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

*Control charts by attributes*

36 Practical Concepts of Quality Control

cision (Reid and Sanders, 2002).

determined by standard formulas;

ent, statistically speaking.

*Control charts for variable*

variation.

*Shewhart control charts*

is the population standard deviation.

whart control charts (Montgomery, 1997)

ics such as the number of defects in the sample or per unit.

According to Ramos (2000), the difficulties are:

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‐

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‐

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

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‐

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‐

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

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).*

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*

The following equations are used to describe the three parameters that characterize the She‐

UCL=*μ* + *kσx*¯ (1)

ble being monitored is measured and recorded (Reid and Sanders, 2002).

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‐ taneously (Werkema, 1995).

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‐ ma, 1995).

*x* ~ *N* (*μ*, *σ*)

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*

and *σ* again involves taking *m* samples (subgroups rational) primary, each containing n ob‐ servations of the quality characteristic considered.

#### Estimation of*μ*:

The (*μ*) mean is estimate through the overall average of the sample (*x*) ¯ as defined in the equation:

$$\bar{\bar{\bar{\chi}}} = \frac{\bar{\bar{\chi}}\_1 + \bar{\bar{\chi}}\_2 + \dots + \bar{\bar{\chi}}\_m}{m} = \frac{1}{m} \sum\_{i=1}^m \bar{\bar{\chi}}\_i \tag{4}$$

where A3 =3 / *c*<sup>4</sup> *n* is a constant tabulated as a function of size *n* of each sample.

*UCL* =*s*¯ + 3*σ*

*LCL* =*s*¯ −3*σ*

^

^

are constants tabulated in function of size *n* of each sample (Panagiotidou and Nenes, 2009;

According Montgomery (2009), various criteria may be simultaneously applied to a control graph for determining whether the process is under control. The basic criterion is one or more points outside the control limits. The additional criteria are sometimes used to increase the sensitivity of the control graphs when there is a small change in the process, so as to re‐

The Shewhart control charts have some rules sensitizers (Montgomery, 2009):

2. Two or three consecutive points outside the warning limits of 2-sigma;

4. A sequence of eight consecutive points of a same side of the center line;

6. Fifteen points in sequence in the area C (both above and below the center line);

3. Four or five consecutive points above of the limits of one-sigma;

5. Six points in a sequence is always increasing or decreasing;

8. Sequence of eight points on both sides of the center line CL;

7. Fourteen points alternately in sequence up or down;

10. One or more points near a limit or control.

*<sup>s</sup>* is a estimative of the standard deviation of the distribution of the *S* and *B*3 and *B*<sup>4</sup>

*<sup>s</sup>* = *B*4*s*¯ (11)

Applications of Control Charts Arima for Autocorrelated Data

http://dx.doi.org/10.5772/50990

39

*<sup>s</sup>* = *B*3*s*¯ (13)

*CL* =*s*¯ (12)

*S* control charts -

where *σ* ^

Werkema, 1995).

*Identification of Process in Control*

*Identification of Process out of Control*

spond quickly to an assignable cause.

9. A standard non-random data;

1. One or more points outside the control limits;

It is understood that the process is controlled to:

a) all points on the chart are within the control limits;

b) the arrangement of points within the control limits is random.

where *x*¯*<sup>i</sup>* , *i* =1,2,...,*m* is the i-ésima sample mean:

$$
\bar{\mathbf{x}}\_i = \frac{\mathbf{x}\_{i1} + \mathbf{x}\_{i2} + \dots + \mathbf{x}\_{in}}{n} \tag{5}
$$

Estimation of *σ*based on sample standard deviation:

The (*σ*) standard deviation is estimate based in the (*s*¯) standard deviation mean as defined by:

$$\bar{S} = \frac{s\_1 + s\_2 + \dots + s\_m}{m} = \frac{1}{m} \sum\_{i=1}^{m} s\_i \tag{6}$$

where *si* , *i* =1,2,...,*m* is the i-ésima sample of the standard deviation:

$$s\_i = \sqrt{\frac{1}{n-1} \sum\_{l=1}^{n} (\boldsymbol{\chi}\_{ij} - \bar{\boldsymbol{\chi}}\_i)^2} \tag{7}$$

It can be shown that the standard deviation sigma must be estimated by*σ* ∧ = *s*¯ *c*4 , where *c*4is a correction factor, tabulated as a function of size n of each sample.

