chapter 3 statistical process control tools and...

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CHAPTER 3

STATISTICAL PROCESS CONTROL TOOLS AND CMM

This chapter discusses the SPC and the control charts in detail. The applicability

of the SPC and the control chart in software industry are discussed and the selection of

the suitable chart for the analysis also discussed. The concepts of CMM and levels of

CMM are discussed.

3.1 Tools for Statistical Process Control

Continuous quality improvement process assumes and even demands that a team

of experts in the field as well as a company leadership actively use quality tools in their

improvement activities and decision making process. Quality tools can be used in all

phases of production process, from the beginning of the product development up to

product marketing and customer support. At the moment there are a significant number of

quality assurance and quality management tools on disposal to quality experts and

managers, so the selection of most appropriated one is not always an easy task.

The underlying concept of statistical process control is based on a comparison of

what is happening today with what has happened previously. If what is happening today

is minutely different from the earlier, it is still in a state of control. A snapshot of the

process typically performs or builds a model of how the process will perform and

calculate the control limits for the expected measurements of the output of the process.

From the earlier calculations/models built it is basically calculated the control limits and

once the control limits have been stabilized then the process is left alone to be understood

alone in the state of statistical control.

Data is collected from the process and compared to the control limits. The

majority of measurements should fall within the control limits and the measurements that

fall outside the control limits are examined to see if they belong to the same population as

our initial model.

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In general, software development processes have the following characteristics:

1. Software development processes are more heavily dependent on human activities

and require creativity than those of manufacturing processes (Caivano, 2005). It

follows that the measurement results of software processes can easily vary and

hence become instable.

2. Involves multiple common causes related to the first point, there are various

factors that can affect the performance of the processes, e.g., tools, methods, type

of software and development environment as pointed out by Florac and Carleton

(1999). These factors bring about a multiple common cause system. Therefore the

data obtained will be a mixed set of common cause results.

3. Hard to obtain a large set of homogeneous data, especially those data which have

important business value. Unlike manufacturing processes where many parts are

daily produced, software development processes usually result in a relatively

small set of work products. Furthermore each process execution is a creative and

unique activity, which makes it hard to obtain large set of homogeneous data.

Despite all these characteristics, SPC is still important in software processes. SPC

can give guidance to stabilize and improve processes. It can also be used to demonstrate

the improvement effects.

The key to overcome these difficulties is to put more emphasis on processes rather

than products. For example, using not only the product data but also process data can

increase the available data significantly. Under the assumption that the definition of

software process involves people, tools and working environment as suggested by

Carleton and Florac, (1999), SPC can help to stabilize human activities. It also helps to

identify and separate multiple common causes.

There are many ways to implement process control, but the key monitoring and

investigating tools include:

1. Histograms

2. Run charts

3. Pareto charts


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4. Cause and effect diagrams

5. Flow diagrams

6. Scatter diagrams

7. Control charts

3.1.1 Histogram

A histogram is a bar graph usually used to present frequency data. Histograms

provide an easy way to calculate the distribution of data over different categories. The

histogram is an effective, practical working tool in the early stages of data analysis.

For constructing a histogram the data are categorized when large data are

available. Then the data are collected and sorted into the categories. The data are counted

for each category and the diagram is drawn. A sample histogram is shown in Figure 3.1.

Each category finds its place on the x-axis and the bars in the histogram will be as high as

the value of the category.

A histogram may be interpreted by asking three questions:

1. How well is the histogram centered? The centering of the data provides

information on the process aim about some mean or normal value.

2. How wide is the histogram? Looking at the histogram width defines the

variability of the process about the aim.

3. What is the shape of the histogram? The data are expected to form a

normal or bell-shaped curve. Any significant change or anomaly usually

indicates that there is something wrong going in the process, which is

causing the quality problem.


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Figure 3.1 Sample histogram

Histograms are limited in their use due to the random order in which samples are

taken and lack of information about the state of control of the process. Because samples

are collected without regard to order, the time-dependent or time–related trends in the

process are not captured. This lack of information on process control may lead to

incorrect conclusions being drawn and hence, inappropriate decisions being made. Still

with these considerations in mind, the histogram’s simplicity of construction and ease of

use make it an invaluable tool in the elementary stages of data analysis.

3.1.2 Run chart

A Run chart is a simple graphic representation that displays data in the order that

they occur and shows a characteristic of a process over time. It is often known as a line

chart or a line graph outside the quality management field. Run charts allow us to

understand objectively if the changes are made to process or system overtime lead to

improvements and do so with minimal mathematical complexity. These methods of

analyzing and reporting data are of greater value to improvement projects and teams than

traditional aggregate summary statistics that ignore time order. Because of this utility and

simplicity, the run chart has wide potential application in software development for

project managers and decision-makers. Run charts also provide the foundation for more


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sophisticated methods of analysis and learning such as Shewhart (Control) charts and

planned experimentation.

Run charts originated from control charts. Shewhart, (1939) initially designed and

developed a system for bringing processes into statistical control by developing ideas,

which would allow for a system to be controlled using control charts. Run charts evolved

from the development of these control charts, but run charts focus more on time patterns

while a control chart focuses more on acceptable limits of the process.

