Statistical Quality Control: Defects, Variables, and Control Charts
1. Defects
1.1 Introduction.
Today, all modern companies understand that building a good level of quality is fundamental to successful management. Achieving this goal is important not only for competition but also for satisfying human needs.
These human needs are constantly evolving; there is ever-increasing demand for higher precision, more accuracy, interchangeability, comfort, and so on. Consumers’ expectations are constantly refined, and any company that doesn’t adapt risks being displaced.
To keep pace, better tools, machinery, and methods are needed. More importantly, these resources must be used more effectively—achieving better quality for the same cost. Statistical quality control is a powerful tool to achieve this.
1.2 Definition of Quality
Quality has two aspects: Quality of Design and Product Quality.
Quality of Design: The degree of agreement between the design and its intended purpose.
Product Quality: The degree of conformity between the product and its design.
The concepts and methods applied to product quality control are generally universal, valid for any product, from toothpaste and soft drinks to tractors and pharmaceuticals.
A good level of quality means a proper design and a product conforming to its design. A defect is a breach of a quality characteristic exceeding a specified limit.
1.4 What Causes Defective Products?
The universal answer is variation in materials, machine conditions, work methods, and inspections. These variations cause defective products. Without these variations, all products would be identical, and defects would not occur.
1.5 Should All Defects Be Treated Equally?
Common sense says no. A slight surface imperfection on a product label is different from a critical defect in a car part. Different criteria should be used to tolerate these defects, resulting in different quality plans for different defect types.
1.6 Classification of Defects
Critical Defects: Violate laws, harm the consumer, or render the product unusable.
Major Defects: Decrease product function or usability, noticeable to the consumer.
Minor Defects: Slightly decrease proper operation; may not be noticed by the consumer, but would be by skilled production and quality control personnel.
Each defect type requires a study resulting in a sample of defects, classified by type and signed by involved parties. Ideally, the sample should include defects at the acceptance/rejection limit.
A population is the total number of units considered.
A sample is a statistically calculated subset of population units, each drawn randomly.
Measuring and calculating a given feature estimates the true population value.
2.2 How Are Variable Values Distributed?
Variations produce different measurements of a variable. The question is how these variations are distributed. Values near the central value are most frequent; frequency drops dramatically as we move away from the central value. This behavior is bell-shaped.
2.3 Types of Variables
There are two types: continuous variables and discrete variables.
Continuous variables are measured; discrete variables are counted.
Continuous variables lead to variable control; discrete variables lead to attribute control.
Quality characteristics are variables represented by a figure (e.g., pin size, wire resistance). Attributes are unmeasured quality characteristics, not usually represented by a figure (e.g., surface imperfections).
Processes and finished lots can be inspected by attributes or variables.
Arithmetic Mean and Standard Deviation
The arithmetic mean measures central tendency, and the standard deviation measures data dispersion around the mean.
4. Continuous (Gaussian or Normal) Distribution
4.1 Understanding the Concept
A histogram is constructed from data. Increasing the data and reducing the class interval gradually results in a continuous frequency distribution. The normal distribution is common; it occurs when variation is caused by many small, independent errors. Its shape is bell-shaped.
5.2 Fishbone (Ishikawa) Diagram: Cause and Effect
5.2.1 What Are Cause and Effect Diagrams?
These diagrams show the relationship between a quality characteristic and the factors it depends on (including causes of quality problems).
6. Control Charts
6.1 What Are Control Charts?
In 1924, Walter A. Shewhart proposed control charts to distinguish variations due to assignable causes from those due to random (non-assignable) causes.
A control chart has a central line, control limits (upper and lower), and plotted characteristic values. If values are within limits without trends, the process is in control. Values outside limits or showing trends indicate an out-of-control process.
Product quality inevitably varies. Variations have different causes, classified as:
Random (Non-Assignable) Causes
These chance variations are inherent to the process; removing them is often impractical or expensive. Within limits, these variations are tolerable and don’t significantly reduce quality. They are considered normal variations and result in the Gaussian distribution.
Assignable Causes
Variations due to assignable causes are abnormal and must be investigated. These are not normal, don’t belong to the process, and are unacceptable.
Assignable causes may lead to defective products (though not always). They contain features outside the quality specifications.
Statistical Quality Control aims to find and remove assignable causes (even if they don’t cause defects).
Points outside control limits or showing trends indicate an out-of-control process due to assignable causes.
Note: Later, we’ll discuss limits and unspecified values. An out-of-control process doesn’t always result in defects. These definitions are for baseline understanding; refer to later developments for a complete understanding of controlled states and specification compliance.
Purpose of a Control Chart
The goal is to keep the process in control. Two charts are used: one for accuracy (X-chart) and one for precision (R-chart).