Expressions for calculating the limits of *<sup>X</sup>*¯ <sup>−</sup>*S*control charts:

*<sup>X</sup>*¯ control charts -

$$\text{LCLL} = \ddot{\bar{\chi}} + 3\bar{\bar{s}} \Big/ c\_4 \sqrt{n} = \ddot{\bar{\chi}} + A\_3 \bar{s} \tag{8}$$

$$\text{CL} = \bar{\bar{\bar{\mathbf{x}}}} \tag{9}$$

$$\text{LCL} \quad \bar{\bar{\text{x}}} \text{ - } \bar{\text{s}} \bar{\text{s}} / c\_4 \sqrt{n} = \bar{\bar{\text{x}}} \text{ - } A\_3 \bar{\text{s}} \tag{10}$$

where A3 =3 / *c*<sup>4</sup> *n* is a constant tabulated as a function of size *n* of each sample.

*S* control charts -

and *σ* again involves taking *m* samples (subgroups rational) primary, each containing n ob‐

as defined in the

(5)

(6)

, where *c*4is a

*<sup>x</sup>*¯*<sup>i</sup>* (4)

)<sup>2</sup> (7)

¯ <sup>+</sup> *<sup>A</sup>*3*s*¯ (8)

¯ - *<sup>A</sup>*3*s*¯ (10)

¯ (9)

∧ = *s*¯ *c*4

The (*μ*) mean is estimate through the overall average of the sample (*x*) ¯

*<sup>x</sup>*¯ <sup>1</sup> <sup>+</sup> *<sup>x</sup>*¯ <sup>2</sup> <sup>+</sup> ... <sup>+</sup> *<sup>x</sup>*¯*<sup>m</sup>*

*<sup>m</sup>* <sup>=</sup> <sup>1</sup>

*xi*<sup>1</sup> + *xi*<sup>2</sup> + ... + *xin n*

The (*σ*) standard deviation is estimate based in the (*s*¯) standard deviation mean as defined

*<sup>m</sup>* <sup>=</sup> <sup>1</sup>

(*xij* - *<sup>x</sup>*¯*<sup>i</sup>*

*<sup>m</sup>* ∑ *i*=1 *m Si*

*s*<sup>1</sup> + *s*<sup>2</sup> + ... + *sm*

, *i* =1,2,...,*m* is the i-ésima sample of the standard deviation:

*si* <sup>=</sup> <sup>1</sup> *<sup>n</sup>* <sup>−</sup><sup>1</sup> ∑ *l*=1 *n*

It can be shown that the standard deviation sigma must be estimated by*σ*

correction factor, tabulated as a function of size n of each sample. Expressions for calculating the limits of *<sup>X</sup>*¯ <sup>−</sup>*S*control charts:

*UCL* = *x*¯

*LCL* = *x*¯

¯ <sup>+</sup> <sup>3</sup>*s*¯ / *<sup>c</sup>*<sup>4</sup> *<sup>n</sup>* <sup>=</sup> *<sup>x</sup>*¯

*CL* = *x*¯

¯ - <sup>3</sup>*s*¯ / *<sup>c</sup>*<sup>4</sup> *<sup>n</sup>* <sup>=</sup> *<sup>x</sup>*¯

*<sup>m</sup>* ∑ *i*=1 *m*

servations of the quality characteristic considered.