Run chart is used to understand the trends and shifts in a process or variation over

time or to identify decline or improvement in a process over time. In a run chart, events

shown on the y-axis are plotted against a time period on the x-axis. A sample run chart is

given in Figure 3.2.

Figure 3.2 Sample run chart

Run chart rules

The key rules indicating special cause variation are:

Rule 1: shift

A shift in the process is six or more consecutive points either all above or all

below the median line. Values that fall on the median line do not count towards or break


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Observations

a shift–skip value on the median and continue counting. The shift rule is shown in

Figure 3.3.

Figure 3.3 Run chart shift rule

Rule 2: trend

A trend is five or more consecutive points all going up or all going down. If two

successive points have the same value, only the first counts towards the trend–like values

do not count towards or break a trend. Second point can be ignored and counting can be

continued with the next point on the chart. The rule 2 is shown in Figure 3.4.

Figure 3.4 Run chart trend rule

Rule 2

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Me

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Observations


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Rule3: runs

A run is a series of points either above or below the median line (points on the

median line are ignored). The number of runs can be easily calculated by counting the

number of times the data line crosses the median line and adding one to that total. A

special table is used to determine the upper and lower limits on this value.

Rule 4: astronomical value

An astronomical value is one that appears highly unusual in a ‘blatantly obvious’

fashion – not just taking the highest or lowest point on the chart. Not a precise definition,

but one that can generally be agreed with colleagues. Figure 3.5 shows the diagram for

applying rule 4.

Figure 3.5 Run chart astronomical value

3.1.3 Pareto chart

Pareto charts are the graphical tool used in Pareto analysis. A Pareto chart is a bar

chart that displays the relative importance of problems in a format that is very easy to

interpret. The Pareto chart is named after Vilfredo Pareto, a 19th

century economist who

postulated that a large share of wealth is owned by a small percentage of the population.

This basic principle translates well into quality problems. A Pareto chart is a series of

bars whose heights reflect the frequency or impact of problems. The bars are arranged in

descending order of height from left to right. The sample Pareto chart is shown in Figure

Rule 4

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Me

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me

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Observations


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3.6. This means the categories represented by the tall bars on the left are relatively more

significant than those on the right. This bar chart is used to separate the “vital few” from

the “trivial many”. These charts are based on the Pareto principle which states that 80

percent of the problems come from 20 percent of the causes. Pareto charts are extremely

useful because they can be used to identify those factors that have the greatest cumulative

effect on the system and thus screen out the less significant factors in an analysis. Ideally,

this allows the user to focus attention on a few important factors in a process.

Check sheets are relatively simple forms used to collect data. They include a list

of nonconformities and a tally of nonconformities. Check sheets should also include the

name of the project for which data are being collected, the shift when the items are

produced, the names of persons collecting the data, dates of data collection and of

production and the location of data collection. Check sheets are not mandatory to

construct Pareto charts.

Figure 3.6 Sample Pareto chart

A Pareto chart is a good tool to use when the investigating process produces data

that are broken down into categories and that counts the number of times each category

occurs. A Pareto diagram puts data in a hierarchical order, which allows the most

significant problems to be corrected first. The Pareto analysis technique is used primarily

to identify and evaluate nonconformities, although it can summarize all types of data. It is

the perhaps the diagram most often used in management presentations.


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Making problem solving decisions isn’t the only use of the Pareto Principle. Since

Pareto charts convey information in a way that enables you to see clearly the choices that

should be made, they can be used to set priorities for many practical applications. Some

examples are:

Process improvement efforts

Skills the division should have

Customer needs

Suppliers

Investment opportunities

3.1.4 Cause and effect diagram

A Cause and effect diagram is a tool that is used to identify, sort and display

possible causes of a specific problem or quality characteristic. The relationship between a

given outcome and all the factors that influence the outcome is graphically illustrated

through this diagram. This diagram is also called as “fishbone diagram “because of the

way it works.

Cause and effect diagram helps the users to identify the possible root cause, the

basic operations, for a specific effect, problem or condition. Using the diagram the team

can take the corrective action by analyzing the existing problems. A cause and effect

diagram is helpful for organizing the known or possible causes of quality or lack of it.

The diagram helps the team in a very systematic way because of its structure.

The advantages of constructing a cause and effect diagram are as follows:

Structured approach is applied to determine the root causes of the problem

or quality characteristic

Encourages group participation and utilizes group knowledge of the

process

Shows possible causes of variation in a process

Improves knowledge of the process by helping the team to learn more

about the factors at work and how they relate

Defines areas where data should be collected for further study


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Constructing a cause and effect diagram is a three step process and discussed below.

Step 1

Writing down the effect to be investigated and draw the 'backbone' arrow to it. In the

example shown below the effect is 'Incorrect deliveries'.

Step 2

Identifying all the broad areas of enquiry in which the causes of the effect being

investigated may lie. For incorrect deliveries the diagram may then become:

Step 3

This step requires the greatest amount of work and imagination form the team to write in

all the detailed possible causes in each of the broad areas of enquiry. Each cause

identified should be fully explored for further more specific causes which, in turn,

contribute to them.

