Six Sigma SPC - Statistical Process Control

Statistical Process Control Overview
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Histogram - Distribution Charts

Zones on Control Charts

X-Bar and R-Bar

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Part of any quality control, ISO9000, quality assurance, or Six Sigma plan must use some type of SPC, (statistical process control), to implement control charts to control the processes. The origins of SPC, or statistical process control, date back to the 1920ís and Dr. Shewhart. Statistical process control was not well received by U.S. manufactures, so Dr. Shewhart first applied his theories in Japan.

The concept is to be able to control your processes by monitoring changes in the process, environment, or labor, using control charts. This is achieved by collecting random data samples from groups or sub-groups of product, (see stratified sampling), at various stages of the product's manufacturing using inspection methods. You can then start plotting the data points on a control chart, which is a line chart. Be aware of any sampling bias problems that may occur.

You must also create a distribution chart to know how your data is grouped as a whole. The following is for data that is called variable. Variable data can be taken in sample sizes, or sub-groups of 2 to 5 pieces thus the data can have a range, (R Bar), and an average or mean also known as X-Bar.


Range is important so that you can tell if the entire process is shifting and by how much, or if the any shifting is just during certain times. Like when  the next shift starts, when material from supplier X starts being pulled from stock, or when the environmental variables change. (humidity or temperature and the like)


This is the arithmetic mean or average. Simply take the total measurements from all the samples and divide by the number of sub-groups.

An example of would be like temperatures taken in  different spots of a large stadium. If you took 4 readings from the ...

Measurement Point Actual Measurement
North 68.5
South 75.5
East 72.3
West 73.4

Your range would be 7.0, (75.5 - 68.5 = 7.0)
Your average, (mean), would be 72.425
(68.5 + 75.5 + 72.3 + 73.4 = 289.7)
(289.7 / 4 = 72.425)

There are 2 type of samples that we deal with. One we will call the observations or sub-groups, and the other we will call the sample. You must decide how many samples you need to take to represent the population for an acceptable result . Think of it as putting the data into a grid. We will call the observations, or sub-groups columns and the samples rows. So the sub-group is a portion of the entire sample you choose. (see stratified sampling and sampling bias)

If we decide to take 30 samples, then this could be put into ...

2 sub-groups of 15 observations (2 rows and 15 columns) 2x15=30
6 sub-groups of 5 observations (6 rows and 5 columns) 6x5=30

Some use 30, which is a number that was acquired from the 1930's and based on how many widgets can be produced in one hour now vs. then, it is not very accurate for today's industries. Motorola at one time used 75 and Juran says to use 125. However many you use, it must be able to be divided by the observations. So if you used 30 as a sample, you could 2 columns of observations and 15 rows of samples, or 2 x 15 = 30. You could also use 5 columns of observations and 6 rows of samples, or 5 x 6 = 30.

Note: try to do this so that you do not need to change the observations. There are advantages and disadvantages to using both 2 and 5, which are the most popular to use. The observation is important because it is the basis for the factor table you use to establish control limits. I just find that if you don't keep changing this, you get better results over the long run.

When Motorola used 75, they wanted it as 5 columns of observations and 15 rows of samples, or 5 x 15 = 75.

If our widget is a gear, there could several parameters that we need to chart, but we are only going to look at diameter for simplicity.

We only take the large sample size once to establish control limits. After that we add 1 sample of how ever many observations we choose, typically 2-5,  to the grid and calculate the mean and range. So for t he example, we will take 15 samples using 3 observations. Our grid would look like this...

Part Number: 54T54
Parameter: - Diameter Blue Print Location: A-2
Minimum - .195
Maximum - .208
Nominal - .201
n Date/Time x1 x2 x3 X-Bar Range
1 4/7/1997 22:54 0.206 0.194 0.203 0.201 0.012
2 4/7/1997 23:55 0.196 0.192 0.223 0.204 0.031
3 4/7/1997 24:55 0.215 0.220 0.238 0.224 0.023
4 4/7/1997 25:55 0.230 0.234 0.229 0.231 0.005
5 4/7/1997 26:55 0.223 0.230 0.227 0.227 0.007
Totals 1.087 0.078
Avg. 0.217 0.016

Nominal is what the process typically runs at. If you have an upper and lower specification  limits, then the nominal is usually the middle of that number, (Max Spec - Min Spec) / 2 + Min Spec or in  this case...
(.208 - .195 = 0.013 / 2 = 0.0065 + .109 = 0.2015)

The totals of the Average at the bottom of 0.0217 and 0..16 are also known as Double X-Bar and Double R-Bar.

Zones on Control Charts

When doing X Bar and Range charts, one needs to establish upper control limits and lower control limits. These limits are +/- 3 sigma from the Double X-Bar.

The X Bar control chart should have sigma zones applied to them in order to look at trends. WECO, (Western Electric Company), rules, are used for trending with the zones. For more information on zones and WECO rules as they apply to X-bar and R bar control charts, please see X-Bar and Range Control Charts.

Distribution Charts

Distribution charts MUST be used with X-Bar and Range charts. Not using these 3 types of charts together can and does lead to a misinterpretation of the data. A distribution chart shows how the data is distributed over the entire range.

The center of the distribution is the mean, (X-Bar or average), of the data being evaluated. The data is stacked in cells around the mean. A typical normal distribution will have stacks that look similar to a bell, called a bell shape curve. See Below.

Example Distribution Chart

Using ZeroRejects, you can easily see where your data is in relation to your specification and design margin.

For addition information...

Process or Product Monitoring and Control

* Reference Quality Planning and Analysis - Juran/Gryan

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Last Updated: Friday, 09-Jun-06 21:09:57 PDT