**3. Statistical Quality Control**

improve the quality of products, only provided information on the quality level of these and pick the items conform, those not complying. The constant concern with costs and produc‐ tivity has led to the question: how to use information obtained through inspection to im‐

The solution of this question led to the recognition that variability was a factor inherent in industrial processes and could be understood through the statistics and probability, noting that could be measurements made during the manufacturing process without having to wait

In 1924, Dr. Walter. A. Shewhart of Bell Telephone Laboratories, developed a statistical graph to monitor and control the production process, being one of the tools of Statistical Quality Control. The purpose of these graphs was differentiate between aleatórias1 causes unavoidable and causes a remarkable process. According to Shewhart (1931), if the random causes were present, one should not tamper with the process, if assignable causes are present, one should detect them and eliminate them. In other words, these graphics monitor

Studies by Johnson and Basgshaw (1974) and Harris and Ross (1991) showed that the graph‐ ics Shewhart and cumulative sums (CUSUM) are sensitive to the presence of autocorrelated data (data that are not independent of each other over time), especially when the autocorre‐

You will need to process the data first and then control them statistically. The presence of autocorrelation in the data leads to growth in the number of false alarms. Alwan and Rob‐ erts (1988) show that many false alarms (signals of special causes) may occur in the presence of moderate levels of autocorrelation, and the resulting measurement system, the dynamics of the process or both aspects, and conventional control charts are used without knowing

Many methods have been proposed to deal with statistical data autocorrelation. The interest in the area was stimulated by the work of Box and Jenkins, published in 1970 work entitled Time Series Analysis: Forecasting and Control, where it was presented among several quan‐ titative methods, methodology used to analyze the behavior of the time series. The method of Box and Jenkins uses the concept of filter composed of three components: component au‐

The reason for monitoring residual processes is that they are independent and identically distributed with mean zero, when the process is controlled and remains independent of pos‐ sible differences in the mean when the process gets out of control. Zhang (1998), the tradi‐ tional graphics Shewhart, CUSUM graphics, the graphics may be applied to the EWMA waste, since the use of graphics residual control has the advantage that they can be applied to autocorrelated data, even if the data is nonstationary processes. When a graph of residual control is applied to a non stationary, it can only be concluded that the process has some deviation in the system because of a non stationary there is no constant average and / or

toregressive (AR), the integration filter (I) component and the moving average (MA).

the change or lack of instability in the process thus ensuring quality products.

lation is extreme, ie tools are not suitable for the process control.

the presence or absence of correlation, much effort can be spent in vain.

prove the quality of products?

32 Practical Concepts of Quality Control

constant variance.

for the completion of the production cycle.

The statistical quality control (SQC) is a technique of analyzing the process, setting stand‐ ards, comparing performance, verify and study deviations, to seek and implement solutions, analyze the process again after the changes, seeking the best performance of machinery and / or persons (Montgomery, 1997).

Another definition is given by Triola (1999), which states that the SQC is a preventive meth‐ od where the results are compared continuously through statistical data, identifying trends for significant changes, and eliminating or controlling these changes in order to reduce them more and more.

SPC charts are designed to detect shifts among natural fluctuations caused by chance noises. For example, the Shewhart chart utilizes the standard deviation (SD) statistic to measure the size of the in-control process variability. By graphically contrasting the observed deviations against a multiple (usually, triple) of SDs, the control chart is intended to identify unusual departures of the process from its normal state (controlled state).

Under certain assumptions, when the observed deviation from the mean exceeds three SDs, it is said that the process is out of control since there is only a probability of 0.0026 for the observation to fall outside the three SD limits given an unshifted mean chance the process mean is shifted. This Shewhart chart scheme is in effect a statistical hypothesis testing that reveals only whether the process is still in-control (Chen and Elsayed, 2000).

To better understand the technical statistical quality control, it is necessary to bear in mind that the quality of a product manufactured by a process is inevitably subject to variation, and which can be described in terms of two types concerned.

