SPECIAL CONTROL CHARTS - mmf2.ogu.edu.trmmf2.ogu.edu.tr/sanagun/Docs/Dok2.pdf · industrial engineering applications and practices: users encyclopedia isbn: 09654599-0-x copyright

INDUSTRIAL ENGINEERING APPLICATIONS AND PRACTICES: USERS ENCYCLOPEDIA ISBN: 09654599-0-X

Copyright 1999 INTERNATIONAL JOURNAL OF INDUSTRIAL ENGINEERING

SPECIAL CONTROL CHARTS

A. Sermet Anagun, PhD

STATEMENT OF THE PROBLEM

Statistical Process Control (SPC) is a powerful collection of problem-solving tools, such as histograms, Pareto

diagrams, cause-effect diagrams, check sheets, stratification, scatter diagrams, and control charts, useful in

achieving process stability and improving capability through the reduction of variability. A control chart, also

known as Shewhart control chart and mostly preferred among the other tools, has an ability to determine whether

there are variation created by causes in the process and to reduce or adjust the variation, if applicable and

required, by taking the necessary corrective actions regarding to the causes that created the variation. When the

control charts are properly used, they may:

• be applied by operators for ongoing control of a process,

• help the process perform consistently and predictably,

• allow the process to achieve higher quality, lower cost, and higher effective capacity,

• provide a common language for discussing the process performance,

• distinguish special from common causes of variation as a guide to local or management action.

In order to use control charts as intended, first, a proper control chart should be selected suitable for the process

characteristics, such as:

• manufacturing type and volume (bulk, continuous, or discrete),

• type of inspection and strategy (destructive or non-destructive testing, in-process or pre-process

inspection),

• cost of inspection,

• inspection time, quality characteristics of the product being produced within the process (quantitative or

qualitative),

• distribution of quality characteristic(s)

• lot or sample size.

Second, to make the process, which is generally defined as a combination of people, machines, and other

equipment, raw materials, methods, and environment that produces products as planned, stable or to keep the

process in control, the causes of variations, if applicable, should be determined and interpret effectively assuming

that the proper control chart has been implemented.

In any process, regardless of how well the process is designed and maintained, a certain amount of inherent or

natural variability, variation due to chance causes, may occur. When chance causes, which are inevitable, difficult

to detect or identify, are in affect, a process is considered to be in a state of statistical control. Any attempt to

adjust for this kind of variation results in over control and is likely to throw the process out of control.

On the other hand, even if a process is in control, variation due to machine and operator performances and

characteristics of incoming materials or other causes may occur within a stable and predictable process. If

unnatural patterns are observed, special causes responsible for the condition must be determined and interpreted

effectively so that these disturbances such as operator fatigue, tool wear, different incoming materials, voltage

fluctuations, or systematic adjustment of the process may be eliminated from the process by taking the necessary

corrective actions.

The control charts are used to detect and eliminate unwanted special causes of variation occurred during a period

of time where a certain amount of products have already been manufactured. It should be recalled that these

special causes of variation have an adverse effect on the overall output of the process, not just individual product

characteristics. In order to obtain the expected benefits of using control charts as a problem-solving tool, the

following issues should be taken into consideration:

• Key characteristics that best indicate the control or out-of-control status of a process should be selected

for control purposes. Thus, quality of many product characteristics may be improved with applying

control charts on the key characteristics as the features which have the greatest influence upon the

product fit, performance, service life as mutually agreed with the customer,



• The control charts must be applied to control a process where the same machine(s), the same machining

method, and the same materials are used.

There are control charts developed which may be categorized into two groups regardless of the characteristics of

the products, either measurable such as length and diameter or countable such as number of defects per unit:

standards are given meaning that population parameters such as mean and standard deviation are known and

there is no need to collect data to establish control limits, and standards are not given meaning that population

parameters are not known and control limits must be derived from at least 20-25 subgroups of data taken from

the process based on a sampling policy.

The commonly known control charts may not be used when:

• production level is low; therefore, an extensive amount of time should be waited to collect a required

number of data, at least 20-25 individuals or subgroups of data, to establish control limits so to construct

a control chart.

• production is run in a discrete form because of unexpected delays due to lack of materials or sudden

changes in production scheduling; therefore, a certain amount of production may be performed by

different personnel and time intervals.

• besides piece-to-piece and time-to-time variation, variation within the piece is needed to be measured

and observed.

• even though the characteristics observed or dimensions measured are similar in terms of machine,

material, and method used, separate control charts should be constructed for each characteristic or

dimension, instead of monitoring similar characteristics on the same chart.

The process should be investigated by the person, who is responsible for statistical process control, to determine

whether the circumstances given above are applicable for processes or characteristics concerned to be able to

select the proper control chart for evaluating the processes and making them stable. When a control chart is

selected without considering the circumstances of the process or characteristics occur, the chart being constructed

may not be proper to evaluate the process because each control chart has its own properties in terms of sample

size, type of characteristic observed, formulation to calculate the control limits, the coefficients used in the

formulation, and the interpretation of the chart constructed.

SOLUTIONS TO THE PROBLEM: SPC USING SPECIAL CONTROL CHARTS

The number of measurements in the sample is an essential criterion for selecting a control chart. Suppose, a set of

samples, for instance, each has seven measurements, are taken from a process based on a sampling policy. Since

the sample size is relatively large, the standard deviation should be preferred instead of the range to represent the

dispersion of the process concerned.

The standard deviation is considered as an effective parameter for representing the variation since it is calculated

using all of the data points. On the other hand, when the sample size is relatively small, the range yields almost as

good as an estimator of the variance. However, for moderate values, the range loses efficiency rapidly, as it

ignores all the information in the sample between the maximum and the minimum measurements. Nevertheless,

the range may be preferred to simplify the calculation; but, for this situation, based on the sample size, the

average and the standard deviation chart should be used to evaluate the process effectively.

Addition, using an improper control chart may often cause a problem such that an interpretation may be done as

the process is out-of-control based on the selected chart even though it is under control indeed or vice versa.

These conclude that the selection of a proper control chart for the purpose of statistical process control requires

an extensive amount of attention in terms of properties of the control charts.

