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Zero Defect ConsultantsStatistical Process Control
Rev No:04, Date: 02.01.2010 1
WELCOME TO ALL
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Training Contents
Day 1:-
Introduction to SPC
Types of Process Controls
Introduction to Statistics
Understanding Mean, Mode, Median, Range, Standard Deviation
Concept of Variation Special cause & common causes
Stable & Unstable Process
Approach towards identification of Special Causes
Histogram An illustration
Normal Distribution
Process Capability
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Training Contents
Day 2:-
Introduction to Control Charts
Types of Control Charts
Understanding application, methodology, interpretations of varioustypes of charts (Variable and Attribute type covered)
Exercises on Control Charts
SPC implementation methodology
Role of Operator in implementing SPC.
Common mistakes done in implementing SPC
Conclusion
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Introduction to SPC
Process:-
Convert input to output using Man, Machine, Material, Method,
Measurement.
Process(Man, M/c, Material,
Method)
Input Output
Measure & give
Feedback
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Process Control:-
Variation in the output of the process is natural. Hence the process needs to
be controlled in order to ensure that output is meeting the customer
requirements.
Tools for Process Control:-
Detection :- A strategy that attempts to identify unacceptable output
after it has been produced and then separate it from the good output.
Prevention :- A future oriented strategy by analysis and action toward
correcting the process itself so that unacceptable parts will not be
produced.
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Need for Process Control
Detection :- Tolerates Waste
Prevention :- Avoids Waste
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Techniques for Process Control
Mistake Proofing :- In this technique 100% process control is achieved by
preventing all types of failures by using modern techniques to get defect
free product. Here causes are prevented from making the effect.
100% Inspection : In this technique 100% checking of all the parameters of
all products has been done to get defect free product. Here only defects are
detected.
Statistical Process Control : In this Statistical technique such as ControlChart, Histogram etc. are used to analyses the process to achieve and
maintain state of statistical control to get defect free product. Causes are
detected and prompting CA before defect occurs
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Statistics
Collection of Data, Analysis and Conclusion
It is a value calculated from or based upon sample data (e.g. a subgroup
average or range) used to make inferences about the process that produces
the output.
E.g. Analysis of rejection data and initiating actions to reduce the rejection
level.
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What is Statistical Process Control
The use of Statistical techniques such as control charts to analyze a process
or its outputs so as to take appropriate actions, to achieve and maintain a
state of statistical control and to improve the process capability.
SPC is
A tool to detect variation
It identifies problems, it does not solve problems
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Introduction to Statistics
Data:
Any facts or numbers or observations made.
Set of observations forms the data.
Types of Data:
Variable data
Attribute data
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Variable DataData generated by
Physically measuring the characteristic using an
instrument
Assigning an unique value to each item
Examples:
Hardness, Strength, Weight, Diameter, etc.
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Attribute DataData generated by
Classifying the items into different groups based on some criteria
No physical measurement is involved
All the items classified into a group will have same value I.e OK
or Not Ok.
Examples:
Sex, Shade Variation, Surface Defects, Go-No GO, etc.
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Statistical Properties of Data
Observations collected needs to be analyzed using various properties.
Statistical properties of data helps in arriving at one value representing
all observations.
Two types of properties
Measure of location
Measure of Dispersion
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Measure of Location
Mean (Average)
Median
Mode
Measure of Dispersion
Range
Standard Deviation
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Mean: Numerical value indicating the central value of data
Sum of all data / Total number of data
Suppose x1
, x2
, - - - xn
be the data, then
Mean = (x1+ x2 + - - -+ xn ) / n = xi /n
Example Hardness Data
Mean:
= (55 + 65 + 59 + 59 + 57 + 61 + 53 + 63 + 59 + 57 + 63 + 55 + 61 + 61 + 57 +
59 + 61 + 57 + 59 + 63) / 20
= 1184 / 20 = 59.2
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Median:
Middle Value
Value which divides data arranged in ascending or descending order
into two equal halves
Case 1: Total number of data is odd
Median: Middle Value
Case 2: Total number of data is even
Median: Average of two middle values
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Median: ExampleProductivity Data
0.97 0.98 0.98 0.99 0.99 0.99 0.99 1.00 1.00 1.00
1.00 1.00 1.01 1.01 1.01 1.01 1.02 1.02 1.02 1.03
Total Number of data: 20 (even)
The middle Values : 1.00 & 1.00 (10th value and 11th value)
Median: Average of 2 middle value
(1.00 + 1.00) / 2 = 1.00
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Mode:
Highest no. of times an observation has occurred (Highest frequency)
Mode: ExampleProductivity Data
0.97 0.98 0.98 0.99 0.99 0.99 0.99 1.00 1.00 1.001.00 1.00 1.01 1.01 1.01 1.01 1.02 1.02 1.02 1.03
0.97 - 1
0.98 - 2
0.99 - 41.00 - 5 - 1.00 is Mode as this occurred more no. of times
1.01 - 4
1.02 - 3
1.03 - 1
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Range: Definition
Range: Maximum value Minimum Value
Example:
5 4 7 3 2
15 9 8 5 2
Maximum Value = 15
Minimum Value = 2
Range = 15 2 = 13
Range: Issues
It depends only on extreme values
Hence affected by outliers
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Range: Issues
0
2
4
6
8
10
12
14
16
1 2 3 4 5 6 7 8 9 10
Range
f C l
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Standard Deviation: Definition
0
2
4
6
8
10
12
14
16
1 2 3 4 5 6 7 8 9 10
Square root of the average squared deviation from mean
Indicates On an average how much each value is away from the Mean
Z D f C l
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Below data indicates the Money held by ten students in a class room.