The *special cause* is a factor that generates variations that affect the process behavior in un‐ predictable ways, it is therefore possible to obtain a standard or a probability distribution.

The *common cause* is defined as a source of variation that affects all the individual values of a process. It results from various sources, without having any predominance over the other.

When these variations are significant in relation to the specifications, it runs the risk of hav‐ ing non-compliant products, ie products that do not meet specifications. The elimination of requiring special causes a local action, which can be made by people close to the process, for example, workers. Since the common causes require actions on the system of work that can only be taken by the administration, since the process is itself consistent, but still unable to meet specifications (Ramos, 2000).

According to Woodall et al (2004), Statistical Quality Control is a collection of tools that are essential in quality improvement activities.

#### *Descriptive Statistics*

According to Reid and Sanders (2002), descriptive statistics can be helpful in describing cer‐ tain characteristics of a product and a process. The most important descriptive statistics are measures of central tendency such as the mean, measures of variability such as the standard deviation and range, and measures of the distribution of data. We first review these descrip‐ tive statistics and then see how we can measure their changes.

*The mean:* To compute the mean we simply sum all the observations and divide by the total number of observations. The equation for computing the mean is:

**Figure 1.** Normal distributions with varying standard deviations (adapted of Reid and Sanders, 2002).

Applications of Control Charts Arima for Autocorrelated Data

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35

**Figure 2.** Differences between symmetric and skewed distributions (adapted of Reid and Sanders, 2002).

of variation, but not showing us how to eliminate it (Ryan, 1989).

In any production process, no matter how well designed or carefully maintained it is, a cer‐ tain amount of inherent or natural variability will always exist. Natural variability is the cu‐ mulative effect of many causes small, essentially unavoidable. When this variation is relatively small, generally considered an acceptable level of performance of the process. In the context of statistical quality control, this natural variability often called "a stable system of special causes" is said to be in statistical control. Control charts are used to examine whether or not the process is under control, ie, indicate only random causes are acting on this process. Synthesize a wide range of data using statistical methods to observe the varia‐ bility within the process, based on sampling data. Can inform us at any given time as the process is behaving, if it is within prescribed limits, signaling thus the need to seek the cause

It was W. A. Shewhart (1931) which introduced control charts in 1924 with the intention to eliminate variations to distinguish them from the common causes and special causes. A con‐ trol chart consists of three parallel lines: a line that reflects the average level of process oper‐ ation, and two external lines called upper control limit (UCL) and lower control limit (LCL),

There are several types of control charts, as the characteristic values or purpose, and we can

calculated according to the standard deviation of a process variable (Shewhart, 1931).

divide them by attribute control charts and control charts for each variable.

*Control Charts*

$$\sum\_{\overline{\mathcal{X}}=1}^{\overline{\mathcal{X}}} \mathcal{X}\_{\overline{i}}$$

where: *x*¯= mean;

*xi* = the observation*i*,*i* =1,2,...,*n*;

*n*= number of observation.

*The range and standard deviation*: There are two measures that can be used to determine the amount of variation in the data. The first measure is the *range*, which is the difference be‐ tween the largest and smallest observations in a set of data. Another measure of variation is the *standard deviation. Standard deviation* is a statistic that measures the amount of data dis‐ persion around the mean.The equation for computing the standard deviation is (Reid and Sanders, 2002),:

$$\sigma = \sqrt{\frac{\sum\_{i=1}^{n} (\alpha\_i - \bar{\alpha})^2}{n-1}}$$

where: *σ*= standard deviation of a sample

*x*¯= the mean;

*xi* = the observation*i*,*i* =1,2,...,*n*;

*n*= number of observation in the sample

Small values of the range and standard deviation mean that the observations are closely clustered around the mean. Large values of the range and standard deviation mean that the observations are spread out around the mean.