In the remaining of this chapter, the control charts, namely special control charts, will be discussed based on

specific examples which are prepared to demonstrate each special control chart in regard to where and when to

use, how to calculate the control limits, and how to evaluate the chart constructed and similarities between the

traditional control charts will be given. Some of the charts being introduced here require a minimum number of

data points to be constructed and evaluated (standards are not given). For the other charts, there is not such a

constraint because the control limits are derived directly from the specification limits (standards are given) and

measurements are plotted on a control chart to evaluate the process concerned.



1. Interrupted Average and Range Chart

The interrupted average and range chart, which is similar to the average and range chart, may be used for discrete

or low volume production. It is also used when there are unexpected delays in process due to lack of materials

and sudden changes in production scheduling. Therefore, the products may be produced by different personnel

and time intervals. In other words, a machine is setup to do an operation for 10 pieces and not perform that

operation again for several days or weeks. Production control considerations might also dictate that more than

one operator-machine combinations may be set up to run an operation simultaneously, so that defining the

process becomes a much more generic than for classical production situations.

The chart is started with different short runs from the process. Generally small samples (e.g., 2 or 3) obtained

from one characteristic of a product that produced within a process are taken based on tight intervals (e.g., 15

minutes). The control chart is filled out as the parts run, filed, then filled out during the next run. This activity

continues until there are not any parts to produce within that time interval or the necessary samples are taken to

construct a chart. For instance, suppose that a drilling operation is being performed for a diameter of a flange

with specification of 0.515±0.005, and an unreasonably long time is required to accumulate necessary number of sub-grouped data because of insufficient number of flange at the time where production is being run. In this case,

the drilling operation is performed as long as the number of parts available at hand, then the machine used for

drilling operation and the personnel who does that operation may be switched to operate on parts with different

specifications depending upon a production scheduling. For these reasons, whereas the parts are processed lot by

lot in terms of time intervals, a separate work order is assigned to each lot and measurements obtained from each

lot are plotted and evaluated separately according to the control limits calculated regarding to the appropriate lot.

Assume that the mentioned drilling operation has been performed and the sub-grouped data, each had three

samples and coded as 0.5XX, have been obtained as given in Table 1.

Table 1. Measurements for a Drilling Operation of a Flange

Work

Order

Date Sample

No.

X1 X2 X3 X R

06 3/10 1 16.0 16.5 16.0 16.2 0.5

2 17.0 16.5 16.0 16.5 1.0

3 16.0 16.0 15.5 15.8 0.5

4 17.0 15.0 15.0 15.7 2.0

5 17.5 16.5 17.0 17.0 1.0

6 17.0 15.6 17.5 16.7 1.9

7 16.0 16.5 16.5 16.3 0.5

17 5/07 8 14.5 14.0 14.0 14.2 0.5

9 15.0 14.5 15.5 15.0 1.0

10 15.5 16.0 15.5 15.7 0.5

11 15.0 15.0 15.0 15.0 0.0

12 16.0 15.5 16.5 16.0 1.0

13 16.5 17.5 18.0 17.3 1.5

20 6/02 14 17.5 17.5 18.5 17.8 1.0

15 16.0 16.5 17.0 16.5 1.0

16 16.0 15.5 16.0 15.8 0.5

17 16.0 16.0 17.0 16.3 1.0

18 16.5 17.0 16.5 16.7 0.5

19 17.0 17.5 16.0 16.8 1.5

20 16.5 16.5 16.5 16.5 0.0

The control limits of the chart for each time interval (run) may be calculated as follows:

for average chart,



kk

kX

kk

kX

RAXLCL

RAXUCL

2

2

−=

+=

for range chart,

k

kR

k

kR

RDLCL

RDUCL

3

4

=

=

where k is the time interval in which the samples are obtained, A2, D3 and D4 are the necessary coefficients for

the sample size of n, k

X is the grand average of the kth time interval which is obtained from the samples’

averages, kR is the average range of the k

th time interval which is obtained based on the differences between a

maximum and a minimum measurements of the samples

Since the measurements obtained from different personnel and time intervals, this chart help production reach a

state of control and monitor variation in process over time. Thus, a proper setup for the process monitored may

be easily determined by focusing on the variation during the intervals to make the process stable and predictable.

In order to determine the control limits for each time intervals, first, grand averages and average range for

intervals should be calculated as follows:

06.17

4.7

731.16

7

2.114

7

7

1

7

1 ======

∑∑== i

Ii

Ii

Ii

IR

R

X

X

75.06

5.4

653.15

6

20.93

6

6

1

6

1 ======

∑∑== i

IIi

IIi

IIi

IIR

R

X

X

79.07

5.5

763.16

7

4.116

7

7

1

7

1 ======

∑∑== i

IIIi

IIIi

IIIi

IIIR

R

X

X

The control limits for the portions of average and range are then calculated using the equations given above and

the coefficients of A2 = 1.023, D4 = 2.574, and D3 = 0 for the sample size of 3. The calculated control limits are

given in Table 2.

Table 2. Control Limits for a Flange Drilling Operation

The Time Intervals

Control Limits I II III

UCLX 17.39 16.30 17.44

LCLX 15.23 14.76 15.82

UCLR 2.73 1.93 2.03

LCLR 0 0 0

As in the average and range chart, subgroup averages on the average chart and subgroup ranges on the range

chart are plotted and the points are connected according to the time intervals where samples are taken. The

control charts for the averages and ranges are given in Figure 1 and Figure 2, respectively.



13

14

15

16

17

18

19

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Figure 1. Average Chart for a Flange

0

0.5

1

1.5

2

2.5

3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Figure 2. Range Chart for a Flange

When the charts are examined, the following results may be obtained:

1. According to the average chart, the first setup is the best among others.

2. Variation between setups in average chart is extensive.

3. There may be an excess tool wear in the second time interval, and few data points are out of control limits.

4. Whereas the ranges are stable, 2/dR may be used for an estimate of standard deviation to perform a process

capability study.

As a conclusion, the machine should be setup and the personnel should be advised based on the setup of the first

interval. For this example, a process capability study may not be performed because the process is not in control.

Therefore, first, necessary corrective actions should be taken to reduce the variation, and then a process

capability index may be calculated to interpret the process. The interrupted average and range chart may be

interpreted and the process capability may be studied in the same manner as the average and range chart.

2. Multiple Variation Chart

The multiple variation chart may be used when specification limits or the population mean and standard deviation

are known. Consequently, there is no need to collect data to establish control limits.

