### STATISTICAL BASIS OF THE CONTROL CHART

```BPT2423 – STATISTICAL PROCESS CONTROL
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Control Chart Functions
Variation
Basic Principles
Choice of Control Limits
 Upper Control Limit (UCL)
 Lower Control Limit (LCL)
Sample Size and Sampling Frequency
Rational Subgroups
Analysis of Patterns
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Understand the concept of variation
Explain the statistical basis of the control chart,
including control limits and rational subgroup
concept
Identify some practical issues in the
implementation of statistical process control
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A control chart enhances the analysis of the process by
showing how the process is performing over time
Serve 2 basic functions:
1. Control charts are decision making tools
Provide an economic basis for making a decision as to
whether to investigate for potential problems; to
adjust the process or to leave the process alone
2. Control charts are problem-solving tools
Point out where improvement is needed and help to
provide a statistical basis on which to formulate
improvement actions
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Control charts have had a long history of use in U.S.
industries; Five reasons for it popularity:
1. Control charts are a proven technique for improving
productivity
2. Control charts are effective in defect prevention
3. Control
charts prevent unnecessary process
4. Control charts provide diagnostic information
5. Control charts provide information about process
capability
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Definition : Where no two items / services are exactly the
same
The goal of most processes is to produce products or provide
services that exhibit little or no variation
Several types of variation are tracked with statistical methods:
1. Within-piece : variation within a single item or surface
2. Piece-to-piece : variation that occurs among pieces
produced at approximately the same item
3. Time-to-time : variation in the product produced at
different times of the day
Variation in a process is studied by sampling the process;
understanding variation and its causes results in better
decisions
Stable and Unstable Variation
The Chart contains:
 Center line that represents the average value of the quality
characteristics corresponding to the in-control state
 Two other horizontal lines, called the upper control limit
(UCL) and the lower control limit (LCL)
 All the sample points on the control chart are connected with
straight-line segments, so that it’s easier to visualize how the
sequence of points has evolved over time
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If the process is in control, nearly all of the sample
points will fall between chosen control limits and no
action is necessary
However, a point that plots outside of the control limits
is interpreted as evidence that the process is out of
control, investigation and corrective action are required
to find and eliminate the causes
Even if all the points plot inside the control limits, if
they behave in a systematic or non random manner,
then this could be an indication that the process is out
of control
If the process is in control, all the plotted points should
have an essentially random pattern
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99.73% of the data under a normal curve falls within ± 3σ;
because of this, control limits are established at ± 3σ from
the centerline of the process
Let w be a sample statistic that measures some quality
characteristic of interest, and suppose that the mean of w
is µw and the standard deviation of w is σw
Formula:
Upper Control Limit (UCL)
Center Line
Lower Control Limit (LCL)
= µw + 3σw
= µw
= µw – 3σw
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In designing a control chart, sample size and the
frequency of sampling must be specify
In general, larger samples will make easier to detect small
shifts in the process
When choosing the sample size, user must keep in mind
the size of the ‘shift’ that are trying to detect
The most desirable situation form the point of view of
detecting shifts would be to take large samples very
frequently (not economic feasible)
General problem is one of allocating sampling effort –
either take small samples at short intervals or larger
samples at longer intervals
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Average run length (ARL) can be use to evaluate the
decisions regarding sample size & sampling frequency of
the control chart
ARL is the average number of points that must be plotted
before a point indicates an out-of-control condition by
using formula:
ARL = 1/p
 where p is the probability that any point exceeds the
control limits
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Example:
Chart with 3σ limits, p=0.0027 : the probability that a
single point falls outside the limits when the process is in
control
ARL = 1/p = 1/0.0027 = 370
which means an out-of-control signal will be generated
every 370 samples, on the average
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Subgroups and the samples composing them must be
homogeneous
A homogeneous subgroup will have been produced
under the same conditions, by the same machine, the
same operator, the same mold and so on
Subgroups used in investigating piece-to-piece variation
will not necessarily be constructed in the same manner
as subgroups formed to study time-to-time variation
Subgroup formation should reflect the type of variation
When constructing variables chart, keep the subgroup
sample size constant for each subgroup taken
Some guidelines to be followed
 The larger the subgroup size, the more sensitive the
chart becomes to small variations in the process
average. This provide a better picture of the process
since it allows the investigator to detect changes in the
process quickly
 While a larger subgroup size makes for a more sensitive
chart, it also increases inspection costs
 Destructive testing may make large subgroup sizes
unfeasible
 Subgroup sizes smaller than four do not create a
representative distribution of subgroup averages
A control chart exhibits a state of control when:
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2/3 of the points are near the center value
A few of the points are on or near the center value
The points appear to float back and forth across the centerline
The points are balanced (in roughly equal numbers) on both
sides of the centerline
There are no points beyond the control limits
There are no patterns or trends on the chart
A process that is not under control or is unstable, displays
patterns of variation - the process need to be investigate
and determine if an assignable cause can be found for the
variation
Patterns in Control Chart:
 Oscillating
 Change / Jump / Shift
 Runs
 Recurring Cycles
 Freaks / Drift
Example : Change / Jump / Shift
Description :
The process begins at one level and jump quickly to another level as the
process continues to operate. Causes for sudden shifts in level tend to reflect
some new and significant difference in the process.
Possible causes :
New operator, new batches of raw material or changes to the process settings
Example : Runs
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