Figure 3.7 Sample cause and effect diagram

This process is continued of branching off into more and more directions until

every possible cause has been identified. The final result shows a sort of a 'mind dump' of


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all the factors relating to the effect being explored and the relationships between them.

Cause and effect diagrams enable a team to focus on the content of the problem. It

focuses the team on causes, not symptoms. It is an effective tool that allows people to

easily see the relationship between factors to study the process, situations and for

planning. The sample cause and effect diagram is shown in Figure 3.7.

Causes in a cause and effect diagram are frequently arranged into the following

four categories

1. Manpower, methods, materials and machinery (recommended for manufacturing)

2. Equipment, methods, materials and people (recommended for administration and

service).

3.1.5 Flow diagram

A process flow chart is simply a tool that graphically shows the inputs, actions

and output of a given system. The purpose of flow chart is to help people to understand

the process. A flow chart illustrates the activities performed and the flow of resources and

information in a process. Two types of flow charts are particularly useful – high level and

detailed. A high level flow chart illustrates how major groups of related activities, often

called “sub processes”, interact in a process. Typically, four to seven sub processes are

shown in a flowchart. By including only basic information, high level flowcharts can

readily show an entire process and its key sub processes.

A detailed flow chart provides a wealth of information about activities at each

step in a sub process. It shows the sequence of the work and includes most or all of the

steps, including rework steps that may be needed to overcome problems in the process. A

quality improvement team can increase the detail to show the individuals performing

each activity or the time required to complete each activity. If necessary, the link between

various points in the sub process and other high level flow charts of the process can also

be shown. An organization pursuing quality improvement is constantly looking for ways

to improve the effectiveness and efficiency of its work.


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“Effectiveness” means producing the required results of output when needed.

“Efficiency” means simply producing those results or outcomes the first time with

minimum resources. In order to generate ideas on how to be more efficient and effective,

it is helpful to define and document how activities are actually performed. Flow charts

are useful for this purpose. Flow charts can be useful to identify activities in a process

that reduce our effectiveness and efficiency. For example, some activities may be

redundant or repeated, other may be unnecessary. Activities may be performed in

sequence, when they could be conducted at the same time to reduce the overall time for

the process. Flow charts can be used to identify conditions that cause delays and

bottlenecks. This can bring focus to problems at various points within the process that

need further evaluation and improvement.

3.1.6 Scatter diagram

A scatter diagram is a tool for analyzing relationships between two variables. One

variable is plotted on the horizontal axis and the other is plotted on the vertical axis. The

pattern of their intersecting points can graphically show the relationship pattern. Most

often scatter diagram is used to prove or disapprove cause and effect relationships. While

the diagram shows relationships, it does not by itself prove that one variable causes the

other. In addition to showing possible cause and effect relationships, a scatter diagram

can show the two variables are from a common cause that is unknown or that one

variable can be used as a surrogate for the other. A sample scatter diagram is shown in

Figure 3.8.

Scatter diagrams can be used to examine theories about cause and effect

relationships and to search for the root causes of an identified problem. Also it can be

used to design a control system to ensure that gains from quality improvement efforts are

maintained.

The following steps are followed when using scatter diagrams:

1. Selecting and defining the two variables to be analyzed in the individual chart.

2. Measuring the two variables, if data has not already been collected.


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3. The set of data pairs should consist of at least thirty, but preferably more like

one hundred data points.

4. Designing the chart by placing the independent variable on the horizontal axis.

The independent variable is the factor believed to be governing relationship

between the two variables.

5. On the vertical axis, inserting the dependent variable that is, the factor

believed to change in proportion to the independent variable.

6. Plotting the data pairs themselves in the chart area.

7. Examining the completed chat, looking for pattern that indicates a connection

between the two variables.

8. If correlation patterns are identified, investigating any third variable

involvement before drawing definite conclusions.

Figure 3.8 Sample scatter diagram

Scatter diagrams generally shows one of six possible correlations between the variables:

Strong Positive Correlation : The value of Y clearly increases as the value of X increases

Strong Negative Correlation : The value of Y clearly decreases as the value of X increases.

Weak Negative Correlation : The value of Y clearly decreases slightly as the value of X

increases.

Complex Correlation : The value of Y seems to be related to the value of X, but the

relationship is not easily determined.

No correlation : There is no demonstrated connection between two variables.


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3.1.7 Control charts

There are two principal types of Quality Assurance: “Sampling inspection of

incoming/outgoing materials (acceptance sampling) and control charts for ongoing

processes (process control). Control charts differentiate between the process being in

control (within an accept range of random variation) and out of control (outside the

acceptable range)” (Antony et al, 2000). A process that is operating with only ‘chance

causes of variation’ present is said to be in statistical control. A process that is operating

in the presence of assignable causes is said to be out of control and the eventual goal of

SPC is the elimination of variability in the process. The control chart can be seen as part

of an objective and disciplined approach enable us to take correct decisions regarding

control of the process, including whether or not to change control parameters. Process

parameters should never be adjusted for a process that is in control as this will result in

degraded process performance (William et al, 2006). The primary purpose of a control

chart is to detect whether a major change or shift is imminent or has occurred in a process

resulting in an alteration of that process. The selection of the appropriate control chart is

vital for the success of the quality of process. The control chart is one of the seven basic

tools of quality control (Tague, 2004).