In contrast to other known chart such as the average and range and the average and standard deviation charts,

piece-to-piece variation, variation within the piece, and time-to-time variation may be examined with this chart.

Piece-to-piece variation concerns itself with different sizes of dimensions such as thickness, diameter, and the

variation within the piece concerns itself with problems such as roundness and taper, the time-to-time variation,

on the other hand, concerns itself with variation between the averages of the subgroups. Since three types of

variation are taken into consideration with this chart, the effects of different raw materials, sources, fixtures,

operators, and spindles may be easily determined and compared to improve the process performance.



Suppose that specifications of a diameter on a shaft are 0.500±0.003 inch, roundness is considered as a key characteristic to evaluate a process regarding to three different variation. Samples, each has four pieces, are taken

from the process and minimum and maximum measurements of each piece are given in Table 3.

Table 3. Measurements of a Diameter of a Shaft (coded as deviations from nominal)

Sample

Number

Measurements

X

Deviation Below

Nominal (-)

Deviation Above

Nominal (+)

1 X1 -0.0005 0.0035

2 X2 -0.0020 0.0030

3 X3 -0.0020 0.0040

4 X4 -0.0005 0.0040

5 X1 -0.0005 0.0035

6 X2 -0.0005 0.0030

7 X3 -0.0020 0.0030

8 X4 0.0000 0.0045

9 X1 -0.0010 0.0030

10 X2 -0.0015 0.0030

11 X3 -0.0010 0.0040

12 X4 -0.0010 0.0040

13 X1 -0.0015 0.0035

14 X2 -0.0005 0.0045

15 X3 -0.0020 0.0030

16 X4 -0.0020 0.0020

In order to construct a multiple variation chart, maximum and minimum values of the characteristic of each

sample in one particular location are measured and plotted for each sample on the same vertical line as deviation

from a nominal value of the characteristic examined. After selecting the proper scale, the specification limits are

drawn on the chart, and the midpoints of each plot (the point between the maximum and the minimum

measurements) are connected. Ranges are not considered with this chart. A multiple variation chart for the data

given in Table 3 is depicted in Figure 3.

Figure 3. A Multiple Variation Chart for the Shaft Example

The multiple variation chart may be interpreted as follows:

• If the distance between averages of each sample in the subgroup varies, this represents piece-to-piece variation, which could be caused due to tool wear, or operator influence. The process is out of control,

-0.005

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004

0.005

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

USL

LSL

CL



whereas either the minimum or the maximum measurement is out of specification limits for the appropriate

pieces.

• If the maximum or the minimum measurement being part of a sample is out of specification limits, the cause may be variation within the piece. The distance between the two measurements for each sample represents the

degree of variation within the piece. There is sudden increase and/or decrease due to excess taper for the

example.

• Variation between the averages of each subgroup to another represents time-to-time variation, which could be caused due to setup error or uncalibrated machine tool/equipment. Even though the process is not in control,

there is not a large variation between samples’ averages. However, necessary corrective actions should be

taken to reduce variation.

3. Print Tolerance Control Chart

As the multiple variation chart, the print tolerance control chart uses specification limits directly and applied

statistics to establish control limits. This chart is not only good for low volume production, but also for high

volume production since the control limits may be calculated without requiring any previous data. The control

limits derived from the specification limits are centered around the nominal of the specification. The control

limits of a print tolerance control chart are determined as follows:

1. Calculate the L factor based on the sample size of n and the assumption of the process mean is normally

distributed within ±3σ limits.

6

2dL =

where, d2 is the coefficient obtained from a table based on the sample size of n.

2. Calculate Rmax, the maximum that the ranges can reach and still maintain capability.

LtoleranceionspecificatTotalR ⋅=max

3. Calculate the control limits;

for average chart,

max2

max2

RAMTDLCL

RAMTDUCL

X

X

−=

+=

for range chart,

max3

max4

RDLCL

RDUCL

R

R

=

=

where, MTD is the mid-tolerance dimension (centerline of the chart), A2, D3 and D4 are the necessary coefficients

for the sample size of n.

Suppose that a shaft is being machined according to specifications of 0.500±0.010 inch. While the production is running, fifteen samples with the size of 5 are taken based on a sampling policy. The obtained data are coded as

values above 0.480 inch and given in Table 4. Let us examine whether the process is in control using the print

tolerance chart.



Table 4. Measurements for a Characteristic of a Shaft

Sample No. X1 X2 X3 X4 X5 X R

1 22 25 23 26 23 23.8 4

2 23 25 24 23 25 24.0 2

3 26 24 25 23 24 24.4 3

4 23 23 24 25 25 24.0 2

5 23 24 23 24 25 23.8 2

6 25 24 23 22 23 23.4 3

7 24 25 25 24 23 24.2 2

8 24 24 25 24 23 24.0 2

9 23 24 24 25 25 24.2 2

10 24 23 23 24 23 23.4 1

11 24 25 25 25 24 24.6 1

12 24 25 26 30 26 26.2 6

13 25 24 23 24 23 23.8 2

14 24 25 25 23 20 23.4 5

15 23 23 21 24 21 22.4 3

The control limits for the measured quality characteristic may be calculated using d2=2.326, A2=0.577,

D4=2.115, and D3=0 for sample size of 5:

1. Calculating L,

3877.06

326.2

6

2 ===d

L

2. Calculating Rmax,

0078.03877.0200.0

max

=⋅=

⋅= LtoleranceionspecificatTotalR

3. Calculating the control limits for the average and the range charts, respectively;

4955.00078.0)577.0(500.0

5045.00078.0)577.0(500.0

=−=

=+=

X

X

LCL

UCL

0.00078.0)0(

0165.00078.0)115.2(

==

==

R

R

LCL

UCL

After calculating the control limits, the actual averages and ranges obtained from the process are plotted and the

chart is interpreted to decide whether the process is in control. The print tolerance charts for averages and ranges

are given in Figure 4 and Figure 5, respectively.

10

12

14

16

18

20

22

24

26

28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

UCL

LCL

MTD

Figure 4. The Print Tolerance Chart for Averages



0

0.005

0.01

0.015

0.02

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

UCL

LCL

CL

Figure 5. The Print Tolerance Chart for Ranges

When the print tolerance charts are examined closely, it can be say that the process is out of control because the

11th and 12

th samples are above the UCL in the averages chart, and the process is setup and operated

approximately 0.004 inch above nominal. For the range chart, on the other hand, all of the samples are within the

control limits and it may be assumed that the variation is not high enough to require corrective actions.