#### *Distribution of the data*

A third descriptive statistic used to measure quality characteristics is the shape of the distri‐ bution of the observed data. When a distribution is symmetric, there are the same number of observations below and above the mean. This is what we commonly find when only normal variation is present in the data. When a disproportionate number of observations are either above or below the mean, we say that the data has a skewed distribution.

**Figure 1.** Normal distributions with varying standard deviations (adapted of Reid and Sanders, 2002).

#### *Control Charts*

measures of central tendency such as the mean, measures of variability such as the standard deviation and range, and measures of the distribution of data. We first review these descrip‐

*The mean:* To compute the mean we simply sum all the observations and divide by the total

*The range and standard deviation*: There are two measures that can be used to determine the amount of variation in the data. The first measure is the *range*, which is the difference be‐ tween the largest and smallest observations in a set of data. Another measure of variation is the *standard deviation. Standard deviation* is a statistic that measures the amount of data dis‐ persion around the mean.The equation for computing the standard deviation is (Reid and

Small values of the range and standard deviation mean that the observations are closely clustered around the mean. Large values of the range and standard deviation mean that the

A third descriptive statistic used to measure quality characteristics is the shape of the distri‐ bution of the observed data. When a distribution is symmetric, there are the same number of observations below and above the mean. This is what we commonly find when only normal variation is present in the data. When a disproportionate number of observations are either

above or below the mean, we say that the data has a skewed distribution.

tive statistics and then see how we can measure their changes.

*x*¯ = ∑ *i*=1 *n xi n*

*xi*

*σ* =

*xi*

where: *x*¯= mean;

Sanders, 2002),:

(*xi* <sup>−</sup> *<sup>x</sup>*¯)2

where: *σ*= standard deviation of a sample

*n*= number of observation in the sample

observations are spread out around the mean.

= the observation*i*,*i* =1,2,...,*n*;

*n* −1

*Distribution of the data*

∑ *i*−1 *n*

*x*¯= the mean;

= the observation*i*,*i* =1,2,...,*n*;

*n*= number of observation.

34 Practical Concepts of Quality Control

number of observations. The equation for computing the mean is:

In any production process, no matter how well designed or carefully maintained it is, a cer‐ tain amount of inherent or natural variability will always exist. Natural variability is the cu‐ mulative effect of many causes small, essentially unavoidable. When this variation is relatively small, generally considered an acceptable level of performance of the process. In the context of statistical quality control, this natural variability often called "a stable system of special causes" is said to be in statistical control. Control charts are used to examine whether or not the process is under control, ie, indicate only random causes are acting on this process. Synthesize a wide range of data using statistical methods to observe the varia‐ bility within the process, based on sampling data. Can inform us at any given time as the process is behaving, if it is within prescribed limits, signaling thus the need to seek the cause of variation, but not showing us how to eliminate it (Ryan, 1989).

It was W. A. Shewhart (1931) which introduced control charts in 1924 with the intention to eliminate variations to distinguish them from the common causes and special causes. A con‐ trol chart consists of three parallel lines: a line that reflects the average level of process oper‐ ation, and two external lines called upper control limit (UCL) and lower control limit (LCL), calculated according to the standard deviation of a process variable (Shewhart, 1931).

There are several types of control charts, as the characteristic values or purpose, and we can divide them by attribute control charts and control charts for each variable.

### *Control charts by attributes*

A control chart for attributes, on the other hand, is used to monitor characteristics that have discrete values and can be counted. Often they can be evaluated with a simple yes or no de‐ cision (Reid and Sanders, 2002).