The statistical argument for control chart relies upon the central limit theorem.

The central limit theorem suggests even if the underlying population from which a series

of observations gathered is not normally distributed, the resulting distribution of averages

from the sample observations will be normally distributed around the average values and

that 99.73 percent of the values will be contained within three sigma deviations of that

average value. Using this theorem, a control chart can be constructed.

A control chart consists of:

Points representing a statistic (e.g., a mean, range, proportion) of measurements of

a quality characteristic in samples taken from the process at different times.

The mean of this statistic using all the samples is calculated (e.g., the mean of the

means, mean of the ranges, mean of the proportions). A center line is drawn at the

value of the mean of the statistic.


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The standard error (e.g., standard deviation/sqrt(n) for the mean) of the statistic is

also calculated using all the samples.

Upper and lower control limits (sometimes called "natural process limits") that

indicate the threshold at which the process output is considered statistically

'unlikely' are drawn typically at 3 standard errors from the center line.

The chart may have other optional features, including:

Upper and lower warning limits, drawn as separate lines, typically two standard

errors above and below the centerline.

Division into zones, with the addition of rules governing frequencies of

observations in each zone.

Annotation with events of interest, as determined by the quality engineer

in-charge of the process's quality control chart.

If the process is in control, all points will plot within the control limits. Any

observations outside the limits or systematic patterns within, suggest the introduction of a

new (and likely unanticipated) source of variation, known as a special cause variation.

Since increased variation means increased quality costs, a control chart "signaling" the

presence of a special cause requires immediate investigation. This makes the control

limits very important decision aids. The control limits indicates about process behavior

and have no intrinsic relationship to any specification targets or engineering tolerance. In

practice, the process mean may not coincide with the specified value of the quality

characteristic.

The purpose of control charts is to allow simple detection of events that are

indicative of actual process change. This simple decision can be difficult where the

process characteristic is continuously varying; the control chart provides statistically

objective criteria of change. When change is detected and considered good its cause

should be identified and possibly become the new way of working, where the change is

bad then its cause should be identified and eliminated. The purpose in adding warning

limits or subdividing the control chart into zones is to provide early notification if

something is incorrect. Instead of immediately launching a process improvement effort to

determine whether special causes are present, the quality engineer may temporarily


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increase the rate at which samples are taken from the process output until it's clear that

the process is truly in control. Note that with three sigma limits, common causes result in

signals less than once out of every forty points for Skewed processes and less than once

out of every two hundred points for normally distributed processes (Wheeler, 2010).

Control charts are categorized into the following two types:

1. Attribute control charts

2. Variable control charts

Attribute control charts are used when collected data or the measurements are

categorized as acceptable or not acceptable. These attributes provide counting data. This

data or actual number or failures or the fraction or percent of failures is charted (Smith,

1991). Variable control charts are applied to data that follow a continuous distribution.

3.1.7.1 Attribute control charts

Stability of the systems can be monitored using attribute control charts. For

constructing attribute charts data count or percentage is accumulated. Attribute charts are

the outcome of an assessment using pass or fail criteria.

Attribute charts consist of:

p-chart (Graph of fraction non-confirming)

np-chart (Graph the number of non-conformities in a sample of n pieces)

u-chart (Draft the defect /n)

c- chart (Graph the defects in a sample of n pieces)

p-charts and np-charts are used to graph non-conformities. c-charts and u-charts

are used to graph non-conforming units. “Non-conformity may be imperfection or the

presence of some non-preferred feature” (Freeman and Mintzas, 1999). A non-

conforming unit on the other hand is defined as, “A non-confirming unit, however, may

fail to meet the assessment of some criteria because of one or more non-conformities”

(Freeman and Mintzas 1999). The accept/reject boundaries must be clearly defined when

attribute control charts are constructed. Also, the accept/reject characteristics should be

agreed with by the customer.


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3.1.7.1.1 p-chart

The p-chart is the most commonly used attribute chart. The p represents the

fraction or percent, of the number of items that are unacceptable (or defective). The p-

chart is most helpful in monitoring and controlling the percentage of defective parts in a

production run (Amsden et al., 1989). It tracks the fraction of nonconforming items in a

sample run. Samples may consist of consecutive items taken at a specified time according

to a random sample plan or 100 percent of the production for a specified time period

(hour, day and week). The acceptable/non-acceptable decision of the process may be

based on one characteristic or several, but a defective (non-acceptable) item is counted as

defective only once, even though it may have several defects (Smith, 1991).

Prior to actual constructing the p-chart, preparation must occur to ensure the

accuracy of the intended data. The process must be clearly defined. Quality control teams

can be effective in developing flow charts and cause and effect diagrams for the process

analysis. The characteristics that will be managed or measured should also be defined.

When individual judgment is used for deciding product conformity, inspectors must all

be consistent and in total agreement as to what exactly is classified as defective (Smith,

1991).

