# Statistical Quality Control

## Statistical Quality Control

### Statistical Quality Control

Chapter 6: Statistical Quality Control

Answers to Discussion Questions in Textbook

• Explain the three categories of statistical quality control (SQC).  How are they different, what different information do they provide, and how can they be used together?

The three categories of SQC are traditional statistical tools, acceptance sampling and statistical process control (SPC).  Traditional statistical tools are descriptive statistics, such as the mean and range, used to describe quality characteristics.  Acceptance sampling is a process of taking a random sample or portion of a batch and deciding whether to accept or reject the whole batch or lot.  SPC is a process that uses samples to determine whether a process is functioning normally or not.  Traditional statistical tools describe the quality characteristics, but do not tell us whether quality is good or bad.  Acceptance sampling tells us whether an entire batch or lot produced should be accepted or rejected after the goods have been produced, while SPC tracks the process over time to ensure it is functioning properly.  These tools can be used together effectively.  We use the traditional statistical tools as inputs into SPC, which is updated frequently enough to ensure that quality problems are caught in a timely manner.  Finally, after a batch has been produced, we use acceptance sampling to determine whether or not the batch can be sold to the customer.

• Describe three recent situations in which you were directly affected by poor product or service quality.

I purchased a bag of flour that I did not open right away.  I placed in it the kitchen cabinet.  Weeks later, I found bugs in many food items in the cabinet.   When I examined the bag of flour, I found dead bugs in the glued seal and inside the bag.  Because of the infestation, I had to throw out a number of food items from the cabinet.  I also had to treat the kitchen with a compound that would get rid of the remaining bugs.

I did not notice that the bag of sliced beef was expired when I purchased it from a grocery store.  A few days later, I opened it and ate some beef.  Within an hour, I became very sick.  I then looked at the package only to determine that it had expired 3 weeks earlier!  I was very upset.  I do accept some responsibility in that I did not always check expiration dates on items I purchased.  However, that package should not have been available for sale.  I now check expiration dates every time.  In addition, when I went to another facility of the same grocery store, I found that every package of sliced beef for sale was expired by at least a week.  So I took all of the packages of beef up to the help desk to give them to a manager and complain.  He apologized saying that he would deal with the problem.

• Discuss the key differences between common and assignable causes of variation.  Give examples.

Common causes of variation are random, which means that there is not a specific reason for the variation, such as a malfunctioning machine.  If you look at 2-liter bottles of a soft drink on a shelf at a retailer, you will notice that they are not filled to the exact same level.  That is to be expected, since a machine cannot fill each one to exactly 2 liters every time.  Assignable causes of variation are not random.  Some examples of assignable causes are a machine in need of repair and defective raw materials from our supplier.

• Describe a quality control chart and how it can be used.  What are the upper and lower control limits?  What does it mean if an observation falls outside the control limits?

Samples of product or services are plotted on the control chart over time.  We then interpret the chart to determine whether the variation in the process is normal or abnormal.  We need to use the chart because most products and services exhibit some variation.  If the variation is normal, the process is assumed to be “in control.”  Therefore, we do not need to fix the process.  If the variation is abnormal, this process is assumed to be “out of control.”  Therefore, we need to determine what is causing this variation, such as a malfunctioning machine or untrained worker.  Control charts do not tell us how many defects are being produced by the process.  The charts tell us whether it is likely that the process has changed.  If the process has changed or is out of control, then it is likely that we are producing defects.

A control chart is a diagram with a center line, an upper control limit and a lower control limit.  The upper control limit is normally equal to the mean of the sample data plus three standard deviations.  The lower control limit is normally equal to the mean minus three standard deviations.  If an observation falls above the upper control limit or below the lower control limit, the process is assumed to be out of control.  This is because the chart is based on the normal distribution, which states that 99.7% of the plots will fall within three standard deviations of the mean.  Since it is highly unlikely that a plot will fall outside three standards deviations of the mean, the process is likely to be out of control.

• Explain the differences between x-bar and R-charts.  How can they be used together and why would it be important to use them together?

The x-bar chart is used to detect variations in the mean of the process, while the R-chart is used to detect changes in the variability of the process.  The x-bar and R-charts are used when the data is a variable, meaning that we can collect data using decimal points, such as 16.5 ounces.  Examples of variables are weight, height and temperature.

