Transcript No Slide Title

Control Charts for Variables
EBB 341 Quality Control
Variation



There is no two natural items in any category
are the same.
Variation may be quite large or very small.
If variation very small, it may appear that
items are identical, but precision instruments
will show differences.
3 Categories of variation

Within-piece variation


Apiece-to-piece variation


One portion of surface is rougher than another
portion.
Variation among pieces produced at the same time.
Time-to-time variation

Service given early would be different from that given
later in the day.
Source of variation

Equipment


Material


Raw material quality
Environment


Tool wear, machine vibration, …
Temperature, pressure, humadity
Operator

Operator performs- physical & emotional
Control Chart Viewpoint

Variation due to
Common or chance causes
 Assignable causes


Control chart may be used to discover
“assignable causes”
Some Terms



Run chart - without any upper/lower
limits
Specification/tolerance limits - not
statistical
Control limits - statistical
Control chart functions

Control charts are powerful aids to
understanding the performance of a process
over time.
Output
Input
PROCESS
What’s causing variability?
Control charts identify variation

Chance causes - “common cause”



inherent to the process or random and not
controllable
if only common cause present, the process is
considered stable or “in control”
Assignable causes - “special cause”


variation due to outside influences
if present, the process is “out of control”
Control charts help us learn more about
processes



Separate common and special causes of
variation
Determine whether a process is in a state of
statistical control or out-of-control
Estimate the process parameters (mean,
variation) and assess the performance of a
process or its capability
Control charts to monitor processes

To monitor output, we use a control chart


we check things like the mean, range, standard
deviation
To monitor a process, we typically use two
control charts


mean (or some other central tendency measure)
variation (typically using range or standard
deviation)
Types of Data

Variable data

Product characteristic that can be measured


Length, size, weight, height, time, velocity
Attribute data
 Product
characteristic evaluated with a discrete
choice
• Good/bad, yes/no
Control chart for variables
Variables are the measurable
characteristics of a product or service.
 Measurement data is taken and
arrayed on charts.

Control charts for variables

X-bar chart


R chart


In this chart, the sample ranges are plotted in order to
control the variability of a variable.
S chart


In this chart the sample means are plotted in order to
control the mean value of a variable (e.g., size of piston
rings, strength of materials, etc.).
In this chart, the sample standard deviations are plotted
in order to control the variability of a variable.
S2 chart

In this chart, the sample variances are plotted in order
to control the variability of a variable.
X-bar and R charts

The X- bar chart is developed from the
average of each subgroup data.


used to detect changes in the mean between
subgroups.
The R- chart is developed from the ranges of
each subgroup data

used to detect changes in variation within
subgroups
Control chart components

Centerline


shows where the process average is centered or
the central tendency of the data
Upper control limit (UCL) and Lower control
limit (LCL)

describes the process spread
The Control Chart Method
X bar Control Chart:
UCL = XDmean + A2 x Rmean
LCL = XDmean - A2 x Rmean
CL = XDmean
R Control Chart:
UCL = D4 x Rmean
LCL = D3 x Rmean
CL = Rmean
Capability Study:
PCR = (USL - LSL)/(6s); where s = Rmean /d2
Control Chart Examples
Variations
UCL
Nominal
LCL
Sample number
How to develop a control chart?
Define the problem

Use other quality tools to help determine the
general problem that’s occurring and the
process that’s suspected of causing it.
Select a quality characteristic to be measured

Identify a characteristic to study - for
example, part length or any other variable
affecting performance.
Choose a subgroup size to be sampled

Choose homogeneous subgroups


Homogeneous subgroups are produced under the
same conditions, by the same machine, the same
operator, the same mold, at approximately the
same time.
Try to maximize chance to detect differences
between subgroups, while minimizing chance
for difference with a group.
Collect the data


Generally, collect 20-25 subgroups (100 total
samples) before calculating the control limits.
Each time a subgroup of sample size n is
taken, an average is calculated for the
subgroup and plotted on the control chart.
Determine trial centerline


The centerline should be the population
mean, 
Since it is unknown, we use X Double bar, or
the grand average of the subgroup averages.
m
X 
X
i 1
m
i
Determine trial control limits - Xbar
chart



The normal curve displays the distribution of
the sample averages.
A control chart is a time-dependent pictorial
representation of a normal curve.
Processes that are considered under control
will have 99.73% of their graphed averages
fall within 6.
UCL & LCL calculation
UCL  X  3
LCL  X  3
  standarddeviation
Determining an alternative value for
the standard deviation
m
R 
R
i 1
i
m
UCL  X  A 2 R
LCL  X  A 2 R
Determine trial control limits - R chart

The range chart shows the spread or
dispersion of the individual samples within
the subgroup.