CL=*μ* (2)

Applications of Control Charts Arima for Autocorrelated Data

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37

LCL=*μ* −*kσx*¯ (3)

where UCL is the upper control limit, CL is the center line or the average of the process, LCL is the lower control limit of the process, and k is the distance the control limits by the center line, which is expressed as a multiple of the *σ*standard deviation. The value of k is 3 most

The control graph is divided into zones (Figure 3). If a data point falls outside the control limits, we assume that the process is probably out of control and that an investigation is

A mean control chart is often referred to as an *<sup>X</sup>*¯chart. It is used to monitor changes in the mean of a process. The *<sup>X</sup>*¯- *<sup>S</sup>* control charts are generally preferred over the *<sup>X</sup>*¯- *<sup>R</sup>*charts when *n*10 or 12, since for larger samples the amplitude sampling R loses the efficiency to estimate*σ*, when compared to the sample standard deviation. The *<sup>X</sup>*¯ control charts is used in order to control the mean of the considered process. The two charts should be used simul‐

The limits of the *<sup>X</sup>*¯- *<sup>S</sup>* control charts are obtained in a similar manner, calculated under the assumption that the quality feature of interest (x) has a normal distribution with (*μ*) mean and (*σ*) standard deviation, ie, in abbreviated form (Panagiotidou and Nenes, 2009; Werke‐

However, satisfactory results are obtained even when this assumption is not true and distri‐ bution of x can only be considered approximately normal. In practice the *μ* and *σ* parame‐ ters are unknown and must be estimated from sample data. The method of estimation of *μ*

warranted to find and eliminate the cause or causes.

**Figure 3.** Control chart (adapted of Reid and Sanders, 2002).

*<sup>X</sup>*¯- *<sup>S</sup> Control Charts*

taneously (Werkema, 1995).

ma, 1995).

*x* ~ *N* (*μ*, *σ*)

widely used.

There are two broad categories of control charts for attributes: those who classify items into compliance or non-compliant, as is the case of graphs of the fraction of the number of faulty or defective, and those who consider the number (amount) of nonconformity existing graph‐ ics such as the number of defects in the sample or per unit.

According to Ramos (2000), the difficulties are:

a) due to the small size of the batch, the approximation of binomial and Poisson by the nor‐ mal distribution may no longer be valid, in which case the limits of control charts can not be determined by standard formulas;

b) the probability distributions Binomial and Poisson may not adequately represent the studied phenomenon. This occurs when the parts are manufactured simultaneously (multi‐ ple mold cavities, for example), in which the incidence of defects or defects is not independ‐ ent, statistically speaking.

### *Control charts for variable*

Control charts for variables monitor characteristics that can be measured and have a contin‐ uous scale, such as height, weight, volume, or width. When an item is inspected, the varia‐ ble being monitored is measured and recorded (Reid and Sanders, 2002).

They may not be used for quality characteristics that cannot be measured because the con‐ trol of the process requires monitoring of the mean and variability of measures. The graph‐ ics control variables used to data that can be measured or which undergo a continuous variation.

Some of the methods suitable for the construction of different control charts are the *Shewhart chart, Chart MOSUM - Moving Sum*, the *EWMA Chart - Weight Exponential Moving Average (Ex‐ ponentially Weighted Moving Averages)* and *CUSUM Chart - Cumulative Sum (Cumulative Sum).*

#### *Shewhart control charts*

The first formal model of control chart was proposed by Dr. Walter A. Shewhart (1931), which now bears his name. Let X a statistical sample which measures a characteristic of the process used to control a production line. Suppose that *μ* is the population mean of X and *σ* is the population standard deviation.

The following equations are used to describe the three parameters that characterize the She‐ whart control charts (Montgomery, 1997)

$$\text{UCL} = \mu + k \sigma\_{\vec{x}} \tag{1}$$

$$\mathbf{CL} = \mu \tag{2}$$

$$\mathbf{LCL} = \mu - k\sigma\_{\overline{x}} \tag{3}$$

where UCL is the upper control limit, CL is the center line or the average of the process, LCL is the lower control limit of the process, and k is the distance the control limits by the center line, which is expressed as a multiple of the *σ*standard deviation. The value of k is 3 most widely used.

The control graph is divided into zones (Figure 3). If a data point falls outside the control limits, we assume that the process is probably out of control and that an investigation is warranted to find and eliminate the cause or causes.

**Figure 3.** Control chart (adapted of Reid and Sanders, 2002).