The x-bar and R-charts should be used together.  Think about preparing a Thanksgiving turkey in the oven.  What can go wrong with the temperature of the oven if it is set at 350 degrees?  The average temperature during cooking could be 250 degrees instead.  On the other hand, the temperature could average 350 degrees, but actually fluctuate during cooking time between 200 and 500 degrees.  Either way, the turkey will not be properly cooked in the oven.  The inaccurate average temperature would have been detected by the x-bar chart.  The changes in the temperature would have been detected by the R-chart.  We use these charts together by plotting the average of the sample on the x-bar chart and the range (high temperature in the sample minus the low temperature) on the R-chart.  Then, we first interpret the R-chart.  If it is out of control, then the process variation is out of control.  The next step would be to investigate the cause of this problem.  There is no need to interpret the x-bar chart if the R-chart is out of control.  If the variation is out of control, it is not possible to make conclusions about the average because the variation would probably change the average.  If the R-chart is in control, then we interpret the x-bar chart.  If it is out of control, then the process average is out of control.

• Explain the use of p-charts and c-charts.  When would you use one rather than the other?  Give examples of measurements for both p-charts and c-charts.

The p-charts and c-charts are both used when the data is an attribute.  Data is an attribute when we ask a yes or no question or count the number of defects.  For example, is the bottle of Coca-Cola full or not?  How many of the Hershey’s kisses in the bag are not covered in foil?  The p-chart is used to determine whether the proportion of defective units in a sample is in control or not.  The c-chart is used to determine whether the number of defects on each item is in control or not.  The key difference is that the sample size for the c-chart is always one.  In other words, each plot represents the number of defects on one item, such as the number of spelling errors in a report.

• Explain what is meant by process capability.  Why is it important?  What does it tell us?  How can it be measured?

Management or regulations set acceptable levels of variation in order to determine if a product is defective or not.  For example, a product is not defective if it is filled to 16 ounces plus or minus one ounce.  For this product, the upper specification limit would be 17 ounces, while the lower specification limit would be 15 ounces.  Process capability tells us whether or not the process itself is capable of manufacturing product that has a high probability of falling within the specification limits (is not defective).  It is important for a company to produce quality products.  Process capability is measured by comparing the specifications to the actual variation in the process.  The process capability index is the width of the specifications divided by the width of the process variation.  If the process capability index is less than one, then the process is not capable of producing within specifications.  The higher the index, the more capable the process.  The index can be used to determine how many defects are produced on average.

• Describe the process of acceptance sampling.  What types of sampling plans are there?  What is acceptance sampling used for?

In acceptance sampling, we determine if a set number of items, such as fifteen out of a batch of 100, are defective or not.  Then we compare the number of defects to a preset maximum number of acceptable defects (c) to determine whether to accept or reject the whole batch.  If the number of actual defects is less than or equal to the preset number, then the entire batch is accepted.  Otherwise, it is rejected.  Acceptance sampling is used to determine whether the level of quality is acceptable or not.  The three types of sampling plans are single sampling, double sampling and multiple sampling.

• Describe the concept of six-sigma quality.  Why is such a high quality level important?

It is referred to as six sigma because the specification limits are six sigma away from the mean.  We know that the normal process variation is three sigma away from the mean.  So, if we divide the width of the specification limits, which is six sigma plus six sigma, divided by the width of the process variation, which is three sigma plus three sigma, then the process capability index is calculated as 2. Therefore, a process with a six-sigma quality level has a process capability index of 2.  This means that the process will make about 3 defective items for every million produced if the process is centered on the mean.  A process is centered on the mean when the process mean and the desired mean are equal.  If they are not equal, then the process has shifted to the left or right, thus causing more defects in the tail of the distribution.  An example will clarify this issue.  Assume that the following data was collected from the process of filling jars of spaghetti sauce that has a six-sigma quality level when the process is centered:

• Process mean = 16.5 ounces
• Desired mean = 16 ounces (what is printed on the jar)
• Process standard deviation = 0.5 ounces
• Upper specification limit = 17 ounces
• Lower specification limit = 15 ounces

This process would produce jars that are filled above 17 ounces since the process mean is greater than the desired mean.  (It might be helpful to show this information on a graph.)

This level of quality is important because customers demand a high level of quality.  In the past, parts per thousand defective was the measure used by companies.  Now the measure is parts per million (ppm) defective.

Web site to visit: http://www.sba.oakland.edu

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#### Statistical Quality Control

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##### Statistical Quality Control

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