Standard Deviation: Example:
5 4 7 3 2
15 9 8 5 2
Step 1:
Calculate Mean
Mean = 6
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Standard Deviation: Example:
5 4 7 3 2
15 9 8 5 2
Step 2:
Take deviations from Mean
-1 -2 1 -3 -4
9 3 2 -1 -4
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Standard Deviation: Example:
Step 2:
Take deviations from Mean
0
2
4
6
8
10
12
14
16
1 2 3 4 5 6 7 8 9 10
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Standard Deviation: Example:
Step 3:
Since some values are positive & rest are negative, while
taking sum they will cancel out.So square the values & Sum
1 4 1 9 16
81 9 4 1 16
Sum = 142
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Standard Deviation: Example:
Step 4:
Standard Deviation = (Sum of Squares / (n -1))
= (142 / (10 -1))
= 15.77 = 3.972
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Variation No two things are exactly alike
It is impossible to produce or process two items exactly alike
Variation is natural.
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Example:
Time to reach office : 9:30 am
It is not possible to reach office exactly at 9:30 everyday
Normally there will be a small variation around 9:30 as follows:
9:31 9:33 9:28 9:29 9:25
9:34 9:26 9:27 9:34 9:31
This small variation is difficult to explain.
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Generally
Looking at the past values, it is possible to give some rangearound 9:30 am (e.g.: 9:30 5 Minutes) to reach office
Normally it is possible to reach office within this range
SupposeA particular day , there is vehicle break down
that day it may not be possible to reach office at 9:30 5 Minutes
Say, you reach office at 9:50 am
In other words
If you reach office too late (beyond normal range of 9:30 5 minutes),
there will be some special reason for that or
it is easy to find out the reason for such variation
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Assignable Cause of Variation (Special Causes)
Variations of large magnitude
Easy to identify the causes of variation
Easy to eliminate the cause of variation
Common Cause of Variation
Variations of small magnitude
Difficult to identify the causes of variation
Difficult to eliminate the cause of variation
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Local Actions
Are usually required to eliminate special causes of Variation
Can usually taken by the people close to the process
Can correct typically about 15% of process problem
Actions on the System
Are usually required to reduce the variation due to common causes
Almost always require management actions for correction
Are needed to correct typically about 85% of process problems
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Stable Process
No Assignable Causes are present
Process is operating under common causes only
Stable process will be predictable & Statistically under control.
If a process is stable & the data follows normal distribution
Then the variation will be Mean 3 x Standard deviation
Unstable Process
Assignable Causes are present
Process is operating under assignable & common causes.
Unstable process is unpredictable & not under Statistical control.
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Methodology to Identify Assignable Causes
The success of any SPC program is not in our ability to collect data, draw
charts etc., but in effectively identifying and eliminating assignable causes.
Assignable causes are those causes that do not allow one to predict the
behavior of processes.
There is no meaning in calculating Process Capability without having a
predictable process.
Many companies have initiated SPC charts. But the charts do not benefit
them. One of the main reason for this is that they have not stopped theprocess when an assignable cause is indicated and eliminated the cause
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Methodology to Identify Assignable Causes
Before starting the SPC data collection, let us do the following steps:
1. Identify the characteristic for which SPC is to be done.
2. Have a brainstorming to list all the causes that may influence the
variation in this characteristic
3. Prepare a Cause & Effect Diagram
4. Prepare a Master Cause Analysis Table (Annexure 1)
5. Prepare a Why-Why Analysis Table (Annexure 2)
6. Identify factors that may affect Average and those that may affect Range
After completion of the above, plan for data collection & implementation ofSPC.
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Master Cause Analysis (Annexure 1)
Sl.No.
Cause Is therea spec?
If so,what is
thespec?
Basis forthe
spec.
Is itchecked andhow?
What isthe
actual?
Diff.in
Spec.Vs
Actual
Actionplan
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Why Why Analysis (Annexure 2)
Sl.No.
Cause WHY WHY WHY WHY WHY
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HISTOGRAM Histogram is a graphical representation of data and shows the
frequency of data.
Histogram provides the easiest way to understand the distribution of
data. It gives the Birds eye view of the variation in Data set.
Portrays the information on location, spread and shape that enables
the user to interpret the Process behavior.
It indicates whether the process is operating under Normal / stable
conditions.
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DEFINITIONSClass:- Is a category with lower and upper boundary value.
