Statistical Quality Control

Statistical Quality Control

Quality expresses the suitability of the specifications and characteristics of the product with the wishes and requirements of the consumer. To achieve this we must:
Ensuring product quality by controlling production processes
Quality Assurance by Product Inspection and Testing
Quality assurance by applying the concept of Total Quality Management
Statistical quality control

The method of statistical quality control depends on the analysis of the results of examinations and tests for quality characteristics using statistical methods.

This method is considered among the most important methods in the field of quality control for industrial products, as it has been used in this field since World War II by international companies.

With the amazing development in computer programs, the importance of this method has increased and important applications have emerged in this field.

Statistical quality control tools

Traditional statistical tools: They are used to describe quality characteristics and include descriptive statistics such as mean, standard deviation, range, central tendency and dispersion in the data, but they are no longer sufficient to judge quality.
Statistical process control: It includes examining a sample of the production process, to ensure that it is working well and falling within the previously agreed upon limits.
Acceptance samples: It is the process of random examination of a quantity of production or a quantity received from the supplier to determine whether the required quality has been achieved in them in order to take a decision to reject or accept the quantity. The information provided by acceptance samples is useful in the decision to accept a lot, but it does not provide us with information about quality problems during the production process.
Traditional Statistical Tools

1. The arithmetic mean or average: It is one of the measures of the central tendency of a set of data, for example, knowing the defective rate in the daily production of one of the plants. The arithmetic mean of the ungrouped data is calculated using the following formula:
= Arithmetic mean, n = = number of observations, xi = observations
2. Range & Standard Deviation

The range (R) is the difference between the largest and smallest value in the data set, and is calculated by base.
The standard deviation (σ) measures how much data is scattered around its mean. For unclassified data, it is calculated using the following rule:
σ = standard deviation, = arithmetic mean, n = number of observations, xi = observation
Small values ​​of range and standard deviation indicate that observations are clustered close to their arithmetic mean, while large values ​​of range and standard deviation mean that observations are spread around the arithmetic mean.
3. Distribution of Data

When the distribution of data (observations) is symmetric, it means that there are an equal number of observations to the right and left of the arithmetic mean.
But when the data are abundantly clustered on the right or left side of the arithmetic mean, the distribution is called an asymmetric (skewed) distribution.
From the foregoing, it is noted that the methods of statistical control of operations employ the three traditional statistical tools to control the quality of products and operations in order to determine the amount of deviation in quality and operations using statistical control panels.
Causes of deviations
1. Random or Natural Causes
They are random deviations that are difficult to avoid and difficult to determine their sources. They are caused by a large number of factors that have little impact on the process, and these deviations are characterized by their limited values.
It is known that it is not possible to reduce these deviations because they occur randomly, and the management of production and operations must accept these deviations or changes, which differ from one production process to another, as some processes are exposed to more deviations than others.
Examples of these deviations include the presence of impurities in the factory atmosphere or sudden temperature changes.
2. Assignable Causes of Variations

Are those changes or deviations whose causes can be identified, treated and eliminated to reduce the variance in operations. The discrepancy in operations can be explained by the following reasons:
Different skills of workers in terms of experience, training, scientific qualification, incentives, stress, negligence, negligence and others.
Differences between machines due to obsolescence, origin, maintenance, lack of spare tools, all of which leads to a decrease in the efficiency of the machines.
Variation in the quality of raw materials, such as differences in the technical specifications of the materials, the chemical composition, and the original origin of those materials.
Statistical control panels for the process

Control charts are widely applied to detect deviation in the production process of goods and services from design characteristics.
Statistical control panels are useful in directing the management's attention to the presence of deviations in the processes that lead to the appearance of a defect in production, such as:
Increase the amount of rework.
Delayed payment of compensation.
Increasing customer complaints.
Increasing the rate of sudden spoilage in production.
Quality control panels are of two types:
Variable adjustment panels, including the R-Chart range panel, and the Chart-X arithmetic mean panel
The traits control panels have a panel to adjust the percentage of defects P-Charts, a panel to adjust the number of defects in the sample C-Chart
Control Charts for Variables

Variable adjustment panels are used to monitor the arithmetic mean and deviations in the process. These panels are useful in studying continuous variables such as weight, temperature, height, and others. They are of two types: range boards and arithmetic mean boards, and they are used together side by side to control the process.
Range panel or R-Chart: It is used to monitor the process variance, and the range is calculated by subtracting the smallest value in the sample data from the largest value, and if this difference falls outside the control limits of the range panel, it can be said that the process is outside the control limits. The central boundary of the range map, R, is calculated as follows
The adjustment limits for the range board are calculated as follows:
Max setting:
Minimum setting:

= Central boundary of the range map
D3, D4 = computed constants that can be extracted from the following statistical table.
sample size (n)

Control Charts for Variables

The arithmetic mean panel Chart-X: It is used to measure the arithmetic mean of the process, and the upper and lower limits of the adjustment can be calculated for this panel as follows:
Max setting:
Minimum setting:
Finding the mean panel using the standard deviation of the process
If the standard deviation of the process is known, the arithmetic mean of the samples and the standard value Z is taken from the tables of the normal distribution. It is possible if the statistical control limits for the arithmetic mean panel are calculated as follows:
Adjustment upper limit:
Minimum setting:
The standard deviation of the sample is calculated as:
So: n = sample size and σ = standard deviation of the process
Control Charts for Attributes

Quality is sometimes measured by non-quantitative attributes such as smell, color, the ability of the product to work, and other descriptive attributes. These panels are used when we can classify the units produced into defective and non-defective units or on the basis of enumerating defects in the sample or unit produced. There are two types of these panels:

P-Charts: Measures the percentage of defective units in the sample.
A panel for adjusting the number of defects in the sample c-Chart: It is concerned with adjusting the number of defects in one unit for not conforming to specifications.
Defective P-Charts Adjustment Panel

This panel is used to control the attributes, as the product's attributes or characteristics are being counted rather than measured, and through which the product can be judged as being good/bad, working/not working.
The panels are based on selecting a random sample, then examining it and calculating the total defective number of samples.
The statistical distribution on which these panels are based is the Binomial Distribution, and for large samples, the normal distribution is a good approximation of the Binomial Distribution.

