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Statistical
Process
Control
What is a process?
Inputs
PROCESS
Outputs
A process can be described as a transformation of
set of inputs into desired outputs.
Types of Measures
• Measures where the metric is composed of a classification
in one of two (or more) categories is called Attribute data.
_ Good/Bad
– Yes/No
• Measures where the metric consists of a number which
indicates a precise value is called Variable data.
– Time
– Miles/Hr
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Population Vs. Sample (Certainty Vs. Uncertainty)
 A sample is just a subset of all possible
values
sample
population
 Since the sample does not contain all the
possible values, there is some uncertainty about
the population. Hence any statistics, such as
mean and standard deviation, are just
estimates of the true population
parameters.
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WHY STATISTICS?
THE ROLE OF STATISTICS ………

LSL
T
USL
Statistics is the art of collecting, classifying,
presenting, interpreting and analyzing numerical
data, as well as making conclusions about the
system from which the data was obtained.
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Descriptive Statistics
Descriptive Statistics is the branch of statistics
which most people are familiar.
It characterizes and summarizes the most
prominent features of a given set of data (means,
medians, standard deviations, percentiles, graphs,
tables and charts.
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Inferential Statistics
Inferential Statistics is the branch of
statistics that deals with drawing
conclusions about a population based on
information obtained from a sample drawn
from that population.
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WHAT IS THE MEAN?
ORDERED DATA SET
The mean is simply the average value of the
data.
-5
-3
-1
xi - 2

mean = x =
=
= -.17
n
-1
12
0
0
n=12
0
0
0
1
3
-6
-5
-4
-3
-2
-1
0
Mean
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1
2
3
4
5
6
4
x
i
= -2
8
WHAT IS THE MEDIAN?
If we rank order (descending or ascending) the data set
,we find the value half way (50%) through the data points
and is called the median value.
ORDERED DATA SET
-5
-3
-1
-1
50% of
data
points
0
Median
0
0
0
0
1
3
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
4
Median
Value
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WHAT IS THE MODE?
If we rank order (descending or ascending) the data
set We find that a single value occurs more often than
any other. This is called the mode.
ORDERED DATA SET
-5
-3
.
-1
-1
0
Mode
Mode
0
0
0
0
1
3
-6
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-5
-4
-3
-2
-1
0
1
2
3
4
5
6
4
10
WHAT IS THE RANGE?
ORDERED DATA SET
The range is a very common metric .
-5
To calculate the range simply subtract the minimum value
in the sample from the maximum value.
-3
-1
-1
Range = x MAX - x MIN = 4 - ( -5) = 9
0
0
0
Range
0
0
1
3
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
4
Range
Min
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Max
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WHAT IS THE VARIANCE/STANDARD DEVIATION?
The variance (s2) is a very robust metric .
The standard deviation(s) is the square root of the variance and is the most
commonly used measure of dispersion.
X =
s
2