Class Width:- Width of the class.
Frequency:- No. of observations falling in the class.
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STEPS FOR CONSTRUCTING HISTOGRAM
Collect Min 50 no. of readings (N). 50 readings should be continuous
data.
Determine Max value and Min value & Calculate Range.
Range = Max Min.
Record the measurement unit (MU) used. This is usually controlled by
the measuring instrument least count.
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STEPS FOR CONSTRUCTING HISTOGRAM
Contd.. Determine No. of classes (k), as below.
No. of Class (k) = N
Determine Class Width (CW), as below
Class Width (CW) = Range / k
Construct the Frequency Distribution Table, as shown in the next
slide.
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Where L1 = Minimum value (1/2*Measurement Unit)
U1 = L1 + Class Width
L2 = U1,
U2 = L2 + Class Width & so on.
Class Tally Frequency
L1 U1
L2 U2
Upto Max value
Total N
Frequency Distribution Table
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STEPS FOR CONSTRUCTING HISTOGRAM
Contd.. Determine the axis for the graph. Place Class on X axis and
Frequency on Y axis.
Mark off the classes, and draw rectangles with heights corresponding
to the measurement frequencies in that class. Title the histogram. Give an overall title and identify each axis.
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HISTOGRAM
GRAPHICAL REPRESENTATION.
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HISTOGRAM
INTERPRETATIONS:NORMAL
Depicted by a bell-shaped curve. Most frequent measurement
appears as center of distribution & less frequent measurements taper
gradually at both ends of distribution.
Indicates that a process is running normally (only common causes are
present).
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HISTOGRAM
INTERPRETATIONS:BIMODAL
Distribution appears to have two peaks. May indicate that data from
more than one process are mixed together
Materials may come from 2 separate vendors
Samples may have come from 2 separate machines.
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HISTOGRAM
INTERPRETATIONS:SKEWED
Appears as an uneven curve; values seem to taper to one side.
Can be skewed left side or right side.
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Normal Distribution
Consider the following data on the case depth(mm) of 9 jobs:2.3 2.7 2.4 2.6 2.5
2.5 2.4 2.5 2.6
Plot of the Data:
0
1
2
3
4
2.2 2.3 2.4 2.5 2.6 2.7 2.8
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Plot of the Data:
0
1
2
3
2.3 2.4 2.5 2.6 2.7
Bell Shaped
Symmetric
Total Area under the curve is 1
Then : Normal Curve & Data follows Normal Distribution
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e o e ect Co sulta tsStatistical Process Control
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Standard Normal Distribution:
If
Data follows Normal Distribution then
(Data - Mean) / SD will follow Standard Normal Distribution
For Standard Normal Distribution:
Mean = 0
SD = 1
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Standard Normal Distribution: Properties
0
1
2
3
4
-3 -2 -1 0 1 2 3
68.26%
95.46%
99.73%
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Normal Distribution: Properties
Between
Mean 1 SD : 68.26 % of Values will lie
Mean 2 SD : 95.46 % of Values will lie
Mean 3 SD : 99.73 % of Values will lie
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Process Capability
A methodology to check whether a process is capable of meeting customerrequirements
Determined by the variation that comes from common causes
Expressed as Process Capability Indices
This needs to be demonstrated when process is being under statistical control
Process Capability Indices
1. Process Performance Index ( Pp & Ppk )
2. Process Capability Index ( Cp & Cpk )
Process Capability Study
1. Process performance Study (Pp & Ppk)
2. Process Capability Study (Cp & Cpk)
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Process Capability Index (Cp)
A methodology to check whether the process have the potential to meet the
customer requirements
Generally
Customer requirements are given as specification on product characteristics
Example
Specification on Heat Treatment Process:
Hardness should be within 55 5 HRC
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Customer Requirement:
Variation allowed by the customer OrVariation acceptable to customer
Example:
Specification: 55 5 HRC
Meaning:
As long as Hardness of the heat treated jobs are between 50 HRC to 60
HRC, Customer is satisfied
Customer requirements are also expressed as
Lower Specification Limit (LSL) = 50 HRC
Upper Specification Limit (USL) = 60 HRC
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Process Capability Index (Cp):
A process have the potential to meet customer requirement, ifTotal variation in process < Allowed variation
Example:
Specification: 55 5 HRC
Allowed variation = 50 HRC to 60 HRC
Total Variation = 52 HRC to 58 HRC
Total Variation < Allowed variation
Hence
Process have the potential to satisfy customer
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Example:Specification: 55 5 HRC
Allowed variation = 50 HRC to 60 HRC
Total Variation = 48 HRC to 62 HRC
Total Variation > Allowed Variation
Then
Process doesnt have the potential to satisfy customer
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Process capability Index Cp:
If the data is normally distributed, then
Total variation : Mean 3 SD
Example:
Mean = 55 HRC & SD = 1HRC
Total Variation = 55 3 x 1 to 55 + 3 x 1
= 52 HRC to 58 HRC
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Process Capability Index Cp: Definition
Ratio of allowed variation to Total variation
Cp = Allowed variation / total variation
= (USL LSL) / ((Mean + 3 SD) (Mean - 3 SD))
= (USL LSL) / 6 SD
A Process has the potential to meet customer requirements if
total variation < allowed variation
6 SD < (USL LSL)
Cp > 1
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Process Capability Index Cp: Example
20 data on acid content (mm) is given in the table below. If the specification onacid content is 0.70 0.2 mm. Check whether the process has the potential to
meet the customer requirement ?