4. PreControl Chart

PreControl (PC) chart, also known as stoplight or target control chart, was developed in the early 60’s as a means

of providing the advantages of control charts on the machine without burdening the operator with knowing how

to construct and interpret control charts. Rather, it represents a series of easily followed rules, which lead the

operator to correct adjustments, or as importantly, to leave the process alone when only random chance is

affecting the output. The concept is based on knowledge of the characteristics of a normal distribution.

The PC chart, applicable to short and long production runs, is easy to use and it is simple to implement in the

facility. Recording data and calculation are nor required; however, plotting and interpretation according to the

rules may be necessary. In order to use PC chart effectively, the process should be centered between specification

limits, with ±3σ equal to or better than specification limits, or a capability index, Cp, of 1.00 or more. If this condition is met, the PC chart will keep the capable process centered and detect shifts that may result in making

some of the parts outside of the specification limits or decreasing the probability of making defectives.

The PC chart uses specification limits to establish PC limits. The PC limits (PCL), as shown in Figure 6, are

established as follows:

4

toleranceionspecificatTotalMTDPCL ±=

where, MTD is the mid-tolerance dimension.



USL

UPCL

LSL

LPCL

MTD

RED

RED

YELLOW

YELLOW

GREEN

GREEN

Figure 6. Graphical Description of PC Limits

The center zone between the upper and the lower PC limits is one-half the print tolerance and is called green

area. The sides between the PC limits and the specification limits are one-fourth of the total tolerance and are

called yellow zones. Outsides the specification limits are called the red zones. As in Figure 1, he zones may be

colored to make the procedure simple, understandable, and applicable.

In PC chart, if the process is capable and the process is centered, around 86% of the parts will fall between the

PC limits (green zones), and around 7% of the parts will fall in each of the outer sections (yellow zones). So,

under the conditions of statistical control, 7% of the points plotted would be expected to occur in each outer zone

or 1 time in 14 on the average.

Let us consider an example of a drilling operation for a diameter of a flange with specifications of 0.500±0.002 inch. The data, 20 individual values, obtained from this operation are given in Table 5.

Table 5. Measurements obtained from a drilling operation for a diameter

Sample No X Sample No X

1 0.5030 11 0.5000

2 0.5015 12 0.5008

3 0.5000 13 0.5000

4 0.5005 14 0.5005

5 0.4995 15 0.5015

6 0.5005 16 0.5012

4 0.5005 17 0.5002

8 0.5015 18 0.5005

9 0.5005 19 0.4982

10 0.5008 20 0.5015

The PC limits, for MTD is 0.500 inch and total specification tolerance is 0.004 inch, may be calculated as

follows:

4

toleranceionspecificatTotalMTDPCL ±=

499.0001.0500.0

501.0001.0500.0

=−=

=+=

LPCL

UPCL



The parts as they produced checked and measurements are usually plotted on the chart. The PC chart for the

drilling operation example is shown in Figure 7.

0.497

0.498

0.499

0.5

0.501

0.502

0.503

0.504

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

USL

LSL

UPCL

LPCL

MTD

Figure 7. PC Chart for the Drilling Operation

The interpretation of the chart may be performed based on the following rules:

1. If the part measured falls outside the specification limits (red zones), the process stopped and reset.

2. If the part measured is between the PC limits and the specification limits (yellow zones), a second part is

tested;

a. If the second part is in the same yellow zone, the process is stopped and reset.

b. If the second part is in the opposite yellow zone, the process is stopped and corrective actions are taken.

c. If the second part is between the PC limits (green zones), continue the process.

3. When five consecutive parts fall within the PC limits, switch to frequency of gaging.

4. When a process is reset for any reason, five consecutive parts must occur within the PC limits before

switching to the frequency of gaging.

Graphical description of the PC rules is depicted in Figure 8.

USL

LSL

UPCL

LPCL

MTD

1 2 2a 2b 2c 3, 4

Figure 8. Graphical Description of PC Rules

When the PC chart constructed for the drilling operation example is evaluated based on the PC rules given above,

the following results may be obtained:

1. Whereas the first sample is out of specification limits, the process should have been stopped and reset (with

rule number of 1).

2. When the 16th sample is taken, the process should have been stopped and reset (with rule number of 2a).



3. When the 20th sample is taken, the corrective actions should have been taken after stopping the process (with

rule number of 2b).

4. The process may be considered in control except for the previous comments.

In addition to the advantages of the traditional control charts such as indicating shifts in process centering and

increases in process spread, assuring that the number of defective parts will not exceed predetermined levels, PC

chart offers some valuable advantages. These advantages are given as follows:

1. Operator control becomes more practical since no recording, calculating, or plotting of data is required.

However, measurements may be plotted if the consumer or management desires to keep a PC chart as

statistical evidence of process control.

2. By adjusting the inspection frequency to a proper level, PC chart may be applied to long production runs.

3. PC chart may be used with attributes by appropriately colored go/no-go gages to initiate PC limits or even

visual characteristics by assigning visual standards to the PC limits.

4. Under the assumption of that the process capability is less than the specification and the process is centered,

the control limits are directly driven from the specification limits without requiring any previous data. This

makes the PC chart simple to understand and easy to implement.

5. Since it works directly with the specification limits, it does not require any statistics background to explain

the chart to the people who will be using the plan.

Despite the advantages, PC chart has some drawbacks. Since it is assumed that the process is capable and

normally distributed within ±3σ limits, central tendency is considered the only problem. Thus, sensitivity to variation in the process is reduced. While the process is monitored, the operator evaluates the process according

to the easily followed rules and tries to keep the process centered. Therefore, assessment of capability may be

difficult.

In addition, some over-control, adjusting the process when it does not need an adjustment, and/or under-control,

not adjusting the process when it does need an adjustment, may result. Finally, the PC chart, unlike the other

control charts, may not be used for problem solving, but instead, may be used for monitoring a process. For that

reason, the PC chart is often used to initially implement some form of statistical process control in the facility and

is soon followed by more sensitive process control charts to detect, if applicable, and control the variation within

the process.