The procedures for creating a p-chart can be split into six steps:

The first is to gather all needed data, selecting the size, frequency and number of

samples. The sample size should be large enough to ensure that most of the samples will

have a nonzero number of defects. The sample size should be big enough to give an

average of five or more defectives per sample. A sample size of at least 50 units is a good

starting point. The calculations and interpretation of the chart are easier when the sample

size is kept constant. When constant sample times are used and the sample size varies, a

single set of control limits can be used as long as the sample sizes do not differ by more

than 25 percent of the average sample size. If the sample sizes differ by more than 25

percent, separate control limit calculations will be needed (Amsden et al., 1989).

Sample frequency should also be a factor in the production schedule. The p-chart

should give an accurate depiction as much as possible of the process at the specific point


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in question. Every item produced in that time should have an equal chance of being

chosen in a sample (Smith, 1991). The number of actual samples should be at least 20.

The data collection timeframe should be long enough to track all possible sources of

variation (Smith, 1991).

The second step in constructing a p-chart is calculating the p (percentage) for each

sample. This is accomplished by dividing the number of defects (np) by the actual sample

size (n) [p = np/n]. If the number of defects in the sample of 50 units were 4, the p would

equal .08 [p = 4/50; .08).

The third step in the p-chart is setting the scale for the chart or graph. After

several p values have been calculated, make the scale from 0 to approximately twice the

largest know p value. This will allow for all the p values to fit on the graph (Smith,

1991).

The fourth step is to plot all p values and connect all plot points with a straight

line (Amsden et al., 1989).

Calculating the p-bar and control limits is the fifth step to the p-chart. The p-bar is

calculated by adding all np’s (total number of defects) and dividing by the total amount

of units in all samples. A solid line should be added to the chart to indicate the p-bar

value.

The Upper Control Limits (UCL) and the Lower Control Limits (LCL) are

calculated using the following formulae

UCL = pbar + 3

LCL = pbar - 3

The last step is the interpretation of the chart. There are three possible situations

to consider. First, all the p values are inside the control limits. If this is the case, this can

be indicative that the process is in statistical control. The inherent variations also will be


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at work with the process. The second situation to consider is that one or two p values are

outside the control limits. The usual practice is to not include these values or any sample

information from these samples. All calculations also need to be refigured including the

p-bars and control limits. The chart then should be re-examined. If any p value is out of

the control limits, the process is not in statistical control. The process and causes need to

be determined and resolved. The p-chart process should begin again to ensure that the

process has been fixed (Amsden et al., 1989).

Figure 3.9 Sample p-chart

If three or more p values are outside the control limits, the process is clearly out-

of-control. causes must be determined immediately to resolve such issues.

Shifts of seven or more points to a higher or lower levels may represent that the

process has been affected somehow or that the exact cause of the defect has somewhat

changed. Trends of a run of seven consecutive points either up or down indicate that

something is causing the defects to change in a gradual manner (Smith, 1991). This type

of pattern indicates that something irregular is affecting the process. The causes could be

anything from poorly trained personnel to inconsistent inspections. If this occurs separate


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p-charts may need to take place at other points in the process. The sample p-chart is

shown in Figure 3.9.

p-charts as well as other factors of SPC have been widely accepted among

“quality practitioners as an aid for monitoring, managing, analyzing and improving

process performance by eliminating causes of variations” (Antony, 2000). The use of this

type of data permits a scientific management style in which decisions are made “based on

facts, rather than guesswork and better products can be produced with less scrap and

rework” (Fine, 1997).

3.1.7.1.2 np-chart

An np-chart is a data analysis technique for determining if a measurement process

has gone out of statistical control. It is sensitive to changes in the number of defective

items in the measurement process. The “np” in np-charts stand for the np (the mean

number of success) of a binomial distribution.

The np control chart consists of:

Vertical axis = the number of defectives for each sub-group

Horizontal axis = the sub-group designation

A sub-group is frequently a time sequence (Example, the number of defectives in

a daily production run where each day is considered a sub-group). Sub-group size must

be constant and also normally large subgroups are needed if the times are equally spaced,

the horizontal axis available can be generated as a sequence.

In addition, horizontal lines are drawn at the mean number of defectives and at the

upper and lower control limits. The distribution of the number of defective items is

assumed to be binomial. This assumption is the basis for the calculating the upper and

lower control limits.

Control limits for the np-chart are calculated on the basis of binomial distribution

and an approximation based on the central limit theorem.


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The control limits are calculated as:

LCL = np - 3

UCL = np + 3

Where n is the number of items and p is the proportion of defective items. Zero

serves as a lower bound on the LCL value. The sample p-chart is shown in Figure 3.10.

Figure 3.10 Sample np-chart

np-charts are used to analyze the results of process improvements. Here it is

consider how the process is running and comparing it how it ran in the past. Finally, np-

charts are used for standardization. Without a control chart, there is no way to know if the

process has changed or to identify sources of process variability.

