# Statistical Quality Control

Statistical quality control is the use of statistical methods in the monitoring and maintaining of the quality of goods and services. When a decision regarding accepting or rejecting a group of parts or items based on the quality found in a sample must be made, a method called acceptance sampling can be used. There is also another method called statistical process control. It uses graphical displays known as control charts to determine whether a process should be continued or adjusted to achieve the desired quality.

Acceptance Sampling

Suppose you are a retailer of technological equipment and you receive a shipment from the manufacturer. You will take a sample of parts and count the defective items. You will accept the entire lot if the number of defective items is low. On the other hand, you will reject the entire lot if the number of defective items is high. Making a good decision corresponds to rejecting poor quality items and accepting high-quality items.

The probabilities of erroneous decisions need to be considered because sampling is being used. The manufacturer will have a problem if the consumer makes an error of rejecting a good-quality lot. The probability of this error is known as the producer’s risk. Similarly, the consumer or purchaser will have a problem if they make the error of accepting poor-quality goods. The consumer’s risk is the name of the probability of this error.

Designing an acceptance sampling plan involves determining a sample size n and an acceptance criterion c.  The c refers to the maximum number of defective items that can be found in the sample and the lot still be accepted. To understand the consumer’s risk and the producer’s risk better, you should assume that a lot has some known percentage of defective items and compute the probability of accepting the lot for a given sampling plan. Many different sampling plans can be evaluated and a sampling plan is selected by varying the assumed percentage of the defective items in a lot. This guarantees that both the producer’s and consumer’s risks are reasonably low.

Statistical Process Control

Statistical process control is a method that uses statistical and sampling techniques to monitor the quality of an ongoing process like production operation. A control chart, which is a graphical display provides a basis for deciding whether the variation in the output of a process is because of common causes or out of the ordinary assignable causes.

A decision can always be made to adjust the process whenever assignable causes are identified. This will bring the output back to acceptable quality levels. We can use the type of data they contain to classify control charts. For example, in situations where a sample mean is used to measure the quality of the output, we use an x-chart. This chart can be used to monitor quantitative data like length, temperature, and weight. On the other hand, the range or R- chart can be used to monitor process variability. An np-chart or a p-chart can be used in cases where the quality of output is measured in terms of the number of defective components in the sample.

A similar procedure is used to construct all control charts. For instance, the mean of the process when the process is in control and producing output of acceptable quality corresponds to the centerline of an x-chart. The scale of measurement for the variable of interest is identified by the vertical axis of the control chart. The upper control limit, which is the upper horizontal line of the control chart, and the lower control limit which is also referred to as the lower horizontal line are chosen. This ensures that when the process is in control, there will be a high probability that the value of a sample mean will fall between the two control limits. The standard procedure used is to set the control limits at three standard deviations below and above the process mean. This process can be periodically sampled. The value of the sample mean is plotted on the control chart as each sample is selected. The process can be continued under the assumption that the quality standards are being maintained if the value of a sample mean is within the control limits. An out of control conclusion points to the need for corrective action to return the process to acceptable quality levels if the value of the sample mean is outside the control limits.

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