5. Two-R Control Chart

Most of the control charts seen in the textbooks have measured piece-to-piece and time-to-time variation. The

range chart measures variation from piece-to-piece in the individuals and moving average charts and between

pieces within in the subgroup in the average and range charts. The time-to-time variation is plotted on the

individual or average portion of the suitable chart. However, there may be another source of variation, called

within-piece variation, that need to be considered for some situations. The traditional control charts such as the

average and range and the average and standard deviation charts do not allow tracking and recording this type of

variation.

Two-R control chart, which is a combination of the individuals control chart and the average and range chart,

allow plotting three different sources of variation. The two-R chart is especially useful in processes where within-

piece or within-group variation is as important as between-piece or between-group variation such as measures of

roundness or concentricity, flatness, surface finish, thickness, a dimensional characteristic common to many parts

processed in a fixture at the same time, or even hardness of a batch of parts run through a heat treat operation.

This chart consists of three separate charts: an individuals or average chart for representing variation between the

individuals or averages of the samples (what is happening to the process center), a range chart for representing

variation piece-to-piece variation (how uniformly the process is behaving batch-to-batch), and a range chart for

representing within piece variation (how uniformly the parts are treated within each batch). For instance, if a

quality characteristic examined is a roundness of a diameter of a shaft, the difference between the maximum and

the minimum readings of a part is called variation within the piece (Rw), and the average of the maximum and the

minimum measures determines the time-to-time variation. The difference between the previous reading and the

most current reading is called the piece-to-piece variation (Rp). The two-R chart is constructed after 20-25 data

points available and interpreted in the same manner as the other control charts. The control limits for the chart

may be calculated as follows:



Individuals chart, pX

pX

REXLCL

REXUCL

2

2

−=

+=

Rw chart,

wR

wR

RDLCL

RDUCL

w

w

3

4

=

=

Rp chart, pR

pR

RDLCL

RDUCL

p

p

3

4

=

=

where, X is the average of the maximum and the minimum readings, wR is the average of ranges for within-

piece variation determined by subtracting the max reading from the min reading, pR is the average of ranges for

piece-to-piece variation determined by subtracting the second plot point from the first plot point, and

4322 and,,/3 DDdE = are the coefficients for the sample size of 2. For the process capability study, an

estimated standard deviation for the process is calculated using 2/ dR for each range type as 22ˆ

wp σσσ += ,

where, 2/ dR pp =σ and 2/ dR ww =σ

Painting is considered as special process. Suppose that a part which affect the speed of an aircraft while the

aircraft is making some sort of movements, such as closing to and leaving from the ground sharply, in the air is

being painted trough a special process. Thickness of the paint shoot up, with specifications of 0.0012-0.0040

inch, is a key characteristic. For this situation, the thickness of the paint through the surface of the part is

measured and recorded as minimum and maximum values. The data, coded as 0.00XX, for this example are

given in Table 6.

Table 6. Measurements for a Painting Process

Sample No. XMin X

Max Xi Rw Rp

1 14 16 15.0 2 -

2 12 16 14.0 4 1.0

3 14 19 16.5 5 2.5

4 13 16 14.5 3 2.5

5 13 16 14.5 3 0.0

6 13 15 14.0 2 0.5

7 14 17 15.5 3 1.5

8 13 15 14.0 2 1.5

9 12 14 13.0 2 1.0

10 13 16 14.5 3 1.5

11 14 16 15.0 2 0.5

12 12 15 13.5 3 1.5

13 13 15 14.0 2 1.5

14 12 14 13.0 2 1.0

15 13 16 14.5 3 1.5

16 14 16 15.0 2 0.5

17 15 16 15.5 1 0.5

18 14 16 15.0 2 0.5

19 15 17 16.0 2 1.0

20 13 14 13.5 1 2.5

The control limits of the 2-R chart for the painting process are calculated using E2=2.660, D4=3.267, and D3=0

for the sample size of 2:



21.119

23,45.2

20

49,525.14

20

5.290====== pw RRX

for individuals chart,

3064.1121.1)660.2(525.14

7436.1721.1)660.2(525.14

=⋅−=

=⋅+=

X

X

LCL

UCL

for wR chart,

0.045.2)0(

0042.845.2)267.3(

=⋅=

=⋅=

w

w

R

R

LCL

UCL

for pR chart,

0.021.1)0(

9541.321.1)267.3(

=⋅=

=⋅=

p

p

R

R

LCL

UCL

The 2-R control charts for individuals, wR , and pR are depicted in Figures 9-11, respectively.

10

11

12

13

14

15

16

17

18

19

20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

UCL

CL

LCL

Figure 9. Individuals Chart

0

1

2

3

4

5

6

7

8

9

10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

UCL

CL

LCL

Figure 10. wR Chart



0

1

2

3

4

5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

UCL

CL

LCL

Figure 11. pR Chart

When both wR and pR control charts are evaluated it may be seen that the process is having a periodic variation

which is one of the special causes and there is a variation between samples, piece-to-piece variation. Variation

within-piece is small; however, most of the samples are accumulated around the centerline. For the individuals

chart, a decreasing trend from the 3rd sample, an increasing trend from the 14

th sample, and, overall, fluctuations

may be observed. Hence, the process requires a close examination for the determination of the reasons cause

special patterns and how to eliminate those by taking what kind of corrective actions.

6. Deviation Charts

The traditional control charts may be applied to continuous and homogeneous processes. In these processes, a

control chart is used for a characteristic of a part. Additional charts are needed if a part has more characteristics.

On the other hand, in order to construct a control chart to be used to detect whether there is a variation, at least

20-25 samples, each may consist of more than one unit, are required to satisfy the normality assumption. In some

situations, the process is often completed before the number of samples required might be obtained and thus the

operators do not have time to calculate the control limits to evaluate the process.

Deviation charts, also known as delta charts, are developed to eliminate these drawbacks. Two types of deviation

charts may be used for different circumstances. One is called deviation from nominal or nominal equals zero

chart (N=0). The second one is called deviation from target chart. These charts will be discussed in details in the

following paragraphs.

6.1. Deviation from Nominal Chart

This chart is used to control process, which produces different parts with similar characteristics. In other words,

the N=0 chart focuses on the process and characteristics rather than dimensions and part numbers (e.g. a machine

produces the same diameter on different parts at different sizes) as long as the following conditions are met:

• The standard deviations must be essentially identical between part numbers,

• The process must be homogeneous,

• The same material must be processed using the same method and machine (one process stream),

• One type of characteristic per part number must be selected,

• The subgroup size must be constant to be able to calculate the control limits for all characteristics or

parts.