3.1.7.1.3 u-chart

A u-chart is an attribute control chart used with data collected in subgroups of

varying size. u-charts show how the process is measured by the number of non-

conformities per item or group of items, changes over time. Non-conformities are

defects or occurrences found in the sample subgroup. They can be described as any

characteristic that is present but should not be or any characteristic that is not present but


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should be. For example, a scratch sent, bubble, missing button and a tear are all

nonconformities. u-chart is used to determine if the process is stable and predictable, as

well as to monitor the effects of process improvement theories.

A u-chart is particularly useful when the item is too complex to be ruled as simply

confirming or nonconforming. A u-chart is appropriate when the area of opportunity for a

defect varies from subgroup to subgroup. This can be seen in the shifting of UCL and

LCL lines that depend on the size of the subgroup. When areas of opportunity are

appropriately measured and a Poisson model applies, u-charts are the tool of choice.

Defects per thousand lines of source code, defects per function point and system failures

per day in steady-state operation are all examples of attributes data that are candidates for

u-charts. Defects per module and defects per test, on the other hand, are unlikely

candidates for u-charts, c-charts or any other charts for that matter. These ratios are not

based on equal areas of opportunity and there is no reason to expect them to be constant

across all modules or tests when the process is in statistical control.

By virtue of their dependence on Poisson distributions, both u-charts and c-charts

must satisfy the following conditions (Wheeler and Chambers, 1992):

They must be based on counts of discrete events

The discrete events must occur within some well-defined, finite region of

space, time or product

The events must occur independently of each other

The events must be rare, relative to the opportunity for their occurrence

Although u-charts may be appropriate for studying software defect densities, the

use of Poisson models is not validated through any of the studies for these situations. For

example, if modules are deliberately made small as the tasks that they perform are

inherently difficult to design and program, then simple defect densities are unlikely to

follow the same Poisson process across all modules. When the opportunities for

observing the event are not constant, as when differently sized portions of code are

examined or tested for defects, the counts must be converted into rates -such as defects


65

per thousand lines of code or defects per thousand source statements, before they can be

compared. The rates are computed by dividing each count (ci) by its area of opportunity

(ai). The rate that results is denoted by the symbol ui. Thus,

Ui = Ci/ai

Once values for Ui have been calculated, they can be plotted as a running record

on a chart. The sample np-chart is shown in Figure 3.11.

No. of Observations

De

fect

De

nsit

y

1413121110987654321

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00

_U=0.06

UCL=0.1271

LCL=0

3

1

1

HP-Grp Maintenance

Figure 3.11 Sample u-chart

The center line and control limits for u-charts are obtained by finding u, the

average rate over all the areas of opportunity and using these formulas:

∑ Ui

Ubar = --------

∑ ai

LCL = Ubar- 3

UCL = Ubar+ 3


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The presence of the area of opportunity ai in the formulas for the control limits for

u-charts means that the limits will be different for each different area of opportunity.

3.1.7.1.4 c-chart

A c-chart is a data analysis technique for determining if a measurement process

has gone out of statistical control. The c-chart is sensitive to changes in the number of

defective items in the measurement process. The “c” in the control chart stands for

“counts” as in defectives per lot.

A c-chart or count chart, is an attribute control chart that displays how the number

of defects or nonconformities, for a process or system is changing over time. The

number of defects collected for the area of opportunity can be either a group of units or

just one individual unit on which defect counts are performed. The c-chart is an

indicator of the consistency and predictability of the level of defects in the process. The

sample c-chart is shown in Figure 3.12.

When constructing a c-chart, it is important that the area of opportunity for a

defect be constant from a subgroup to subgroup since the chart shows the total number

of defects. When the number of items tested within subgroup changes, then a u-chart

should be used, since it shows the number of defects per unit rather than total defects.

Figure 3.12 Sample c-chart


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3.1.7.2 Variable control charts

Control limits for attributes data are often computed in ways quite different from

control limits for variables data. Variables data (sometimes called measurement data) are

usually measurements of continuous phenomena.

Examples : measurements of length, weight, volume and

speed.

Software examples : elapsed time, effort expended, years of

experience, memory utilization and cost of

rework.

The following are the types of variable control charts:

X-bar chart

R-chart

s-chart

Individual chart

Moving Range chart

X-bar chart is based on the average of a subgroup. Subgroups of 2 to 30 samples

may be used when computing the control limits for the X-bar chart when based on the

range.

R-chart takes into account the range of a subgroup. Subgroup sizes may be as

small as 2 or as large as 30. It is important to construct and interpret an R-chart before the

X-bar chart. When the R chart indicates that process variation is in control, analyze the

X-bar chart is analyzed otherwise X-bar is not meaningful.

s-chart takes into account the standard deviation of a subgroup. There is no limit

to the subgroup size. Process variability can be controlled by either a R-chart or a

standard deviation chart (s chart) depending on how the population standard deviation is

estimated. s-chart is used to determine whether the standard deviation has changed.


68

Individual chart displays each value. A subgroup size is used to compute the

limits, with value of 2 being most common, although the subgroup size may be as large

as 30.