0.85 0.75 0.80 0.65 0.75 0.60 0.80 0.70 0.75 0.60
0.80 0.75 0.70 0.70 0.75 0.75 0.85 0.60 0.50 0.65
Specification = 0.70 0.2 mm
USL = 0.90 mm
LSL = 0.50 mm
Mean = 0.715
SD = 0.092
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Process Capability Index Cp: Example
Cp = (USL LSL) / 6 SD
= (0.90 0.50) / (6 x 0.092)
= 0.4 / 0.552 = 0.72
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Process Capability Index (Cp): Issues
Cp checks only whether the process has the potential to meet the
requirements
Cp never checks whether the Process is actually meeting requirements
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Process Capability Index (Cpk): Definition
Cpk = Min [Ppl, Ppu]
Cpl = (Mean LSL) / 3 SD
Cpu = (USL - Mean) / 3 SD
Cpk checks whether the process is centered at the middle of the specification
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Process Capability Index Cpk: Graphical Representation
0
0.2
0.4
0.6
0.8
1
1.2
1 2 3 4 5 6 7
USLLSL
Mean + 3 SD3 SD
ba
c d
Cpl = a / c = (Mean LSL ) / 3 SD
Cpu = b / d = (USL - Mean ) / 3 SD
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0
0.2
0.4
0.6
0.8
1
1.2
5 6 7 8 9 10 11
126
Mean + 3 SD- 3 SD
42
3 3
Example:
USL : 12 LSL: 6
Mean : 8 SD : 1
Cpu = 4 / 3 = 1.33
Cpl = 2 / 3 = 0.66
Cpk = Min [1.33,0.66] = 0.66
Cpk < 1, performance is not OK
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0
0.2
0.4
0.6
0.8
1
1.2
6 7 8 9 10 11 12
126
Mean + 3 SD- 3 SD
33
3 3
Example:
USL : 12 LSL: 6
Mean : 9 SD : 1
Cpu = 3 / 3 = 1
Cpl = 3 / 3 = 1
Cpk = Min [1 , 1] = 1
Cpk = 1
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0
0.2
0.4
0.6
0.8
1
1.2
6 7 8 9 10 11 12
126
Mean + 3 SD- 3 SD
33
3 3
Conclusion:Cpu = 3 / 3 = 1
Cpl = 3 / 3 = 1
Cpk = Min [1 , 1] = 1
Cp = (USL LSL) / 6 SD = 6 /6 = 1When Mean is at middle of
Specification [(USL + LSL) / 2] then
Cpu = Cpl = Cpk = Cp
Otherwise
Cpk < Cp
When Cpk < Cp
Performance is not optimum
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Statistical Process Control
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Process Performance Study Is a Short term Study performed during New Product Development.
Indices are represented as Pp & Ppk.
Standard deviation calculated using first principle formulae - Sigma(n-1).
Acceptance value for Pp / Ppk is Min 1.66
Process Capability Study
Is a Long term Study performed to monitor the ongoing Production.
Indices are represented as Cp & Cpk.
Standard deviation calculated using sigma = Rbar/d2
Acceptance value for Cp / Cpk is Min 1.33
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Statistical Process Control
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A tool for process control and improvement
A statistical tool to know that process is stable or in control
A statistical tool to detect the presence of Assignable Causes in the process.
PROPERLY USED CONTROL CHARTS CAN : Be used by operators for ongoing control of a process
Help the process perform consistently and predictably
Allow the process to achieve Higher Quality, Lower unit cost, Higher
effective capability
Provide common language for discussing the performance of the process
Distinguish special from common causes of variation, as a guide to local action
or action on the system
CONTROL CHARTS
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Statistical Process Control
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For Normally distributed Data
If the process is stable then
Variation will be between Mean 3 x SD
Theory of Control Charts
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Statistical Process Control
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A Graphical Tool with three horizontal lines
1. Lower Control Limit (LCL)
2. Center Line (CL)
3. Upper Control Limit (UCL)
Control Charts
UCL
CL
LCL
Control Chart
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Statistical Process Control
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UCL & LCL are such that
if the value lies between UCL & LCL then the process is stable or in
control
Control Charts
UCL
CL
LCL
Control Chart
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Statistical Process Control
Rev No:04, Date: 02.01.2010 72
For Normal Data
UCL =Mean + 3 x SD
CL = Mean
LCL = Mean 3 x SD
Control Charts
UCL
CL
LCL
Control Chart
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Statistical Process Control
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1. Calculate the Control Limits from past data