If these conditions are met, data from multiple parts may be merged so that the number of samples required to

construct a control chart is easily obtained. The deviation from nominal chart is identical to the average and range

chart, except the centerline of the chart is the nominal of the specification, which is coded “0”, and the plotting

points for the characteristics concerned are the deviations from the nominal, which are determined by subtracting

the nominal value of the proper characteristic from the reading, such as +0.002, above the nominal, or –0.005,

below the nominal. With this arrangement, since the plotting points are deviations, a machine capability study



may be easily performed assuming the same parts produced within the process and the same characteristics

measured. Besides, the same patterns will be seen on the charts as if actual sizes were being plotted. Therefore,

the chart is interpreted in the same manner as the traditional control charts.

In order to evaluate the process properly, a range chart should be constructed at first. In the deviation from

nominal chart, the ranges are calculated and plotted identical to the range chart for averages or individuals

depending on the sample size of n. The plotting point is the difference between the maximum and minimum

readings for the range chart for the averages. A moving range is calculated for the individual chart by subtracting

the current reading from the last reading and using the absolute value of that difference.

Once the ranges are calculated, a test depending on the standard deviations of the parts measured should be

performed to decide whether the characteristics of the parts might be examined on the same chart. The test is

performed using the following steps:

1. of ratio a CalculateTotal

Part

R

R

where, PartR is the average range of the part suspected and TotalR is the average of the total of all the parts

including the suspected part.

2. Use the following criterion to make decision about the suspected part(s)

>

≤=

chartthatforusedbeshouldchartseparatea3.1

partsotherthewithchartsametheonplottedbemaypartthatfortsmeasurementhe3.1

Total

Part

R

R

Assume that the parts are being manufactured on a process with specifications of 2.015±0.002, 1.831±0.002 and 4.6795±0.002 inches, respectively. The parts have different specifications; however, they are made of the same material, and manufactured on a process using the same method. Thus, a control chart, for instance, deviations

from nominal chart, may be a candidate to control that process in which the parts are being processed. Suppose

the parts are processed based on schedule and the coded measurements obtained from the process are given in

Table 7.

Table 7. Measurements for the Parts A, B, and C

Sample No. Part Nominal X Deviation R

1 A 15 14.9 -0.1 -

2 A 15.1 0.1 0.2

3 A 15.3 0.3 0.2

4 A 14.8 -0.2 0.5

5 B 31 31.0 0.0 -

6 B 31.4 0.4 0.4

7 B 31.2 0.2 0.2

8 B 30.9 -0.1 0.3

9 B 30.8 -0.2 0.1

10 C 79.5 79.2 -0.3 -

11 C 79.8 0.3 0.6

12 C 78.8 -0.7 1.0

13 C 78.9 -0.6 0.1

14 C 79.3 -0.2 0.4

15 C 79.8 0.3 0.5

The plot points for the averages or individuals are simply represented by deviations from nominals and plotted as

in the traditional control charts. The control limits are calculated using proper formula depending on the sample

size of n; average chart for n≥2 and individuals chart for n=1. Whereas each sample has a single measurement, the control limits are determined as if a control chart being used for individuals:



The control limits using E2=2.660, D4=3.267 and D3=0 are given:

375.012

5.4

3and0533.0

15

80.0==

−=−=

−== ∑∑

m

RR

m

XX

for individuals chart,

051.1375.0)660.2(0533.0

9442.0375.0)660.2(0533.0

−=⋅−−=

=⋅+−=

X

X

LCL

UCL

for range chart,

0.0375.0)0(

225.1375.0)267.3(

=⋅=

=⋅=

R

R

LCL

UCL

The control charts are depicted in Figure 12 and Figure 13 for individuals and ranges, respectively.

-1.5

-1

-0.5

0

0.5

1

1.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

UCL

CL

LCL

Figure 12. A Deviations from the Nominal Chart for Averages

0

0 .2

0 .4

0 .6

0 .8

1

1 .2

1 .4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

UCL

CL

LCL

Figure 13. A Deviations from the Nominal Chart for Ranges

There is a decreasing trend during manufacturing of parts B especially starting from the 6th sample, and a large

variation may be seen within the process when the parts C are started to manufactured. The latter result, which

may also be seen in the range chart, shows evidence that the parts C would be different in terms of material,

machine, and method so that the test should have been performed before starting to construct the charts. Indeed,

the ratio of Totalc RR / (0.52/0.375=1.39) is greater than 1.3, which implies that the parts C should be plotted on

a separate chart. Hence, the deviations from the nominal charts should be constructed and reevaluated after



eliminating the parts C. For a capability study of the deviations from nominal chart, the steps as in the other

control charts are followed, but actual limits of each different characteristic and its tolerance are used for making

decision whether a process is capable.

6.2. Deviation from Target Chart

Many processes in manufacturing need to position their process output close enough to one of the specification

limits for a purpose of economics such as extension of tool life and longer solution deterioration times. In this

case, the parts are produced within the print tolerance; however, some parts such as bearings, pistons, bores, and

keys are processed either above or below nominal to be on the safe side especially for assembly. A control chart

named deviation from the target chart is developed for the situations where similar characteristics of a part or

different parts are processed and leading a target for each characteristic instead of being around nominal is

important or required.

In order to use the deviation from the target chart, the properties mentioned for the deviation from the nominal

chart must also be satisfied because of the similarities between them. Addition, the parts or characteristics of the

parts being plotted on the same chart could have different sample averages. For this case, the chart may not be

easily evaluated. Besides, a result may be obtained that the process in which the parts are being manufactured is

out of control due to large variation between the sample averages while the process is statistically under control

(Type I error). Therefore, a test should be performed to determine which deviation chart is appropriate for the

process or the parts concerned. The steps for the test are given below:

1. Calculate standard deviations of the parts taken into consideration and F1 value,

1

)( 2

−

−= ∑

n

XXs

i

and

n

tF

v;2/1

α=

where, n is the number of samples for each part concerned, vt ;2/α is the value obtained from t distribution

for significance level of α and degree of freedom of v=(n-1).