*x*¯ ¯ =

, *i* =1,2,...,*m* is the i-ésima sample mean:

Estimation of *σ*based on sample standard deviation:

*x*¯*i* =

*s*¯ =

Estimation of*μ*:

38 Practical Concepts of Quality Control

equation:

where *x*¯*<sup>i</sup>*

by:

where *si*

*<sup>X</sup>*¯ control charts -

$$\text{LCLL = \bar{s} + \Im \hat{\sigma}\_s = B\_4 \bar{s}} = B\_4 \bar{s} \tag{11}$$

$$\text{CL} = \text{\textquotedblleft} \tag{12}$$

$$LCL \ = \bar{\text{s}} - 3\stackrel{\wedge}{\sigma}\_s = B\_3 \bar{\text{s}} \tag{13}$$

where *σ* ^ *<sup>s</sup>* is a estimative of the standard deviation of the distribution of the *S* and *B*3 and *B*<sup>4</sup> are constants tabulated in function of size *n* of each sample (Panagiotidou and Nenes, 2009; Werkema, 1995).

#### *Identification of Process in Control*

It is understood that the process is controlled to:

a) all points on the chart are within the control limits;

b) the arrangement of points within the control limits is random.

#### *Identification of Process out of Control*

According Montgomery (2009), various criteria may be simultaneously applied to a control graph for determining whether the process is under control. The basic criterion is one or more points outside the control limits. The additional criteria are sometimes used to increase the sensitivity of the control graphs when there is a small change in the process, so as to re‐ spond quickly to an assignable cause.

The Shewhart control charts have some rules sensitizers (Montgomery, 2009):


Typical patterns of behavior are non-random (Lourenço Filho, 1964):

a) *Periodicity* - increases and decreases at regular intervals of time. The periodicity appears as one of the operating conditions of the process suffers periodic changes or when regular exchange of machines or operators.

We can see the existence of unit root if the values of the autocorrelation function begin near to unit and decline slowly and gradually as increases the distance (number of lags, k) be‐ tween the two sets of observations to which they concern, calling himself, not stationary and follows a random walk. If these coefficients decline rapidly as this distance increases, there

A common assumption in many time series techniques is that the data are stationary. A sta‐ tionary process has the property that the mean, variance and autocorrelation structure do not change over time. A process is considered stationary if its statistical characteristics do

Stationarity is a assumption in time series analysis. It means that the main statistical proper‐ ties of the series remain unchanged over time. More precisely, a process {*Yt*} is said to be completely stationary or strict sense stationary (abbreviated as *SSS*) if the process *Yt* and

A big reason for using a stationary data sequence instead of a non-stationary sequence is that non-stationary sequences, usually, are more complex and take more calculations when

Where a series submit over time variation in your parameters, so, we have a series non-sta‐ tionary, which when submitted to differentiation process becomes stationary. If the time ser‐ ies is not stationary, we can often transform it to stationarity with one of the following way:

The differenced data will contain one less point than the original data. Although you can

b) If the data contain a trend, we can fit some type of curve to the data and then model the

c) For non-constant variance, taking the logarithm or square root of the series may stabilize the variance. For negative data, you can add a suitable constant to make all the data positive before applying the transformation. This constant can then be subtracted from the model to

In according of Cochrane (2005), The building block for our time series models is the white

difference the data more than once, one difference is usually sufficient.

obtain predicted (i.e., the fitted) values and forecasts for future points.

noise process, which I'll denote*εt*. In the least general case,

, for all n, will be the same

http://dx.doi.org/10.5772/50990

41

Applications of Control Charts Arima for Autocorrelated Data

*Yt*+*n* have the same statistics for any*n*. So, the characteristics*Y*(*t*+*n*)

forecasting is applied to a data series (Beusekom, 2003).

a) Difference the data, by create the new series

is a series of characteristics of stationary (Morettin and Toloi, 2006; Russo et al. 2006).

*Stationary Processes*

not change with time.

*Non-Stationary Processes*

as*Yt*.

*Yt* = *Xt* − *Xt*−<sup>1</sup>

*White noise*

*ε<sup>t</sup>* ~*i*.*d*.*d*.*N* (0,*σε<sup>t</sup>*

2 )

residuals from that fit.

b) *Trend* - when the points are directed substantially upwards, or downwards. The general trend indicates a gradual deterioration of a critical process. This "decay" can be a tool wear and operator fatigue.

c) *Shift* - changes in performance of the process. The cause of the change can be introduction of new machinery, new operators, new methods or even a quality program, which usually brings motivation and improves performance.
