Introduction to Quality Engineering
Chaper Eighteen
Sampling Theory and Practice
Copyright © March 7, 2024 by Robert Wayne Atkins, P.E.
All Rights Reserved.
The following information is Chapter Eighteen in my book: Introduction to Quality Engineering.
Sampling is critical in order to make valid decisions about the quality of a product or a service. Therefore sampling is discussed in many of the chapters of this book because sampling is extremely important to the topic discussed in those chapters.
However, in order to provide a reasonable summary of all the different aspects of sampling, sampling is reviewed by itself in this chapter. However, no equations are included in this chapter. The actual equations that are relevant to sampling are included in each of the chapters that are based on sampling.
The Primary Objective of Sampling
Samples are collected and analyzed for lots of different reasons. And there are a lot of different ways to collect sample data.
However, the primary objective of sampling is to collect enough data so that the characteristics of the sample are very close to the characteristics of the population from which it was drawn. If the characteristics of the sample are not very close to the characteristics of the population then any statistics that are calculated based on an analysis of the sample will not be representative of the population. Therefore any decisions made on those statistics will probably not be good decisions.
Definitions
- Population: A population contains 100% of everything that is of importance to us. A population may also be called a group or a lot or a batch. Depending on how the group is defined, it may be possible to examine everything in the group. For example, if a group is defined to be a box of 100 parts then it may be feasible to examine every part in the box.
- Sample: A sample contains something less than 100% of everything that is of importance to use. For example, if a group is defined as 2,000,000 parts then it will probably not be feasible to examine every part. In this situation it would be desirable if we could select a sample from the 2,000,000 parts and make a decision based on a careful examination of the parts in the sample.
- Representative Sample: This type of sample should be collected in an unbiased manner so that the characteristics of the sample are very close to the characteristics of the population from which it was selected.
- Specimen: If a specimen is collected from the population of interest, but it is not possible to verify that the characteristics of the specimen are approximately the same as the population, then the specimen is not a representative sample.
Sampling Methods
- Simple Random Sampling: Each item in the population has an equal chance of being included in the sample. This is usually the preferred method for selecting an unbiased sample.
- Systematic Sampling: Items are selected from the sample based on some sequence such as every tenth part on an assembly line.
- Stratified Sampling: The population is divided into groups and all the items in a group have the same characteristic. The sample is selected so that the sample has the same ratio of characteristics as the population. If a company makes hats in three sizes: 20% small, 50% medium, and 30% large, then a sample of hats should contain the same ratio of sizes as the population.
- Quota Sampling: This is similar to stratified sampling. The population is divided into groups and all the items in a group have the same characteristic. A decision is then made on how you wish for each group to be represented in your sample. If you want each group to be equally represented then you randomly select the same number of items from each group. But if you want the different groups to be represented in some other manner, then you randomly select items from each group based on the percent representation that you want that group to have in the final sample.
- Convenience Sampling or Opportunity Sampling: Items that are easy to access are included in the sample. This is not a good way to collect a sample. For example, if a box of 200 neatly stacked parts is opened, and 5 parts are randomly selected from the top layer of parts, then this would be a convenience sample because the parts below the top layer had no opportunity to be included in the sample.
- Judgment Sampling: The analyst selects items from the population using his judgment to draw a sample that appears to be representative of the population that the analyst can see. For example, a box contains 200 randomly jumbled apples. The analyst selects 5 apples from the box and the analyst uses his judgment to select the sample so that the 5 apples in his sample have the same general appearance as the remaining apples in the box. In other words, the analyst does not pick all the best apples or all the worst apples. Instead the analyst attempts to pick a sample that closely matches the mixture of apples the analyst can see in the box.
Minimum Sample Size
The number of items included in a sample can have a significant impact on whether or not the sample is representative of the population.
Generally, as the number of items in the sample increases, the more closely the sample will be representative of the population.
However, there is a point of diminishing marginal returns. At some point the sample will be large enough to very closely approximate the characteristics of the population. When this happens then adding more items to the sample will not improve its ability to represent the population. However, adding more items will increase the time required to collect the sample, and the amount of money invested to collect the sample, and the time to analyze the results of the sample, and the cost to analyze the results of the sample. And this extra time and money will have been wasted because the extra word did not improve the final results of the sampling process.
However, adding more items to a sample will make a very trivial change in the final statistics that are calculated based on the sample. But the trivial change in the final statistics may have no impact on the decision that is made based on the sample data.
For example, assume that the average weight of a part must be between 0.749 grams and 0.757 grams in order for it to perform correctly in every application in which it will be used.
1. A sample of 50 parts shows an average weight of 0.7532 grams.
2. A sample of 125 parts shows an average weight of 0.7531 grams.
3. A sample of 600 parts shows an average weight of 0.7533 grams.
4. A sample of 4,000 parts shows an average weight of 0.7530 grams.
In the above example, each time more parts were included in the sample the final answer changed by a trivial amount. But the answer had statistically stabilized after 50 parts had been weighed because the weight of the parts was almost in the center of the acceptable range of weights for this part. Collecting more data did not change the final decision for these parts. Collecting more data only cost more money and it delayed making a decision that could have been made sooner.
Future chapters include the equations that can be used to calculate the minimum sample size based on what is being measured.
Where Sampling is Done
When making decisions about quality, different sampling strategies are appropriate based on where the sample data will be collected. In a manufacturing environment there are three primary places where sampling needs to be done.
- Receiving Dock: Purchased raw materials and component parts should be inspected at the receiving dock before those items are transferred into the Raw Materials Warehouse. Acceptance sampling plans are designed to be used at the receiving dock and the sample size will usually be based on the documented historical level of quality that each supplier has demonstrated in the past.
- Manufacturing Area: Each step in the manufacturing process should be verified before products are moved to the next step in the manufacturing process. Control charts are used at manufacturing processes and the sample size will usually be relatively small compared to the sample sizes used at the receiving dock. The reason is because samples will be collected all day long in manufacturing and not just at one time when a shipment is received. If 10 parts are measured each hour in manufacturing then 80 parts will be measured during one 8-hour shift, and 400 parts will be measured in a 5-day workweek.
- Finished Goods: When a product has completed the last step in the manufacturing process, and before it is transferred into the Finished Goods Warehouse (or shipped to a Distribution Center), it should be inspected to verify that it is an acceptable product and that it meets all the relevant quality standards. This is the last chance for a company to detect inferior quality before it is sold to a customer. The sample size should be based on what is being measured and the normal variability of those measurements.
The sampling procedures are different for each of the above scenarios. The sampling procedure that is recommended for each area should only be used in that area because it was designed to be used in that type of situation. A sampling procedure should not be used in an area where it will not work correctly.
Conclusion
Sampling plans that are used in research and development, and in psychology, and it demographic sampling, and in political assessments, are appropriate for those situations.
The sampling plans that are used in quality applications are appropriate for quality type decisions. The sampling plans recommended in this book are representative, efficient, non-biased, accurate, precise, valid, and cost-effective.
Sampling procedures, sampling plans, and minimum sample sizes will be discussed in detail in the other chapters of this book that are based on the different types of samples that should be collected.
Click here to read more information about the book: "Introduction to Quality Engineering."
Grandpappy's e-mail address is: RobertWayneAtkins@hotmail.com