Xi
n
X
(

=
=
-2
12
= -.17
- X)
61.67
=
= 5.6
n -1
12 - 1
2
i
-6 -5 -4 -3 -2 -1 0
1
2
3
4
5
DATA SET
-5
-3
-1
-1
0
0
0
0
0
1
3
6
4
Xi - X
( Xi - X )
2
-5-(-.17)=-4.83 (-4.83)2=23.32
-3-(-.17)=-2.83 (-2.83)2=8.01
-1-(-.17)=-.83
(-.83)2=.69
-1-(-.17)=-.83
(-.83)2=.69
0-(-.17)=.17
(.17)2=.03
0-(-.17)=.17
(.17)2=.03
0-(-.17)=.17
(.17)2=.03
0-(-.17)=.17
(.17)2=.03
0-(-.17)=.17
(.17)2=.03
1-(-.17)=1.17
(1.17)2=1.37
3-(-.17)=3.17
(3.17)2=10.05
4-(-.17)=4.17
(4.17)2=17.39
61.67
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Statistical Process Control (SPC)
• Measures performance of a process
• Uses mathematics (i.e., statistics)
• Involves collecting, organizing, & interpreting
data
• Objective: Regulate product quality
• Used to
– Control the process as products are produced
– Inspect samples of finished products
CONTROL CHART
Functions of a Process Control System are
 To signal the presence of assignable causes
of variation
 To give evidence if a process is operating in
a state of statistical control
CONTROL CHART
Variable Values
Essential features of a control chart
Upper Control Limit
Central Line
Lower Control Limit
Time
Control Chart Purposes
• Show changes in data pattern
– e.g., trends
• Make corrections before process is out
of control
• Show causes of changes in data
– Assignable causes
• Data outside control limits or trend in
data
– Natural causes
• Random variations around average
Quality Characteristics
Variables
Attributes
1. Characteristics that
1. Characteristics for which
you focus on defects
you measure, e.g.,
2. Classify products as either
weight, length
‘good’ or ‘bad’, or count #
2. May be in whole or in
defects
– e.g., radio works or not
fractional numbers
3. Continuous random 3. Categorical or discrete
variables
random variables
CONTROL CHART
Types of Control Charts for Attribute Data
Description
Type
Sample Size
Control Chart for proportion non p Chart
conforming units
May change
Control Chart for no. of non
conforming units in a sample
Control Chart for no. of non
conformities in a sample
np Chart
Must be constant
c Chart
Must be constant
Control Chart for no. of non
conformities per unit
u Chart
May Change
Control Chart Types
Control
Charts
Variables
Charts
R
Chart
Attributes
Charts
`X
Chart
P
Chart
C
Chart
X Chart
• Type of variables control chart
– Interval or ratio scaled numerical data
• Shows sample means over time
• Monitors process average and tells whether
changes have occurred. These changes may due
to
1. Tool wear
2. Increase in temperature
3. Different method used in the
second shift
4. New stronger material
• Example: Weigh samples of coffee & compute
means of samples; Plot
R Chart
• Type of variables control chart
– Interval or ratio scaled numerical data
• Shows sample ranges over time
– Difference between smallest & largest values in inspection
sample
• Monitors variability in process, it tells us the loss or gain in
dispersion. This change may be due to:
1. Worn bearing
2. A loose tool
3. An erratic flow of lubricant to machine
4. Sloppiness of machine operator
• Example: Weigh samples of coffee & compute ranges of samples;
Plot
Construction of X and R Charts
• Step 1: Select the Characteristics for applying a control
chart.
• Step 2: Select the appropriate type of control chart.
• Step 3: Collect the data.
• Step 4: Choose the rational sub-group i.e Sample
• Step 5: Calculate the average ( X) and range R for each
sample.
• Step 6: Cal Average of averages of X and average of
range(R)
Construction of X and R Charts
• Steps 7:Cal the limits for X and R Charts.
• Steps 8: Plot Centre line (CL) UCL and
LCL on the chart
• Steps 9: Plot individual X and R values on
the chart.
• Steps 10: Check whether the process is in
control (or) not.
• Steps 11: Revise the control limits if the
points are outside.
X Chart
Control Limits
UCL = x  A R
x
2
LCL = x - A R
x
2
Sub group average X = x1 + x2 +x3 +x4 +x5 / 5
Sub group range R = Max Value – Min value
From
Tables
R Chart Control Limits
UCL R = D 4 R
LCL R = D 3 R
From Tables
Problem8.1 from TQM by
V.Jayakumar Page No 8.5
p Chart for Attributes
• Type of attributes control chart
– Nominally scaled categorical data
• e.g., good-bad
• Shows % of nonconforming items
• Example: Count # defective chairs &
divide by total chairs inspected; Plot
– Chair is either defective or not
defective
p Chart
• p = np / n where p = Fraction of Defective
np = no of Defectives
n = No of items
inspected in sub group
p= Avg Fraction Defective = ∑np/ ∑n = CL
p Chart
Control Limits
p (1 - p )
UCLp = p  z
n
p (1 - p )
LCLp = p - z
n
z = 3 for
99.7% limits
Purpose of the p Chart
Identify and correct causes of bad quality
The average proportion of defective
articles submitted for inspection,over a
period.
To suggest where X and R charts to be
used.
Determine average Quality Level.
Problem
• Problem 9.1 Page no 9.3 TQM by V.Jayakumar
np CHART
P and np are quiet same
Whenever subgroup size is variable,p chart is used. If sub
group size is constant, then np is used.
FORMULA: Central Line CLnp = n p
Upper Control Limit, UCLnp = n p +3√ n p (1- p )
Lower Control Limit, LCLnp = n p -3 √ n p (1- p )
Where p = ∑ np/∑n =Average Fraction Defective
n = Number of items inspected in subgroup.
Problem
• Problem No 9.11 page No 9.11 in TQM
by V.Jayakumar
c Chart
• Type of attributes control chart
– Discrete quantitative data
• Shows number of nonconformities (defects) in a unit
– Unit may be chair, steel sheet, car etc.
– Size of unit must be constant
• Example: Count no of defects (scratches, chips etc.)
in each chair of a sample of 100 chairs; Plot
c Chart
Control Limits
UCLc = c  3 c
LCLc = c - 3 c
Use 3 for
99.7% limits