2. Plot the values in the chart
3. If the values are within the limits, the process is stable. Otherwise not.
Control Charts: Working
Control Chart
0
2
4
6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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Statistical Process Control
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Continuous Data
Xbar & R Chart ( or Median R Chart)
Individual X & Moving range Chart
Xbar & S Chart ( or Median S Chart)
Types of Control Charts
Discrete Data
Control Chart for Defectives
p chart
np chart
Control Chart for Defects
c chart
u chart
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Statistical Process Control
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Determine
Characteristics to becharted
Are the
data
variable
Is interest inmonitoring %
bad parts
Is it
homogeneou
s in nature
Selection Procedure for Control Charts
Use X-MR Chart
Is interest in
monitoring
nonconformiti
es
Is the sample
size constantIs the sample
size constant
Can sub group
avg. easily
calculated
Is the
subgroup size
9 or more
Is s,
calculated
easily
Use Xbar-S Chart
Use Median Chart
Use Xbar-R Chart
Use Xbar-R Chart
Use U Chart
Use C Chart
Use p ChartUse np Chart
Y
N
N
N
N
N
N
N
N
Y
Y
Y
Y
Y
Y
Y
Y
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Statistical Process Control
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Xbar R Chart: Methodology
Conduct initial study :-
Decide on the Total Number of Samples N . (N >19)
Decide on Sub Group Size n. (n > 3)
Decide on Frequency of Sampling. ( eg: Once in a hour, Once in 2 hours,
Once in every 50 items, etc.) Collect Data & Calculate Control Limits
Plot Control Chart
Calculate Process Capability Indices (Pp/Ppk).
If Capable, Set Control Limits for Ongoing study.
Monitor Process through plotting control chart.
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Statistical Process Control
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Xbar R Chart: Example
Process: Turning Characteristic: Inner diameter (4.90 5.10)
Sample Size N: 9 Sub Group Size n: 4
Frequency of Sampling: Once in a Hour
Step 1: Collect Data
Sample No. Hour x1 x2 x3 x4
1 8:00 5.00 5.01 4.98 5.00
2 9:00 5.01 4.98 5.00 5.00
3 10:00 5.02 5.01 5.00 5.00
4 11:00 5.00 5.00 5.00 5.00
5 12:00 4.98 4.98 5.01 4.99
6 13:00 5.02 4.99 5.00 4.98
7 14:00 4.99 4.99 4.98 4.98
8 15:00 5.00 5.01 5.02 5.00
9 16:00 4.98 5.00 5.01 4.98
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Statistical Process Control
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Xbar R Chart: Example
Step 2: Calculate Sub Group Mean & Range
Sample No. Hour x1 x2 x3 x4 Mean Range
1 8:00 5.00 5.01 4.98 5.00 4.998 0.03
2 9:00 5.01 4.98 5.00 5.00 4.998 0.03
3 10:00 5.02 5.01 5.00 5.00 5.008 0.024 11:00 5.00 5.00 5.00 5.00 5.00 0.00
5 12:00 4.98 4.98 5.01 4.99 4.990 0.03
6 13:00 5.02 4.99 5.00 4.98 4.998 0.04
7 14:00 4.99 4.99 4.98 4.98 4.985 0.01
8 15:00 5.00 5.01 5.02 5.00 5.008 0.02
9 16:00 4.98 5.00 5.01 4.98 4.993 0.03
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Statistical Process Control
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Xbar R Chart: Example
Step 3: Calculate Control Limits for R Chart
Center Line
CL = Mean = Rbar = Sum of all Range Values / Total Number
of Values
= 0.21 / 9 = 0.0233
Upper Control Limit
UCL = Mean + 3 SD = D4 Rbar, For n = 4, D4 = 2.282
= 2.282 x 0.0233 = 0.053
Lower Control Limit
LCL = Mean - 3 SD = D3 Rbar, For n = 4, D3 = 0
= 0 x 0.0233 = 0.0
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Statistical Process Control
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Xbar R Chart: Example
Step 4: Plot R values in R chart as shown below:
R Chart
0
0.02
0.04
0.06
1 2 3 4 5 6 7 8 9
Step 5: If any value is beyond Control Limits, Do Homogenization
Homogenization:
Remove the out of control value
Recalculate the limits
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Statistical Process Control
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Xbar R Chart: Example
Step 6: Calculate Control Limits for Xbar Chart
Center Line
CL = Mean = Xdoublebar = Sum of all Means / Total Number
of Values
= 44.975 / 9 = 4.997
Upper Control Limit
UCL = Mean + 3 SD = xdoublebar + A2Rbar, For n = 4, A2 = 0.729
= 4.997 + 0.729 x 0.0233 = 5.014
Lower Control Limit
LCL = Mean - 3 SD = xdoublebar - A2Rbar, For n = 4, A2 = 0.729
= 4.997 - 0.729 x 0.0233 = 4.98
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Statistical Process Control
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Xbar R Chart: Example
Step 7: Plot mean values in xbar chart as shown below:
xbar Chart
4.96
4.98
5
5.02
1 2 3 4 5 6 7 8 9
Step 8: If any value is beyond Control Limits, Do Homogenization
Step 9: If all values are within limit, Calculate Standard deviation
.
