Statistical Quality Control


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Control Charts
What are control charts?
A control chart is a popular statistical tool for monitoring and improving quality. Originated by Walter Shewhart in 1924 for the manufacturing environment, it was later extended by W. Edward Deming to the quality improvement in all areas of an organization (a philosophy known as Total Quality Management, or TQM).

The purpose of control charts
The success of Shewhart's approach is d on the idea that no matter how well the process is designed, there exists a certain amount of nature variability in output measurements.

When the variation in process quality is due to random causes alone, the process is said to be in-control. If the process variation includes both random and special causes of variation, the process is said to be out-of-control.

The control chart is supposed to detect the presence of special causes of variation.

In its basic form, the control chart is a plot of some function of process measurements against time. The points that are plotted on the graph are compared to a pair of control limits. A point that exceeds the control limits signals an alarm.

An alarm signaled by a control chart may indicate that special causes of variation are present, and some action should be taken, ranging from taking a re-check sample to the stopping of a production line in order to trace and eliminate these causes. On the other hand, an alarm may be a false one, when in practice no change has occurred in the process. The design of control charts is a compromise between the risks of not detecting real changes and of false alarms.

Assumptions underlying Control Charts
The two important assumptions are:

The measurement-function (e.g. the mean), that is used to monitor the process parameter, is distributed according to a normal distribution. In practice, if your data seem very far from meeting this assumption, try to transform them.
Measurements are independent of each other.
Constructing a 3-sigma ("Shewhart-type") control chart
During a stable s e of the process:

Determine the process parameter that you want to monitor (such as the process mean, or spread).
Create the centerline of the plot, according to the target value of your monitored parameter.
Group the process measurements into subgroups (samples) by time period. The points to be plotted on the plot, are some function of the process measurements within each subgroup, which estimate the target value.
For example, if you are monitoring your process mean, then the points on the plot should be the sample-means, computed at regular intervals. Denote the point at time t as Xt
Create upper and lower control limits (UCL,LCL) according to the following formula:
UCL = CL + 3 s
LCL = CL - 3 s
where s is the standard deviation of Xt.
For the example above, Xt may be daily means of process measurements. If each daily sample comprises of n measurements, then the standard deviation of Xt is equal to the process standard deviation divided by the root of n

After the control limits have been set, continue to plot the points on the graph, as a function of time. When a point exceeds the control limits, it indicates that the process is out of control, and action should be taken (of course, there is a slight chance that is is a false alarm).

Uses of Control charts:-

Control chart is a device for describing in a precise manner what is meant by statistical control. Its uses are :-

1-It is a proven technique for improving productivity.
2-It is effective in defect prevention.
3-It prevents unnecessary process adjustments.
4-It provides diagnostic information.
5-It provides information about process capability.

Types of control charts
Control charts for Attributes.

p chart
c chart
u chart

Control charts for Variables.

X bar chart
R chart

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Acceptance Sampling

What is it and what are acceptance sampling plans?
Acceptance sampling is a procedure used for sentencing incoming batches. The most widely used plans are given by the Military Standard tables, which were developed during World War II.

Types of acceptance sampling plans
Sampling plans can be categorized across several dimensions:

Sampling by attributes vs. sampling by variables: When the item inspection leads to a binary result (either the item is conforming or nonconforming) or the number of nonconformities in an item are counted, then we are dealing with sampling by attributes. If the item inspection leads to a continuous measurement, then we are sampling by variables.

Incoming vs. outgoing inspection: If the batches are inspected before the product is shipped to the consumer, it is called outgoing inspection. If the inspection is done by the consumer, after they were received from the supplier, it is called incoming inspection.

Rectifying vs. non-rectifying sampling plans: Determines what is done with nonconforming items that were found during the inspection. When the cost of replacing faulty items with new ones, or reworking them is accounted for, the sampling plan is rectifying.

Single, double, and multiple sampling plans: The sampling procedure may consist of drawing a single sample, or it may be done in two or more steps. A double sampling procedure means that if the sample taken from the batch is not informative enough, another sample is taken. In multiple sampling, additional samples can be drawn after the second sample.