2. Use the following criterion to determine which chart is appropriate,

⋅≤

⋅>=−

ChartNominalthefromDeviationuse

ChartTargetthefromDeviationuseTargetNominal

1

1

sF

sF

For instance, a nominal value of part A is coded as 24.5, and a target value as 25.9. Seven samples are taken and

standard deviation is calculated as 1.32. To make a decision, F1 value should be determined. The required F1

value for α =0.05 is

92.07

4469.2

7

6;025.01 ===

tF

Since 2144.132.192.01 =⋅=⋅ sF and 24.5-25.9 = -1.4 = 1.4 > 1.2144, deviation from the target chart should

be selected.

As mentioned before, the control limits for the deviation from the target charts are calculated and the charts are

constructed based on either average and range charts or individuals and moving range charts depending on the

sample size. Let us suppose an example to introduce this particular control chart. Parts A, B, and C have different

specifications, but they are made of the same material and machined using the same method. The data, each is

having three pieces, obtained from the process are coded and given in Table 8.



Table 8. Measurements for the Parts A, B, and C

Sample No Part X1 X2 X3 X Target ′X

R

1 A 93 95 91 93 90 3 4

2 A 89 92 83 88 90 -2 9

3 A 94 93 89 92 90 2 5

4 A 96 89 97 94 90 4 8

5 A 88 86 93 89 90 -1 6

6 A 91 92 87 90 90 0 5

7 B 59 55 58 57.3 58 -0.7 4

8 B 63 57 59 59.7 58 1.7 6

9 B 59 61 60 60 58 2 10

10 B 60 55 56 57 58 -1 5

11 B 58 54 56 56 58 -2 4

12 B 60 55 59 58 58 0 5

13 C 36 30 33 33 30 3 6

14 C 29 33 31 31 30 1 4

15 C 32 26 29 29 30 -1 6

16 C 34 29 27 30 30 0 7

The control limits for the deviation from the target charts; using A2=1.023, D4=2.575, and D3=0 for the sample

size of n=3 and the number of sample m=16, are calculated as follows:

375.516

86and5625.0

16

9======

∑∑m

RR

m

XX

for average chart,

937.4375.5)023.1(5625.0

062.6375.5)023.1(5625.0

−=⋅−=

=⋅+=

X

X

LCL

UCL

for range chart,

0.0375.5)0(

841.13375.5)575.2(

=⋅=

=⋅=

R

R

LCL

UCL

The control charts are given in Figure 14 and Figure 15, respectively.

-6

-4

-2

0

2

4

6

8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

UCL

CL

LCL

Figure 14. A Deviation from the Target Chart for Averages



0

2

4

6

8

10

12

14

16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

UCL

CL

LCL

Figure 15. A Deviation from the Target Chart for Ranges

When both average and range charts are examined, it may be said that process is in control; however, variation

for the parts A is larger than the others. Therefore, the variation for the parts A should be decreased.

7. Short Run Chart

The traditional control charts such as the average and range charts and the average and standard deviation charts

only allows for one quality characteristic to plotted at a time. When two characteristics are tracked, two charts are

needed to evaluate the process. Deviation chart based on either nominal or target values may be used to plot two

or more characteristics on a chart when the dispersion expressed in range or standard deviation are essentially

identical between characteristics.

The range or standard deviation is used in the formulas for calculating the upper and lower control limits on both

portions of the control charts. When the dispersion expressed in range or standard deviation calculated using the

samples taken from the process is large, artificially high limits may occur so that true out of control conditions

could be masked. In order to make the evaluation process as effective as it should be, short run chart, also known

as Z and W for the average and range (standard deviation) charts and Z and W for the individuals and moving

range charts, may be used to somehow eliminate the effect of having the range or the standard deviation in the

calculation of the control limits. With the short run chart, a plot point is calculated on a control chart that is

independent of the range or standard deviation using a series of inequalities.

A range value determined either a difference between the maximum and the minimum readings observed as in the

average and range charts, or a difference between successive readings as in the individuals and range charts, lies

between the upper and the lower control limits. This may be shown in the following inequality:

RiR UCLRLCL <<

or

RDRRD i 43 <<

Now, if the last inequality is divided by R , which may be a historical or a target value for the process concerned,

or a current value determined based on the samples taken, the following result is obtained:

43 DR

RD i <<

The control chart will have D3 as the lower control limit, D4 as the upper control limit, and 1.00 for the central

line because it occurs when R = R . Therefore, the plot point for each range, Ri, is

R

RR ii =



where, Ri is the ith range calculated from the current subgroup for a characteristic of a part, and R is the

historical average or target value for the same part. The control limits D3 and D4 are independent of R ; however,

they are a function of sample size n, which must be constant.

With the short run chart, samples obtained from two or more characteristics of a part or different parts may be

plotted. If two or more characteristics of a part, two diameters with different dimensions on the same part, or

more than two characteristics of different parts, two diameters with different dimensions on the different parts or

two different characteristics such as length and width, on the same part, are monitored, the plot points are

determined based on the historical average or target value of the processes from which the samples are taken. For

instance, if there are data obtained from two parts, part A and part B, related to a characteristic, two diameters

with same or different dimensions, then the plot points for part A and part B may be calculated using AR and

BR , respectively. This enables us to plot different parts with similar characteristics generated from similar

processes with different averages and dispersions.

The same approach is followed to determine the plot points for the average portion of the chart. The control

limits for average portion of the average and range chart also contain R in the formula. An average of a

subgroup, X , falls between the control limits. This may be shown in the following inequality:

XXUCLXLCL <<

or

RAXXRAX 22 +<<−

If we subtract X from the inequality and divide by R , we will get:

XRAXXXXRAX −+<−<−− )()()( 22

22

)(A

R

XXA <

−<−

As in the range chart, the control limits –A2 and A2 are dependent of sample size n, which must be constant. The

central line is equal to 0.0 because it occurs when 0=− XX , which is the ideal situation. The plot point of each

average, iX , is:

R

XXX i

−=

where, R is the average range obtained from the ranges of the samples, X is the historical or current grand

average, and X is the average of a subgroup.

The short run chart may be applied to individuals by making the necessary modifications. The ranges are

calculated considering the difference between the successive data points, also called moving range. Therefore, if

there is not any data available to compute a historical range average, sum of the ranges derived from the samples

is divided by m-1, where, m is the number of samples. The control limits for the range portion of the chart are

calculated using D3 and D4 for sample size of 2 and the plot points are determined in the same manner as in the

average and range chart. However, in contrast to the average and range chart, the control limits for the

individuals are calculated using E2 for the sample size of 2. The control limits independent of the X and R , and

plot points determined based on the concept discussed are given as:

The control limits;

REXXREX 22 +<<−

XREXXXXREX −+<−<−− )()( 22



22 ER

XXE <

−<−

The plot points;

R

XXX i

−=

where, R is the average historical or current range and X is the average of the individuals or a historical value.