= Rbar/d2 , Where d2 is constant
= 0.023/2.059
= 0.0111
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Statistical Process Control
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Xbar R Chart: Example
Step 10: Calculate Process Capability Indices
Cp = Tol / 6 = 0.2 / 6 * 0.0111 = 3.03
Cpk = Min { (Xbar LSL)/3 , (USL-Xbar)/3 }
Cpk = Min { 2.91 , 3.09 } = 2.91.
Process is Capable.
Step 11: If Capable, Set the Control limits for ongoing monitoring.
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Statistical Process Control
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Control Chart Constants
n d2 A2 D3 D4 E2
2 1.128 1.880 0 3.268 2.66
3 1.693 1.023 0 2.574 1.77
4 2.059 0.729 0 2.282 1.46
5 2.326 0.577 0 2.114 1.29
6 2.534 0.483 0 2.004 1.18
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Statistical Process Control
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Xbar R Chart: Exercise
Sample
Number
x1 x2 x3 Sample
Number
x1 x2 x3
1 6.0 5.8 6.1 11 6.2 6.9 5.0
2 5.2 6.4 6.9 12 6.7 7.1 6.2
3 5.5 5.8 5.2 13 6.1 6.9 7.4
4 5.0 5.7 6.5 14 6.2 5.2 6.8
5 6.7 6.5 5.5 15 4.9 6.6 6.6
6 5.8 5.2 5.0 16 7.0 6.4 6.1
7 5.6 5.1 5.2 17 5.4 6.5 6.7
8 6.0 5.8 6.0 18 6.6 7.0 6.8
9 5.5 4.9 5.7 19 4.7 6.2 7.1
10 4.3 6.4 6.3 20 6.7 5.4 6.7
The following are the data on Time (in Minutes) to Process Transactions in aBPO company. Construct an Xbar R chart to monitor the process
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Statistical Process Control
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Individual X & Moving Range Chart: Example
Sample Number Data
1 2.5
2 2.3
3 2.8
4 2.6
5 2.4
6 2.9
7 2.1
8 2.5
For Short Run Process / Bulk Material Processing. Can be used when thetesting method is a destructive type.
Not possible to collect data in Sub Groups
Process: Heat Treatment Characteristic: Case Depth
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Statistical Process Control
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Individual X & Moving Range Chart: Example
Sample Number Date MR
1 2.5
2 2.3 0.23 2.8 0.5
4 2.6 0.2
5 2.4 0.2
6 2.9 0.5
7 2.1 0.8
8 2.5 0.4
Step 2: Calculate Moving Range
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Statistical Process Control
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Individual X & Moving Range Chart: Example
Step 3: Calculate Control Limits for M R Chart
Center Line
CL = Mean = MRbar = Sum of all MR Values / Total Number
of Values
= 2.8 / 7 = 0.4
Upper Control Limit
UCL = Mean + 3 SD = D4 MRbar, For n = 2, D4 = 3.268
= 3.268 x 0.4 = 1.3072
Lower Control Limit
LCL = Mean - 3 SD = D3 MRbar, For n = 2, D3 = 0
= 0 x 0.4 = 0.0
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Statistical Process Control
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Step 4: Plot MR values in MR chart as shown below:
MR Chart
0
0.5
1
1.5
1 2 3 4 5 6 7
Step 5: If any value is beyond Control Limits, Do Homogenization
Individual X & Moving Range Chart: Example
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Statistical Process Control
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Step 6: Calculate Control Limits for Individual X Chart
Center Line
CL = Mean = Xbar = Sum of all Data / Total Number
of Values
= 20.1 / 8 = 2.512
Upper Control Limit
UCL = Mean + 3 SD = xbar + E2MRbar, For n = 2, E2 = 2.66
= 2.512 + 2.66 x 0.4 = 3.58
Lower Control Limit
LCL = Mean - 3 SD = xbar - E2MRbar, For n = 2, E2 = 2.66
= = 2.512 - 2.66 x 0.4 = 1.45
Individual X & Moving Range Chart: Example
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Statistical Process Control
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Step 7: Plot individual values in Individual X chart as shown below:
Individual X Chart
0
1
2
3
4
1 2 3 4 5 6 7 8
Step 8: If any value is beyond Control Limits, Do Homogenization
Individual X & Moving Range Chart: Example
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Statistical Process Control
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Individual X & Moving Range Chart: Exercise
The data given below are surface Finish values of 30 jobs after chromiumplating. Construct an Individual X & Moving Range chart to monitor the
process?
0.078 0.079 0.077 0.076 0.074 0.072 0.069 0.075 0.078 0.077
0.075 0.078 0.08 0.081 0.08 0.079 0.082 0.073 0.078 0.074
0.072 0.075 0.068 0.073 0.074 0.081 0.076 0.08 0.074 0.07
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Statistical Process Control
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Control Charts for Defectives: np Chart
Used when sample size is constant
Used to measure no of nonconforming items in an inspection.
