Statistical Process Control

© Production & Industrial Engineering
Mechanical Engineering Department
Statistical Process Control
Operations Management
by
R. Dan Reid & Nada R. Sanders
2nd Edition © Wiley 2005
PowerPoint Presentation by Naveen Banshiwal

Sources of Variation in Production
and Service Processes
 Common causes of variation
 Random causes that we cannot identify
 Unavoidable
 Cause slight differences in process variables like diameter,
weight, service time, temperature, etc.
 Assignable causes of variation
 Causes can be identified and eliminated
 Typical causes are poor employee training, worn tool,
machine needing repair, etc.

Measuring Variation: The
Standard Deviation
Small vs. Large
Variation

Process Capability
 A measure of the ability of a process to meet
preset design specifications:
 Determines whether the process can do what we
are asking it to do
 Design specifications (tolerances):
 Determined by design engineers to define the
acceptable range of individual product
characteristics (e.g.: physical dimensions, elapsed
time, etc.)
 Based upon customer expectations & how the
product works (not statistics!)

Relationship between Process
Variability and Specification Width

Three Sigma Capability
 Mean output +/- 3 standard deviations
falls within the design specification
 It means that 0.26% of output falls
outside the design specification and is
unacceptable.
 The result: a 3-sigma capable process
produces 2600 defects for every million
units produced

Six Sigma Capability
 Six sigma capability assumes the process is
capable of producing output where the mean
+/- 6 standard deviations fall within the
design specifications
 The result: only 3.4 defects for every million
produced
 Six sigma capability means smaller variation
and therefore higher quality

Process Control Charts
Control Charts show sample data plotted on a graph with Center
Line (CL), Upper Control Limit (UCL), and Lower Control Limit
(LCL).

Types of Control Charts
 Control chart for variables are used to
monitor characteristics that can be measured,
e.g. length, weight, diameter, time, etc.
 Control charts for attributes are used to
monitor characteristics that have discrete
values and can be counted, e.g. % defective,
number of flaws in a shirt, number of broken
eggs in a box, etc.

Control Charts for Variables
 Mean (x-bar) charts
 Tracks the central tendency (the average
value observed) over time
 Range (R) charts:
 Tracks the spread of the distribution over
time (estimates the observed variation)

x-bar and R charts
monitor different parameters!

Constructing a X-bar Chart:
A quality control inspector at the Cocoa Fizz soft drink company has
taken three samples with four observations each of the volume
of bottles filled. If the standard deviation of the bottling operation
is .2 ounces, use the data below to develop control charts with
limits of 3 standard deviations for the 16 oz. bottling operation.
Time 1 Time 2 Time 3
Observation 1 15.8 16.1 16.0
Observation 2 16.0 16.0 15.9
Observation 3 15.8 15.8 15.9
Observation 4 15.9 15.9 15.8

Step 1:
Calculate the Mean of Each Sample
Observation 1 15.8 16.1 16.0
Observation 2 16.0 16.0 15.9
Observation 3 15.8 15.8 15.9
Observation 4 15.9 15.9 15.8
Sample means
(X-bar)
15.875 15.975 15.9

Step 2: Calculate the Standard
Deviation of the Sample Mean
x
σ .2
σ .1
n 4
 
   
 

Step 3: Calculate CL, UCL, LCL
 Center line (x-double bar):
 Control limits for ±3σ limits (z = 3):
15.875 15.975 15.9
x 15.92
3
 
 
 
 
x x
x x
UCL x zσ 15.92 3 .1 16.22
LCL x zσ 15.92 3 .1 15.62
    
    

An Alternative Method for the X-bar
Chart Using R-bar and the A2 Factor
Use this method when
sigma for the process
distribution is not
known. Use factor A2
from Table 6.1
Factor for x-Chart
A2 D3 D4
2 1.88 0.00 3.27
3 1.02 0.00 2.57
4 0.73 0.00 2.28
5 0.58 0.00 2.11
6 0.48 0.00 2.00
7 0.42 0.08 1.92
8 0.37 0.14 1.86
9 0.34 0.18 1.82
10 0.31 0.22 1.78
11 0.29 0.26 1.74
12 0.27 0.28 1.72
13 0.25 0.31 1.69
14 0.24 0.33 1.67
15 0.22 0.35 1.65
Factors for R-Chart
Sample Size
(n)