3.1.7.2.1 XmR chart

An individual and moving range (XmR) chart is a pair of control charts for

processes with a subgroup size of one. This is used to determine, if a process is stable and

predictable and also it creates a picture of how the system changes over time. The

individual (X) chart displays individual measurements. The moving range (mR) chart

shows variability between one data point and the next. XmR charts reflect data which do

not lend themselves to form subgroups with more than a one measurement. The only

condition that needs to be checked before using the XmR control chart is that the average

count per sample is greater than one.

For example, a process repeats itself frequently or it appears to operate differently

at different times. In such case, grouping the data might mask the effects of such

differences. One can overcome this problem by using an XmR chart whenever there is no

rational basis for grouping the data.

An individual chart is equivalent to X-bar but reported to single observation, not

to samples. Used when the nature of the process is such that it is difficult or impossible to

group measurements into subgroups. This occurs frequently in low volume production

and in situations in which continuously varying quantities within the process are process-

related variables. The solution is to artificially create subgroups from the data and then

calculate the range of each subgroup. This is done by creating rolling groups (most often

pairs) of data through time and using the pairs to determine the range R. The resulting

ranges are called moving ranges.

Control limits in XmR chart are calculated from moving range (mR). A range is

based on the absolute value of consecutive differences in observations. The first step in

calculating control limits is to estimate the average of the moving range.


69

Count the number of time periods, n.

Calculate the absolute value of the difference of every consecutive value,

call this moving range.

Add the moving ranges and divide by n-1 to get the average moving range.

The second step in constructing a XMR chart is calculating the UCL. UCL is also

called as Upper Natural Process Limit (UNPL). The upper control limit is average of the

observations plus a constant ‘E’ times the average moving range. The constant ‘E’

depends on how many consecutive observations are included in the moving range. The

value of the correction factor is chosen so that 99% of the data fall within the control

limits.

If the moving range is calculated from 2 consecutive time periods then the

correction factor E is 2.66.

The Upper Control Limit can be calculated as:

UCL = Average of observations + 2.66 * Average of moving range

The third step in constructing a XMR chart is calculating the LCL. LCL is also

called as Lower Natural Process Limit (LNPL). The lower control limit is average of

observation minus 2.66 times the average range.

The Lower Control Limit is calculated as:

LCL = Average of observations – 2.66 * Average of moving range

The last step is the interpretation of the chart. There are four possible situations to

consider. First, if any one of the values of Xi is outside the control limit of UCL. The

second situation to consider is that any of two out of three successive values should be in

2 sigma and 3 - sigma limits. If four out of five successive values of Xi are in 1 sigma

and 2 sigma means the process is totally out of control. The last step for interpreting the

XMR chart is to consider that the eight successive values are on the same side of the

centerline. In mR chart if any point Xi is outside the UCL then the process is said to be

out of control. The sample XmR-chart is shown in Figure 3.13.


70

Figure 3.13 Sample XmR chart

3.2 CMM Practices

Capability Maturity Model broadly refers to the process improvement approach

that is based on a process model. CMM also refers specifically to the first such method,

developed by the Software Engineering Institute in the mid -1980’s, as well as the family

of process model that is followed. A process model is a structured collection of practices

that describe the characteristics of effective process, the practices that included are those

proven by experience to be effective.

X CHART- USER REQUIREMENT SPECIFICATION

-80.00

-60.00

-40.00

-20.00

0.00

20.00

40.00

60.00

80.00

100.00

120.00

PROJE

CT1

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CT3

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PROJECTS

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LNPLx

ULPNx

Sigmax

2Sigmax

3Sigmax

1-Sigmax

2-Sigmax

3-Sigmax

mr CHART-USER REQUIREMENT SPECIFICATION

0.0010.0020.0030.0040.0050.0060.0070.0080.0090.00

100.00

PROJE

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71

CMM can be used to assess an organization against a scale of five process

maturity levels. Each level ranks the organization according to its standardization of

processes in the subject area being assessed. The subject areas can be as diverse as

software engineering, systems engineering, project management, risk management,

system acquisition, information technology service and personnel management.

The CMM is a way to develop and refine an organization’s process. The first

CMM was for the purpose of developing and refining software development processes. A

maturity model is a structural collection of elements that describe characteristics of

effective process. A maturity model provides:

a place to start

the benefit of a community’s prior experiences

a common language and a shared vision

a framework for prioritizing actions

a way to define what improvement means for the organization

A maturity model can be used to benchmark for accessing different organizations

for equivalent comparison. It describes the maturity of the company based upon the

project the company is dealing with and its clients.

In the 1970s, technological improvements made computers more widespread,

flexible and inexpensive. Organizations began to adopt more and more computerized

information systems and the field of software development grew significantly.

Unfortunately, the influx of growth caused growing pains; project failure became more

commonplace not only because the field of computer science was still in its infancy, but

also because projects became more ambitious in scale and complexity. In response,

individuals such as Edward Yourdon and Gerald Weinberg, published articles and books

with research resulted in an attempt to professionalize the software development process.

Humphrey’s, (1989) CMM was described in the book managing the Software

Process. The CMM as conceived by Humphrey is based on the earlier work of Phil

Crosby. Active development of the model by the SEI began in 1986. The CMM was


72

originally intended as a tool to evaluate the ability of government contractors to perform

a contracted software project. Though it comes from the area of software development, it

has been continued and can continue to be widely applied as a general model of the

maturity of processes in software (and other) organizations.