Sample Number Number of Defectives
1 47
2 42
3 48
4 58
5 32
6 38
7 53
8 68
9 45
10 37
Example: Inspection results of video of the month shipment to customers for 10consecutive days are given in table. The number of inspection each day
is constant and is equal to 1000. Construct np chart to control the
defectives?
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Statistical Process Control
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np Chart : Calculation of Control Limits
CL = MeanUCL = Mean + 3 SD
LCL = Mean 3 SD
pbar = Sum of Defectives / total Number Inspected
= 468 /(1000*10) = 0.0468
Mean = npbar = 1000 x 0.0468 = 46.8
SD = npbar(1-pbar) = (1000 x (0.0468 x (1-0.0468))) = 6.67
CL = npbar = 1000 x 0.0468 = 46.8
UCL = Mean + 3 SD = 46.8 + 3 x 6.68 = 66.84
LCL = Mean - 3 SD = 46.8 - 3 x 6.68 = 26.76
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Rev No:04, Date: 02.01.2010 95
Plot the number of defectives in np chart as shown below:
np Chart
0
20
40
60
80
1 2 3 4 5 6 7 8 9 10
If any value is beyond Control Limits, Do Homogenization
np chart: Example
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Rev No:04, Date: 02.01.2010 96
np Chart: Exercise
Sample
Number
Number of
Defectives
Sample
Number
Number of Defectives
1 3 11 6
2 6 12 9
3 4 13 5
4 6 14 6
5 20 15 7
6 2 16 4
7 6 17 5
8 7 18 7
9 3 19 5
10 0 20 0
The following are the data on defectives in payment of dental insurance
claims. Control the dental insurance payment process with np chart.Sample Size is 300
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Rev No:04, Date: 02.01.2010 97
Control Charts for Defectives: p Chart
Used when sample size is not constant
Sample Number Number Inspected Number of Defectives
1 500 5
2 550 6
3 700 8
4 625 9
5 700 7
6 550 8
7 450 10
8 600 6
9 475 9
10 650 6
Example: The daily inspection results for electric carving knives are givenbelow. Construct a control chart to monitor the process ?
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Rev No:04, Date: 02.01.2010 98
p Chart : Calculation of Control Limits
CL = Mean
UCL = Mean + 3 SD
LCL = Mean 3 SD
pbar = Sum of Defectives / Total Number Inspected
= 74 / 5800 = 0.0128
Mean = pbar = 0.0128
SD = pbar(1-pbar) / ni
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Rev No:04, Date: 02.01.2010 99
p Chart : Calculation of Control Limits
SampleNumber
NumberInspected
Number ofDefectives
p LCL UCL
1 500 5 0.010 0 0.028
2 550 6 0.011 0 0.027
3 700 8 0.011 0 0.026
4 625 9 0.014 0 0.0265 700 7 0.010 0 0.026
6 550 8 0.015 0 0.027
7 450 10 0.022 0 0.029
8 600 6 0.010 0 0.027
9 475 9 0.019 0 0.028
10 650 6 0.009 0 0.026
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Rev No:04, Date: 02.01.2010 100
Plot the proportion of defectives in p chart as shown below:
p Chart
0
0.01
0.02
0.03
0.04
1 2 3 4 5 6 7 8 9 10
If any value is beyond Control Limits, Do Homogenization
p chart: Example
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p Chart: Exercise
Sample
Number
Number
Inspected
Number of
Defectives
Sample
Number
Number
Inspected
Number of
Defectives
1 171 31 11 181 38
2 167 6 12 115 33
3 170 8 13 165 26
4 135 13 14 189 15
5 137 26 15 165 16
6 170 30 16 170 35
7 45 3 17 175 12
8 155 11 18 167 6
9 195 30 19 141 50
10 180 36 20 159 26
Daily inspection results for the model 305 electric range assembly line are
given in the table. Construct a control chart to monitor the process?
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Rev No:04, Date: 02.01.2010 102
Control Charts for Defects: c Chart
Used when sample size is constant
Day Number of nonconformities
1 8
2 19
3 14
4 18
5 11
6 16
7 8
8 15
9 21
10 8
Example: A leading bank has compiled the data in the table showing thecount of nonconformities for 100 accounting transactions per
day. Construct a control chart to monitor the process?
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Rev No:04, Date: 02.01.2010 103
c Chart : Calculation of Control Limits
CL = Mean
UCL = Mean + 3 SD
LCL = Mean 3 SD
cbar = Sum of nonconformities / Total Number Inspected
= 138 /100 = 13.8
Mean = cbar = 13.8
SD = cbar = 13.8 = 3.71
CL = cbar = 13.8
UCL = Mean + 3 SD = 13.8 + 3 x 3.71 = 24.93
LCL = Mean - 3 SD = 13.8 - 3 x 3.71 = 2.67
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Rev No:04, Date: 02.01.2010 104
Plot the number of nonconformities in c chart as shown below:
c Chart
0
10
20
30
1 2 3 4 5 6 7 8 9 10
If any value is beyond Control Limits, Do Homogenization
c chart: Example
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Rev No:04, Date: 02.01.2010 105
c Chart: Exercise
Day Number of
Nonconformities
Day Number of Nonconformities
1 22 11 15
2 29 12 10
3 25 13 33
4 17 14 23
5 20 15 27
6 16 16 15
7 34 17 17
8 11 18 17
9 31 19 19
10 29 20 22
100 product labels are inspected every day for surface nonconformities.