Step 1: Calculate the Range of
Each Sample and Average Range
Observation 1 15.8 16.1 16.0
Observation 2 16.0 16.0 15.9
Observation 3 15.8 15.8 15.9
Observation 4 15.9 15.9 15.8
Sample ranges
(R)
0.2 0.3 0.2
0.2 0.3 0.2
R .233
3
 
 

 Center line:
 Control limits for ±3σ limits:
 
 
2x
2x
15.875 15.975 15.9
CL x 15.92
3
UCL x A R 15.92 0.73 .233 16.09
LCL x A R 15.92 0.73 .233 15.75
 
  
    
    

Control Chart for Range (R-Chart)
Center Line and Control Limit
calculations:
4
3
0.2 0.3 0.2
CL R .233
3
UCL D R 2.28(.233) .53
LCL D R 0.0(.233) 0.0
 
  
  
  
Factor for x-Chart
A2 D3 D4
2 1.88 0.00 3.27
3 1.02 0.00 2.57
4 0.73 0.00 2.28
5 0.58 0.00 2.11
6 0.48 0.00 2.00
7 0.42 0.08 1.92
8 0.37 0.14 1.86
9 0.34 0.18 1.82
10 0.31 0.22 1.78
11 0.29 0.26 1.74
12 0.27 0.28 1.72
13 0.25 0.31 1.69
14 0.24 0.33 1.67
15 0.22 0.35 1.65
Factors for R-Chart
Sample Size
(n)

Control Charts for Attributes –
P-Charts & C-Charts
 Use P-Charts for quality characteristics that
are discrete and involve yes/no or good/bad
decisions
 Percent of leaking caulking tubes in a box of 48
 Percent of broken eggs in a carton
 Use C-Charts for discrete defects when there
can be more than one defect per unit
 Number of flaws or stains in a carpet sample cut from a
production run
 Number of complaints per customer at a hotel

Constructing a P-Chart:
A Production manager for a tire company has inspected the
number of defective tires in five random samples with 20
tires in each sample. The table below shows the number of
defective tires in each sample of 20 tires.
Sample Sample
Size (n)
Number
Defective
1 20 3
2 20 2
3 20 1
4 20 2
5 20 1

Step 1:
Calculate the Percent defective of Each Sample
and the Overall Percent Defective (P-Bar)
Sample Number
Defective
Sample
Size
Percent
Defective
1 3 20 .15
2 2 20 .10
3 1 20 .05
4 2 20 .10
5 1 20 .05
Total 9 100 .09

Step 2: Calculate the Standard
Deviation of P.
p
p(1-p) (.09)(.91)
σ = = =0.064
n 20

CL p .09 
 Center line (p bar):
 
 
p
p
UCL p z σ .09 3(.064) .282
LCL p z σ .09 3(.064) .102 0
    
      

Constructing a C-Chart:
The number of
weekly customer
complaints are
monitored in a
large hotel.
Develop a three
sigma control limits
For a C-Chart using
the data table On
the right.
Week Number of
Complaints
1 3
2 2
3 3
4 1
5 3
6 3
7 2
8 1
9 3
10 1
Total 22

Calculate CL, UCL, LCL
 Center line (c bar):
UCL c c 2.2 3 2.2 6.65
LCL c c 2.2 3 2.2 2.25 0
z
z
    
      
#complaints 22
CL 2.2
# of samples 10
  

SQC in Services
 Service Organizations have lagged behind
manufacturers in the use of statistical quality control
 Statistical measurements are required and it is more
difficult to measure the quality of a service
 Services produce more intangible products
 Perceptions of quality are highly subjective
 A way to deal with service quality is to devise
quantifiable measurements of the service element
 Check-in time at a hotel
 Number of complaints received per month at a restaurant
 Number of telephone rings before a call is answered
 Acceptable control limits can be developed and charted

Conclusion
 In this slide we discuss about variable
type control charts.
 We study control chart and understood
application in the industry and various
department.

Statistical Process Control

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