The individuals version of the short run chart is also applicable to the situations where two or more

characteristics of a part or more than two parts with similar characteristics may be monitored at the same time.

Suppose a process in which similar characteristics of two shafts are manufactured. While shaft A has

specifications of 9.6310±0.0025 inch, specifications of shaft B are 12.2150±0.003 inch. Based on the records, historical or target average and range for the shaft A are 9.6315 and 0.00082 inches, so for the shaft B 12.2158

and 0.00053 inches. The shafts are sampled with the size of 3 as they manufactured based on the schedule. The

data, coded as 9.63XX and 12.21XX, obtained from the process and statistics calculated for the short-run chart

are given in Table 9.

Table 9. Measurements for Shaft A and Shaft B

Sample No. A A B B B B A A A A A A

X1 20 12 57 55 63 59 12 12 22 29 12 21

X2 18 14 60 59 60 63 19 18 21 22 16 20

X3 13 21 61 58 58 60 14 10 17 27 10 15

X 17 15.7 59.3 57.3 60.3 60.7 15 13.3 20 26 12.67 18.67

R 7 9 6 4 5 4 7 8 5 7 6 6

Target X 15 15 58 58 58 58 15 15 15 15 15 15

Target R 8.2 8.2 5.3 5.3 5.3 5.3 8.2 8.2 8.2 8.2 8.2 8.2

RXX /)( − .244 .081 .251 .126 .440 .503 .000 .204 .610 1.34 .284 .450

RR / .854 1.10 1.13 .755 .943 .755 .854 .976 .610 .854 .732 .732

The control limits are determined directly using ±A2 =1.02 for the averages, D3 =0 and D4 =2.57 for the ranges according to the sample size of 3. In short run charts, since the calculated overall average and average range

should be the same as the historical values of related statistics, the centerline for each characteristic or part is

determined as:

for average chart,

00==

−=

−

ii

ii

i

ii

RR

XX

R

XX

for range chart,

1==i

i

i

i

R

R

R

R

The short run charts are given in Figure 16 for the average portion and in Figure 17 for range portion of the chart,

respectively.



-1 .5

-1

-0 .5

0

0.5

1

1.5

1 2 3 4 5 6 7 8 9 10 11 12 13

UCL

CL

LCL

Figure 16. Short Run Chart for Averages

0

0.5

1

1.5

2

2.5

3

1 2 3 4 5 6 7 8 9 10 11 12 13

UCL

CL

LCL

Figure 17. Short Run Chart for Ranges

The process may be considered in control whereas variation within the process based on ranges is small. On the

other hand, when the averages are concerned, the process in out of control due to the plotting point of the 10th

sample is above the upper control limits and large variation especially for shaft A. Therefore, the sources of

variation should be determined and eliminated from the process with taking proper corrective actions.

CONCLUDING REMARKS

Control charts, mostly preferred tool for statistical process control, are used for two basic purposes. As a

judgement point of view, the control charts are used to provide guidance whether a process has been operating in

a state of statistical control and signal the presence of special causes of variation so that corrective actions may be

taken. On the other hand, for the operation point of view, control charts are used to maintain the state of

statistical control by extending control limits as a basis for real time decisions.

When a control chart, which is a graphical means of displaying both time-to-time and piece-to-piece variation

exhibited by a key characteristic, is used properly, it provides potential benefits. The control charts may help the

process perform consistently and predictably in terms of higher quality, lower cost, and higher effective capacity.

Besides, they provide a common language for discussing the processes performances and distinguish special from

common causes of variation as a guide to local or management action.

In order to obtain the benefits discussed; however, a control chart suitable for the product and/or process

characteristics should be selected. In this chapter, besides the traditional control charts, alternative control charts,

which may be used when special circumstances, such as low volume of production, importance of additional



sources of variation, and so on, occur, are introduced to people who work as responsible for statistical process

control in such manufacturing processes and come across problems related to issues of statistical process control.

As a conclusion, it should be recalled that an extensive amount of attention is required for selecting a proper

control chart among the ones given here and applying the selected chart for the key characteristic(s) or

process(es) concerned. The practitioners may use the information based on the properties of the discussed special

control charts given in Table 10 as a guide to determine the proper control chart for their interests.

Table 10. The properties of special control charts

Special Control Charts

Property 1 2 3 4 5 6 7

Sample size 2 2 2 1 or 2 1 1 or 2 1 or 2

Quality

characteristic

1 1 1 1 or 2 1 1 1

Number of

characteristics

1 1 1 1 1 1 or 2 1 or 2

Type of

variation

observed

1 and 2 1, 2, and 3 1 and 2 1 and 2 1, 2, and 3 1 and 2 1 and 2

Control limits 1 2 1 2 1 1 3

Manufacturing

volume/type

1 1 1 1 or 2 1 1 1

Type of

inspection

1 1 1 1 1 or 2 1 or 2 1 or 2

Inspection

strategy

1 1 1 2 1 1 1

Type of data

used

1 2 1 1 4 2 or 3 5

Interpretation 1 2 1 2 3 4 4

Sample size: 1- One, 2- More than one,

Quality characteristic: 1- Measurable, 2- Countable,

Number of characteristics: 1- One, 2- More than one,

Type of variation observed: 1- Time-to-time, 2- Piece-to-piece, 3- Variation within the piece,

Control limits: 1- Calculated using raw data, 2- Drawn directly using specification limits, 3- Drawn directly

using coefficients for sample size of n,

Manufacturing volume/type: 1- Low/Discrete, 2- High/Continuous,

Type of inspection: 1- Non-destructive, 2- Destructive,

Inspection strategy: 1- In-process, 2- Pre-process,

Type of data used: 1- Raw, 2- Deviation from nominal, 3- Deviation from target, 4- Min/Max values,

5- Pre-processed,

Interpretation: 1- Similar to Xbar-R, 2- Rule-based, 3- Similar to individuals and moving range,

4- Similar to Xbar-R but separately for each characteristic.

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