Military Standard 105E (ISO 2859, ANSI/ASQC Z1.4)
The original version of the standard (MIL STD 105A) was issued in 1950. The last revision (MIL STD 105E) was issued in 1989, but canceled in 1991. The standard was adopted by the International Standards Organization as ISO 2859.

The tables give inspection plans for sampling by attributes for a given batch size and acceptable quality level (AQL). An inspection plan includes: the sample size/s (n), the acceptance number/s (c), and the rejection number/s (r). The single sampling procedure with these parameters is as follows: Draw a random sample of n items from the batch. Count the number of nonconforming items within the sample (or the number of nonconformities, if more than one nonconformity is possible on a single item). If the number of nonconforming items is c or less, accept the entire batch. If it is r or more then reject it. In most cases r =c+1
(for double and multiple plans, there are several values for the sample sizes, acceptance, and rejection numbers).

The standard includes three types of inspection (normal, tightened, and reduced inspection). The type of inspection that should be applied depends on the quality of the last batches inspected. At the beginning of inspection, normal inspection is used. The types of inspection differ as follows:

Tightened inspection (for a history of low quality) requires a larger sample size than in under normal inspection.

Reduced sampling (for a history of high quality) has a higher acceptance number relative to normal inspection (so it is easier to accept the batch)
There are special switching rules between the three types of inspection, as well as a rule for discontinuation of inspection. These rules are empirically d.

Supplier and Consumer Risks
The supplier risk is the risk that the a batch of high quality (according to the AQL) is rejected. The consumer risk is the risk that a batch of low quality will be accepted.

The military standard plans assure a supplier risk of 0.01-0.1 (depending on the plan). The only way to control the consumer risk is by changing the inspection level.

Acceptance Limit (c)
The upper limit on the number of non-conforming items in a sample, that would still lead to the acceptance of the entire lot. If the number of non-conforming items in the sample exceeds this number, the entire batch must not be accepted.

Average Outgoing Quality Limit (AOQL)
The highest/worst possible average percent of non-conforming items in the process, after employing some inspection scheme. This measure is usually used in rectifying inspection, where the inspection procedure changes the outgoing rate of non-conforming items in the batch or process, relative to the incoming rate. For example, by removing the non-conforming items that are encountered during inspection. Note that this is only the worse possible average percent of non-conforming items, and therefore there is still a possibility that the percent non-conforming of a single batch will exceed this limit.

Average Run Length (ARL)
The mean (average) of the run length. This is the average number of samples that are taken until an alarm is signaled by the control chart.

Acceptable Quality Level (AQL)
The maximal percent of nonconforming items (or the maximal number of nonconformities per 100 items), which is considered, for inspection purposes, as a satisfying process mean.

The AQL is generally specified by the authority responsible of sampling. Different AQLs may be designated for different types of defects. It is common to use an AQL of 1% for major defects, and 2.5% for minor defects.

Values of AQL that are 10% or less are suitable for percent nonconforming or nonconformities per 100 items. Values of AQL over 10% are only suitable for nonconformities per 100 items.

Alarm Zones
The alarm zones in an ordinary 3-sigma control chart are beyond the upper control limit (3,infinity), and below the lower control limit (-infinity, -3).

To specify the alarm zone as the area between the warning limits and control limits, enter a=2, b=3.
In a control chart with 0.001 probability control limits (3.09 "sigma") and 0.025 warning limits (2.24 "sigma"), the rule "two consecutive points between the control and warning limits" is given by: k=2, a=2.24, b=3.09.
Alarm zones are usually symmetric around the center line. For example: and .
A batch is a collection of items from which a sample will be drawn, for deciding on its conformance to the acceptance inspection. A batch should include items of the same type, size, etc. and that were produced under the same production conditions and time.

The batch size is the number of items in a lot or a batch.

Clearance Number
The number of consecutive items (or batches, in Skip lot sampling) that must be found conforming, in order to quit the screening phase (100% inspection) when applying continuous sampling