UCL Control Charts 1.0 Introduction A control chart (or Shewhart control charts) is a graphical tool for monitoring the activity of an ongoing process. The values of quality characteristic (eg. Length, diameter, surface roughness, tool wear) are plotted on vertical axis and the horizontal axis represents sample number or subgroup number. Three lines are indicated on the control chart. The centre line, which typically represents the average value of the characteristic being plotted, is an indication of where the process is centred. Two limits, the upper control limit and lower control limit, are used to make decisions regarding the process. If the points plot within the control limits and do not exhibit any identifiable pattern, the process is said to be in statistical control. If a point plots outside the control limits or if an identifiable pattern the process is said to be out of statistical control. 1.1 Benefits 1. A control chart indicates when something may be wrong so that corrective action can be taken. 2. The pattern of the plot on a control chart diagnoses possible causes and hence indicates possible remedial actions. 3. With the help of control chart, we can calculate capability of the process under study. 4. Control chart provides a base line for instituting and measuring quality improvement. 1.2 Causes of variation Variability is a part of any process. Several factors such as methods, equipment, people, materials, and policies influence variability. Environmental factors also contribute to variability. The causes of variation are divided into two categories: common causes and special causes. Control of a process is achieved through elimination of special causes. Improvement of a process is accomplished through the reduction of common causes. Special causes are not inherent in the process. They can be due to use of wrong tool, an improper raw material, or an operator error. Common causes are inherent in the process. They occur due to natural variation in the process and are not in the control of the operator. They cannot be completely eliminated but minimized. 2.0 Design of control charts Design of control charts involves (i) selection of control limits, (ii) frequency of sampling, (iii) selection of sample size, and (iv) criteria for rational sampling (i) Selection of control limits: Control limits normally chosen are corresponding to 3-sigma limits. In this case, the expected value of the quality characteristic of samples is taken as the centre line value. The Upper Control Limit Lower Control Limit Centre Line Sample number Quality Characteristic Figure-1
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UCL
Control Charts
1.0 Introduction
A control chart (or Shewhart control charts) is a graphical tool for monitoring the activity of an ongoing process.
The values of quality characteristic (eg. Length, diameter, surface roughness, tool wear) are plotted on vertical
axis and the horizontal axis represents sample number or subgroup number.
Three lines are indicated on the control chart. The centre line, which typically represents the average value of the
characteristic being plotted, is an indication of where the process is centred. Two limits, the upper control limit
and lower control limit, are used to make decisions regarding the process. If the points plot within the control
limits and do not exhibit any identifiable pattern, the process is said to be in statistical control. If a point plots
outside the control limits or if an identifiable pattern the process is said to be out of statistical control.
1.1 Benefits
1. A control chart indicates when something may be wrong so that corrective action can be taken.
2. The pattern of the plot on a control chart diagnoses possible causes and hence indicates possible
remedial actions.
3. With the help of control chart, we can calculate capability of the process under study.
4. Control chart provides a base line for instituting and measuring quality improvement.
1.2 Causes of variation
Variability is a part of any process. Several factors such as methods, equipment, people, materials, and policies
influence variability. Environmental factors also contribute to variability. The causes of variation are divided into
two categories: common causes and special causes. Control of a process is achieved through elimination of
special causes. Improvement of a process is accomplished through the reduction of common causes. Special
causes are not inherent in the process. They can be due to use of wrong tool, an improper raw material, or an
operator error. Common causes are inherent in the process. They occur due to natural variation in the process
and are not in the control of the operator. They cannot be completely eliminated but minimized.
2.0 Design of control charts
Design of control charts involves (i) selection of control limits, (ii) frequency of sampling, (iii) selection of sample
size, and (iv) criteria for rational sampling
(i) Selection of control limits: Control limits normally chosen are corresponding to 3-sigma limits. In this
case, the expected value of the quality characteristic of samples is taken as the centre line value. The
Upper Control Limit
Lower Control Limit
Centre Line
Sample number
Qu
alit
y C
har
acte
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Figure-1
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upper control limit is drawn at a distance of 3-sigma on one side of the centre line and the lower
control limit is drawn at a distance of 3-sigma on the other side of the centre line.
) (1)
UCL= )+3.σ (2)
LCL= )-3.σ (3)
Sometimes probability limits can also be used. It is based on the degree of probability that the
sample statistic falls outside the control limits. Take for example, 3 sigma limits cover 99.73% of
the area under normal distribution curve. That means the probability of a statistic value falling
outside the range is 0.27%. If we want the probability to be (let us say) 0.10%, the upper control
limit is calculated at a distance of 3.29 sigma (obtained from normal distribution tables).
Sometimes the control limits are calculated based on allowable errors such as (i) type I error and
(ii) type II error. Type I error results from inferring that a process is out of control when it is
actually in control. Type II error results from inferring that a process is in control when it is really
out of control.
(ii) Frequency of sampling: Collecting large samples at frequent intervals provides a lot of information.
However, it may not be practicable. Collection of small samples at frequent intervals also does not
provide complete information. Selection of frequency depends on current state of process. If the
current state of process is stable, the frequency can be decreased. Otherwise, it may be increased.
Similarly if the inspection involved is destructive in nature, we cannot afford frequent sampling. If the
process involves greater variability, we have to increase the frequency. Finally the cost of sampling
and inspection also influence the frequency of sampling.