The model identifies five levels of process maturity for an organization. Within

each of these maturity levels are s Key Process Areas (KPA) which characterize that level

and for each KPA there are five definitions identified:

1. Goals

2. Commitment

3. Ability

4. Measurement

5. Verification

The KPAs are not necessarily unique to CMM, representing - as they do - the

stages that organizations must go through on the way to becoming mature. The

assessment is supposed to be led by an authorized lead assessor. One way in which

companies are supposed to use the model is first to assess their maturity level and then

form a specific plan to get to the next level. Skipping levels is not allowed.

3.2.1 Levels of the CMM

There are five levels of CMM which are described in details.

Level 1 - Initial

Processes are usually ad hoc and the organization usually does not provide a stable

environment. Success in these organizations depends on the competence and heroics of

the people in the organization and not on the use of proven processes. In spite of this ad

hoc, chaotic environment, maturity level 1 organizations often produce products and

services that work; however, they frequently exceed the budget and schedule of their

projects.


73

Organizations are characterized by a tendency to over commit, abandon processes

in the time of crisis and are not able to repeat their past successes again. Software project

success depends on having quality people.

Level 2 - Repeatable

Software development successes are repeatable. The processes may not repeat for

all the projects in the organization. The organization may use some basic project

management to track cost and schedule.

Process discipline helps ensure that existing practices are retained during times of

stress. When these practices are in place, projects are performed and managed according

to their documented plans.

Project status and the delivery of services are visible to management at defined

points (for example, at major milestones and at the completion of major tasks).

Basic project management processes are established to track cost, schedule and

functionality. The minimum process discipline is in place to repeat earlier successes on

projects with similar applications and scope. There is still a significant risk of exceeding

cost and time estimate.

Level 3 - Defined

The organization’s set of standard processes, which is the basis for level 3, is

established and improved over time. These standard processes are used to establish

consistency across the organization. Projects establish their defined processes by the

organization’s set of standard processes according to tailoring guidelines.

The organization’s management establishes process objectives based on the

organization’s set of standard processes and ensures that these objectives are

appropriately addressed.

A critical distinction between level 2 and level 3 is the scope of standards, process

descriptions and procedures. At level 2, the standards, process descriptions and

procedures may be quite different in each specific instance of the process (for example,


74

on a particular project). At level 3, the standards, process descriptions and procedures for

a project are tailored from the organization’s set of standard processes to suit a particular

project or organizational unit.

Level 4 - Managed

Using precise measurements, management can effectively control the software

development effort. In particular, management can identify ways to adjust and adapt the

process to particular projects without measurable losses of quality or deviations from

specifications. At this level organization sets a quantitative quality goal for both software

process and software maintenance.

Sub processes are selected that significantly contribute to overall process

performance. These selected sub processes are controlled using statistical and other

quantitative techniques.

A critical distinction between maturity level 3 and maturity level 4 is the

predictability of process performance. At maturity level 4, the performance of processes

is controlled using statistical and other quantitative techniques and is quantitatively

predictable. At maturity level 3, processes are only qualitatively predictable.

Level 5 - Optimizing

Focusing on continually improving process performance is done through both

incremental and innovative technological improvements. Quantitative process-

improvement objectives for the organization are established, is continually revised to

reflect changing business objectives and used as criteria in managing process

improvement. The effects of deployed process improvements are measured and evaluated

against the quantitative process-improvement objectives. Both the defined processes and

the organization’s set of standard processes are targets of measurable improvement

activities.

Process improvements to address common causes of process variation and

measurably improve the organization’s processes are identified, evaluated and deployed.


75

Optimizing processes that are nimble, adaptable and innovative depends on the

participation of an empowered workforce aligned with the business values and objectives

of the organization. The organization’s ability to rapidly respond to changes and

opportunities is enhanced by finding ways to accelerate and share learning.

A critical distinction between maturity level 4 and maturity level 5 is the type of

process variation addressed. At maturity level 4, processes are concerned with addressing

special causes of process variation and providing statistical predictability of the results.

Though processes may produce predictable results, the results may be insufficient to

achieve the established objectives. At maturity level 5, processes are concerned with

addressing common causes of process variation and changing the process (that is, shifting

the mean of the process performance) to improve process performance (while

maintaining statistical probability) to achieve the established quantitative process-

improvement objectives.

3.3 Chapter Summary

This chapter discussed the methods of several common statistical process control

charts. The definition of a statistical control chart is that it is a graphical device for

monitoring a measurable characteristic of a process for the purpose of showing whether

the process is operating within its limits of expected variation. A major objective of a

statistical process control chart is “to detect quickly the occurrence of assignable causes

of process shifts so that the process can be investigated and corrective actions undertaken

before many nonconforming units are manufactured”. Also the concepts of CMM and the

practices followed in CMM were discussed. This chapter provides a basis for the

identification of methodology for the selection of a control chart used for SPC analysis in

this research. The next chapter discusses the research methodology used to guide the

research.


chapter 3 statistical process control tools and...

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