The standard deviation of the defective ratios is calculated using the following rule: n = sample size, the arithmetic mean of the defect.
Defective P-Charts Adjustment Panel

The arithmetic mean of the defect is calculated using the following equation:
Since:
d = defective in each sample

N = the sum total of the samples' observations

The upper and lower bounds of the defective percentage plate shall be calculated as follows:
the highest rate
minimum
The following example shows how to calculate the adjustment limits for the defective ratio panel when the sample size is fixed.
Defective C-Charts Adjustment Panel
In many cases, it is not possible to withdraw samples from the process with an equal size for many reasons such as the lack of sufficient production, a change in the size of the production meal, or non-compliance with the administration’s instructions. In this case, the average sample size should be calculated () as follows:
= total sample size ÷ number of samples or
So: k = number of samples, si = sample size i
After calculating the mean sample size ( ), it should be replaced by (n), so the new form of the standard deviation rule becomes as follows:
Then the adjustment limits of the defective proportions panel are calculated. The following example shows how to calculate the adjustment limits in case the sample size is different.
C-Charts:
It is one of the attributes control panels. It expresses real numbers, not percentages. This panel provides the necessary information about changes in the number of defects from the outputs of the production process.
The administration resorts to this type of panel when it is interested in controlling the number of defects in each production unit, for example, the number of typographical errors on each page of a daily newspaper, or the number of defects in each square meter of carpet, or the number of air bubbles in each glass beaker. ...
These panels are based on the Poisson distribution since the variance is equal to the arithmetic mean in this distribution.
Acceptance Samples

Before we touch on acceptance samples, it is necessary to distinguish between two types of examination due to their relationship to acceptance samples, namely: comprehensive examination and examination by samples.

Comprehensive Examination: Comprehensive inspection is to conduct the necessary tests for all units within the production quantity or batch, as the quality of the product is measured and compared with the established quality standards, and in this way, the defective units are sorted from the valid units.

The disadvantages of a comprehensive examination are:

Mental fatigue, causes them to make mistakes during the examination process, such as rejecting valid units or accepting defective units.
The high cost of examination per unit, especially when there are no defective units.
It takes a long time.
Sampling examination: Acceptance samples are a type of examination that involves drawing random samples from a “lot” or “batch production” and examining them according to specific specifications and standards, and it is economical compared to a comprehensive examination. Attributes and variables can be examined with acceptance samples, but attributes are the most common in the industry.
Acceptance sampling is not a substitute for process control, because the most likely approach is to build statistical quality control with the aim of eliminating the acceptance sampling process because it is expensive and does not add value to the product. One of the tools used in developing a sample examination plan is the process characteristics curve.
Operating Characteristics Curve (OC)

The Process Characteristics Curve is one of the sampling testing tools that aims to help the organization distinguish between good and bad batches and helps in determining the sample size (n) and acceptance level (c), i.e. calculating the probability of accepting or rejecting batches of different quality levels. Acceptance samples also affect both parties: the producer and the customer.

Producer: Each party wishes to avoid the cost of accepting or rejecting the batch. The producer usually bears the responsibility to compensate all defective units in the rejected batch or bear the cost of the new batch sent to the customer.

for a customer. Therefore, the producer wants to avoid the mistake of the customer rejecting a good batch on the grounds that it is inferior, which is called producer's risk.
The customer: On the other hand, the customer wants to avoid the mistake of accepting a bad batch on the basis that it is good, and this is called the customer's risk.
Acceptance Quality Level (AQL): The lowest level of acceptable quality, i.e. acceptance of batches that have this level or better but not less than that.
If the acceptable quality level is equal to 15 defectives in a batch consisting of 1000 units, then the quality level is {100 * (15/1000)} = 1.5%. If the defective percentage is 1.5% or less, the batch is considered acceptable.
Lot Tolerance Percent Defective (LTPD) Lot Tolerance Percent Defective: Indicates the quality level of an accepted or rejected lot. If it is agreed that the permissible defective percentage is 60 defects in a batch of 1,000 units, then the LTPD equals 60/1000 = 6%.
Operating Characteristics Curve (OC)

Producer's Risk (α): It is the probability of rejecting a good batch thinking that it is bad, and it occurs as a result of a random sample containing a defective percentage higher than the defective percentage in the entire batch. Usually, product risk α is determined in sampling plans with a product risk level of 5%, meaning Another is that the probability of rejecting a good payment is 5%.
Customer's risk (β): It is the probability of accepting a bad batch thinking that it is good, and it occurs as a result of a random sample containing a defective percentage less than the defective percentage in the entire batch, and the customer's risk is often specified in sampling plans at the level of β = 0.10 or 10%.