The data for the past 20 days is given below. Construct a suitable controlchart to monitor the nonconformities
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Rev No:04, Date: 02.01.2010 106
Control Charts for Defects: u Chart
Used when sample size is not constant
Lot Number Number Inspected Number of Defects
1 10 45
2 10 51
3 10 36
4 9 48
5 10 42
6 10 5
7 10 33
8 8 27
9 8 31
10 8 22
Example: The inspection results for the surface finish of rolls of whitepaper for 10 lots is given below. Construct a control chart to
monitor the process ?
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Rev No:04, Date: 02.01.2010 107
u Chart : Calculation of Control Limits
CL = Mean
UCL = Mean + 3 SD
LCL = Mean 3 SD
ubar = Sum of Defects / Total Number Inspected
= 340 / 93 = 3.66
Mean =ubar = 3.66
SD = (ubar / ni)
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Rev No:04, Date: 02.01.2010 108
u Chart : Calculation of Control Limits
LotNumber
NumberInspected
Number ofDefects
u LCL UCL
1 10 45 4.5 2.80 5.47
2 10 51 5.1 2.86 5.47
3 10 36 3.6 2.70 5.47
4 9 48 5.3 2.83 5.575 10 42 4.2 2.77 5.47
6 10 5 0.5 1.09 5.47
7 10 33 3.3 2.66 5.47
8 8 27 3.4 2.56 5.69
9 8 31 3.9 2.63 5.69
10 8 22 2.8 2.44 5.69
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Rev No:04, Date: 02.01.2010 109
Plot the defects / unit (u) of in u chart as shown below:
u Chart
0
2
4
6
1 2 3 4 5 6 7 8 9 10
If any value is beyond Control Limits, Do Homogenization
u chart: Example
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u Chart: Exercise
SampleNumber
Number ofbottles
inspected
Number ofDefects
SampleNumber
Number ofbottles
Inspected
Number of Defects
1 40 45 11 52 55
2 40 40 12 52 74
3 40 33 13 52 43
4 40 43 14 52 61
5 40 62 15 40 43
6 52 79 16 40 32
7 52 60 17 40 45
8 52 50 18 40 33
9 52 73 19 40 50
10 52 54 20 52 28
Construct a suitable control chart for the data in the table for empty bottle
inspections of a soft drink manufacturer?
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Rev No:04, Date: 02.01.2010 111
Select the
characteristics &
Process for SPC
Plan for Data
Collection
Is Process
Capable
Is Process
Stable
Establish Control
Limits
Improve the Process
Find
Assignable
Cause & Fix it
N
N
Y
Y
Perform MSA Study
Collect Data & Plot
Control Chart
Prepare Reaction PlanOngoing Process
Control
SPC Implementation:-
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Rev No:04, Date: 02.01.2010 112
Collect Data
Is Process in
Control (I.e no
special cause)
Refer Reaction Plan
N
Y
Plot on the Control
Chart
Take Corrective Action
Take Disposition
action, if required
Operator Role in SPC:-
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Rev No:04, Date: 02.01.2010 113
REACTION PLAN
Process:-
Parameter:-
Doc No.:-
Rev. No./Date:-
Chart Condition Possible
Causes
Corrective
Action
Disposition Action
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Rev No:04, Date: 02.01.2010 114
Common Mistakes in SPC Implementation
SPC Chart plotting is just a show put up to customers and outsiders. No realanalysis is done.
Companies do Fill SPC Charts, even though 100% inspection is being done.
Hi we are implementing SPC charts Hence we are going to reduce
Rejection. Just by plotting charts, rejection level doesnt reduce.
Charts are plotted at the end of the shift & analyzed.
Inspection frequency & the SPC chart plotting frequency is different.
Management involvement is very less. It is just to meet the customer /
certification requirement.
Charts & Capability indices show positive sign. But rejections/reworks are
increasing????
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Rev No:04, Date: 02.01.2010 115
Common Mistakes in SPC Implementation
Reaction plans are not prepared. Even if available, not referred / used.
Corrective actions are not initiated even if the process is not stable / not
capable.
Process Capability results are not considered as feedback for new product
development.
Poor awareness at Operator level on usage of SPC charts & Its interpretation
So would you like to avoid above mistakes????
&
Get Maximum benefit of SPC???
Zero Defect ConsultantsStatistical Process Control
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Zero Defect ConsultantsStatistical Process Control
7/29/2019 SPC Course Material
117/117
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