how we do it using the same set of data.

STEP 5 Plot the X Chart and R Chart

First, plot X chart with data of each X (average) against the subgroup with control limit line for X chart as shown below

Notice that the X value fluctuated around the CL and all points are within the UCL and LCL. Such condition means that the process is stable. The UCL and LCL is calculated from the same data as X chart plot. It will always be within the UCL and LCL. The data use to calculate the UCL and LCL must be a process that is stable therefore the UCL and LCL can be use to evaluate abnormal situation

Next, plot R chart with data of each R (range) against the subgroup with control limit line for R Chart

Similarly, the R chart point also falls within the UCL and LCL.

Control Chart - Part 2

This is the second part

Process Illustration

Below is an illustration process distribution against time

Process (a) is very stable and under control within it limit. But process (b) and (c) is not under control. Process (b) distribution is same but the distribution shift against time. Process (c) distribution is variance varies with time.

In manufacturing, process variations or shifts are our biggest problems. If we could control the process in order to maintain always as (a) the defective will be at its minimum and productivity will be at its highest. But it is quite impossible to capture the distribution of the process every lots or everyday, it would take too much time and cost. Control chart is one of the best alternatives.

How Control chart works?

a)Process distribution shift towards UCL

The X chart display upwards shift similar to the distribution shift but the R chart does show normal movement since the variance of the distribution remain same.

b) Process distribution shift towards UCL

The R chart shows a tendency of moving upwards as the distribution variance becomes bigger. The X chart movement also shows slightly similar movement as the distribution.

The control chart can tell us the changes in the process distribution as time goes by. We can also monitor if our process is becoming more stable or not. Our action and improvement plan on the process shows any improvement can also be seen. It is a very useful tool in prediction of process stability. To plot a control chart we do not need to take many samples as compare to plotting the distribution of the process. 

Type of Control Chart
There are several type of chart that we could implement depending on the type of data and control measurement. Please refer below table




Variable control charts are more sensitive than attribute control charts. Therefore, variable control charts may alert us to quality problems before any actual "reject" (as detected by the attribute chart) will occur. The variable control charts are of trouble that will sound an alarm before the number of rejects increases in the production process.

Attribute control charts have the advantage of allowing for quick summaries of various aspects of the quality of a product, that is, the engineer may simply classify products as acceptable or unacceptable, based on various quality criteria. Thus, attribute charts sometimes bypass the need for expensive, precise devices and time-consuming measurement procedures. Also, this type of chart tends to be more easily understood by managers unfamiliar with quality control procedures; therefore, it may provide more persuasive (to management) evidence of quality problems. 

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.