(iii) Selection of sample size: The degree of shift expected to take place will influence the choice of sample
size. Large shifts in the process parameter can be detected by smaller sample sizes than those
needed to detect smaller shifts. Alternatively, if it is important to detect slight changes in the
process, we require smaller sample sizes.
(iv) Rational sampling: The rational sample is chosen in such a manner that the variation within it is
considered to be due only to common causes. So, the samples are selected such that if special causes
are present they will occur between the samples. Therefore, the differences between the samples
will be maximized, and differences within samples will be minimized. Selection of sample
observations is done by either instant of time method (sample is consecutive units of production at
an instant of time) or by sampling interval method (sample is taken from all the units that are
manufactured since last sample is taken.
2.1 Rules for identifying an out of control process
Rule-1: A process is assumed to be out of control if a single point plots outside the control limits.
Rule-2: A process is assumed to be out of control if two out of three points plot outside 2σ warning limits
on the same side of centre line.
Rule-3: A process is assumed to be out of control if four out of five consecutive points fall beyond 1σ limit
on the same side of centre line.
Rule-4: A process is assumed to be out of control if nine or more consecutive points fall on one side of
centre line.
Rule-5: A process is assumed to be out of control if there is a run of six or more consecutive points
steadily increasing or decreasing.
UCL
Rule-6: A process is assumed to be out of control if there are fifteen points in a row within 1σ limits.
Rule-7: A process is assumed to be out of control if there fourteen points in a row alternatively up and
down.
Rule-8: A process is assumed to be out of control if there is unusual non random pattern in the data.
2.2 Control charts for variables
2.2.1 Control charts for variables
Variables are quality characteristics that are measurable on a numerical scale. Examples of variables include
length, diameter, surface roughness and tool wear. The objective of control charts for variable is to control the
mean value of the quality characteristic as well as its variability. The mean gives an indication of central tendency
of a process, and the variability provides an idea of the process dispersion. Therefore, we need information about
both these statistics to keep a process in control.
2.2.2 Advantages and disadvantages
1. Variables provide more information than attributes which deal with qualitative information such as
whether an item is nonconforming. Variables show the degree to which a quality characteristic is
nonconforming.
2. as costs are due to fixed cost of the measuring instruments and variable unit costs in the form of cost of
inspection.
3. These types of charts are best suited for control of process at shop floor environment.
2.2.3 Selection of characteristics for investigation
As it is not possible to maintain control charts separately for all the quality characteristics of a product due to
involvement of large number of characteristics in a product, the management restrict the study to selected
number of quality characteristics. In most industries, the quality characteristics are selected by using Pareto
analysis.
2.3 Design of Control charts for Mean ( ) and Range (R)
The following steps are used to develop the control charts:
Step-1: Using a preselected sampling scheme and sample size, record measurements of the selected quality
characteristic on the appropriate forms.
Step-2: For each sample, calculate the sample mean and range using the following formulas:
(4)
R = Xmax - Xmin (5)
Where Xi represents the ith observation, n is the sample size, Xmax is the largest observation and Xmin is the smallest
observation.
Step-3: Obtain and draw the center line and the trial control limits for each chart. For chart, the center line is
given by
(6)
Where ‘m’ represents the number of samples. For the R-chart, the center line is found from
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(7)
The control limits for and R charts are given by
(8)
= (9)
(10)
The constants A2, D3 and D4 can be obtained from tables.
Step-4: Plot the values of the range on the control chart for range, with the center line and the control limits
drawn. Determine whether the points are in statistical control. If not, investigate the special causes associated
with the out of control points and take appropriate remedial action to eliminate special causes.
An R-chart is analyzed before to determine out of control situations. An R-chart reflects process
variability, which should be brought to control first. Once the variability is in control, we can focus our attention
on
Step-5: Delete the out-of-control points for which remedial actions have been taken to remove special causes and
use the remaining samples to determine the revised center line and control limits for the
These limits are known as revised control limits. The cycle of obtaining information, determining the trial limits,
finding out of control points, identifying and correcting special causes, and determining revised control limits then
continues.
Step-6: Implement the control charts.
The charts should be implemented for future observations, using the revised center line and control limits. The
charts should be displayed in a conspicuous place where they will be visible to operators, supervisors and
managers. Statistical process control will be effective only if everyone is committed to it—from the operator to
the chief executive officer.
2.4 Errors in Making Inferences from Control Charts:
There are two types of errors – Type I and Type II --- that can occur when making inferences from control charts.
Type I errors result from inferring that a process is out of control when it is actually in control. The probability of
Type I error is denoted by ‘α’. Consider a process that is in control. Suppose a point plots outside the control
limits for this process. We tend to conclude that the process is out of control. However, we know very well that
we are covering only ‘3σ’ limits (99.73%) for control charts. We are not covering the rest 0.27% of the points.
Thus there is a chance of getting a point out of this 0.27% though a process is in control.
Similarly Type II errors result from inferring that a process is in control when it is really out of control. The
probability of Type I error is denoted by ‘β’. If no observations fall outside the control limits, we conclude that
the process is in control. Suppose however, that a process is actually out of control. Perhaps the process mean
has changed. In this case, the sample statistic may be within limits and the process is out control. This is type II