Both charts are necessary; neither is sufficient alone. If either chart shows an out-of-control state, the process is declared out of control.
To understand a process and whether it’s under control, we need to know the chance variation. Control charts provide this knowledge. Small samples are taken at regular intervals to minimize variation factors between units.
Samples of 3 to 10 units are common (3 to 6 is best; 5 is ideal).
Several control chart types exist, depending on purpose and variable characteristics. Control limits are calculated using specific formulas.
6.2 Types of Control Charts
There are two types: one for continuous values and one for discrete values. Several alternatives exist for each type.
Figure 6.2.1 X – R Chart
Used for continuous values (length, weight, concentration). X represents the average of a small sample; R is the range. An X-chart is used with an R-chart to control variation within a subgroup.
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Figure 6.2.2 np, p Charts
Used when the quality characteristic is the number of defective units or the fraction defective. For a constant sample size, an np-chart shows the number of defective units; a p-chart shows the fraction defective for variable sample sizes. Other attribute charts include c and u charts.
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6.3 Specifications, Tolerances, Discrepancies
Before creating control charts, we must distinguish between specifications, tolerances, and control limits with no specified value.
Specifications and tolerances are given by the customer or product designer. Specifications indicate desired product characteristics. Tolerances define acceptable limits; exceeding these limits results in a defective product.
In summary:
a) Specifications and tolerances are criteria not necessarily related to production processes; they usually relate to product design or aesthetics.
b) If a product’s measured variable exceeds specifications, it’s declared defective.
Regarding the machine and process producing the product, we ask:
Can the machine produce the product within the specified tolerances? Will it produce defect-free products?
Process Capability and Machine Capacity address this question.
6.4 Process Capability and Machine Capacity
Process Capability study involves several steps. First, process data is collected and used to calculate Natural Process Limits (also called Process Limits Without Specified Value). These are the normal limits of the process when operating normally. Exceeding these limits indicates an out-of-control process (if only one point).
Specifications and tolerances are transformed into Process Limits With Specified Values.
These Process Limits With Specified Values are the specifications, translated to a form comparable to the process limits without specified values.
If the process can operate within the specified value limits, it will comply with specifications and produce no defects. The specified value limits “defend” the specifications.
Comparing limits with and without specifications determines whether the process can produce flawless products.
If the range of specified value limits is greater than the range of unspecified value limits, the process is capable of meeting specifications. If the range of specified value limits is less than the range of unspecified value limits, the process is incapable of meeting specifications, and defects will occur.
Analogy: If specified value limits are a shoebox and unspecified value limits are shoes, the process is capable if the box is larger than the shoes. If the shoes are larger than the box, the process is incapable.
7. How to Make a Control Chart
Figure 7.1 X – R Chart
The chart has three areas: data recording, accuracy control (arithmetic mean plot), and precision control (ranges).
7.2.2 Calculation of Limits With Specified Values
These limits are another expression of the specifications. They are translated to limits for subgroups of n units (e.g., n=5).
A specification is a value determined by the client or designer, with a tolerance within which the product is satisfactory.
7.6 Process Capability Index
This index determines whether a process can meet specifications. A sample is taken, a frequency distribution and histogram are created, and the standard deviation is calculated. The Process Capability Index is then calculated.
A very wide Process Capability Index (e.g., over 2) means the specified value limits are very broad and can accommodate assignable causes without producing defects. This is undesirable because the variability may be noticed by the customer, leading to a poor product image or loss of supplies/money. The specified value limits should be narrowed.
In other words, “we must shrink the shoebox.”
8.2 Standard for Selecting Random Samples
This rule formalizes the method of extracting representative units from a lot or population using random number tables.
8.3.2 Acceptable Quality Level (AQL)
8.3.2.1 Use
AQL, along with a sample size code, identifies sample plans.
8.3.2.2 Definition of AQL
AQL is the percentage of defective batches with a ~95% probability of being approved.
8.3.3 Normal, Tightened, and Reduced Inspection
Production begins with normal inspection. After a sequence of approved lots, reduced inspection may be used. Tightened inspection is used if two out of five lots are rejected. Prolonged use of tightened inspection requires process improvement.
8.3.4 Sampling Plan
A sampling plan indicates the number of units to inspect in each lot and the acceptance/rejection criteria.
8.3.5 Inspection Level
The inspection level determines the relationship between lot size and sample size. Level II is common.
8.3.6 Types of Sampling Plans
Single, double, and multiple sampling plans exist. The choice depends on administrative difficulties and average sample size. Multiple sampling usually has a smaller average sample size than double sampling, but greater administrative difficulty.