If the product shows a wide spread, then the
individuals within the subgroup are not similar to
each other.
Equal averages can be deceiving.
Calculated similar to x-bar charts;

Use D3 and D4 (appendix 2)
Example: Control Charts for Variable Data
Sample
1
2
3
4
5
6
7
8
9
10
Slip Ring Diameter (cm)
1
2
3
4
5
5.02 5.01 4.94 4.99 4.96
5.01 5.03 5.07 4.95 4.96
4.99 5.00 4.93 4.92 4.99
5.03 4.91 5.01 4.98 4.89
4.95 4.92 5.03 5.05 5.01
4.97 5.06 5.06 4.96 5.03
5.05 5.01 5.10 4.96 4.99
5.09 5.10 5.00 4.99 5.08
5.14 5.10 4.99 5.08 5.09
5.01 4.98 5.08 5.07 4.99
X
4.98
5.00
4.97
4.96
4.99
5.01
5.02
5.05
5.08
5.03
50.09
R
0.08
0.12
0.08
0.14
0.13
0.10
0.14
0.11
0.15
0.10
1.15
Calculation
From Table above:
 Sigma X-bar = 50.09
 Sigma R = 1.15
 m = 10
Thus;
 X-Double bar = 50.09/10 = 5.009 cm
 R-bar = 1.15/10 = 0.115 cm
Note: The control limits are only preliminary with 10 samples.
It is desirable to have at least 25 samples.
Trial control limit


UCLx-bar = X-D bar + A2 R-bar = 5.009 +
(0.577)(0.115) = 5.075 cm
LCLx-bar = X-D bar - A2 R-bar = 5.009 (0.577)(0.115) = 4.943 cm
UCLR = D4R-bar = (2.114)(0.115) = 0.243
cm
 LCLR = D3R-bar = (0)(0.115) = 0 cm
For A2, D3, D4: see Table B, Appendix n = 5

3-Sigma Control Chart Factors
Sample size
n
2
3
4
5
6
7
8
X-chart
A2
1.88
1.02
0.73
0.58
0.48
0.42
0.37
R-chart
D3
0
0
0
0
0
0.08
0.14
D4
3.27
2.57
2.28
2.11
2.00
1.92
1.86
X-bar Chart
5.10
UCL
5.08
X bar
5.06
5.04
5.02
5.00
CL
4.98
4.96
LCL
4.94
0
1
2
3
4
5
6
Subgroup
7
8
9
10
11
R Chart
UCL
0.25
Range
0.20
0.15
CL
0.10
0.05
LCL
0.00
0
1
2
3
4
5
6
Subgroup
7
8
9
10
11
Run Chart
6.70
6.65
Mean, X-bar
6.60
6.55
6.50
6.45
6.40
6.35
6.30
0
5
10
15
Subgroup number
0.35
0.3
Range, R
0.25
0.2
0.15
0.1
0.05
0
0
5
10
15
Subgroup number
20
25
20
25
Another Example
of X-bar & R chart
Given Data (Table 5.2)
Subgro
up
X1
X2
X3
X4
X-bar
UCL-X-bar
X-Dbar
LCL-X-bar
R
UCL-R
R-bar
LCL-R
1
6.35
6.4
6.32
6.37
6.36
6.47
6.41
6.35
0.08
0.20
0.0876
0
2
6.46
6.37
6.36
6.41
6.4
6.47
6.41
6.35
0.1
0.20
0.0876
0
3
6.34
6.4
6.34
6.36
6.36
6.47
6.41
6.35
0.06
0.20
0.0876
0
4
6.69
6.64
6.68
6.59
6.65
6.47
6.41
6.35
0.1
0.20
0.0876
0
5
6.38
6.34
6.44
6.4
6.39
6.47
6.41
6.35
0.1
0.20
0.0876
0
6
6.42
6.41
6.43
6.34
6.4
6.47
6.41
6.35
0.09
0.20
0.0876
0
7
6.44
6.41
6.41
6.46
6.43
6.47
6.41
6.35
0.05
0.20
0.0876
0
8
6.33
6.41
6.38
6.36
6.37
6.47
6.41
6.35
0.08
0.20
0.0876
0
9
6.48
6.44
6.47
6.45
6.46
6.47
6.41
6.35
0.04
0.20
0.0876
0
10
6.47
6.43
6.36
6.42
6.42
6.47
6.41
6.35
0.11
0.20
0.0876
0
11
6.38
6.41
6.39
6.38
6.39
6.47
6.41
6.35
0.03
0.20
0.0876
0
12
6.37
6.37
6.41
6.37
6.38
6.47
6.41
6.35
0.04
0.20
0.0876
0
13
6.4
6.38
6.47
6.35
6.4
6.47
6.41
6.35
0.12
0.20
0.0876
0
14
6.38
6.39
6.45
6.42
6.41
6.47
6.41
6.35
0.07
0.20
0.0876
0
15
6.5
6.42
6.43
6.45
6.45
6.47
6.41
6.35
0.08
0.20
0.0876
0
16
6.33
6.35
6.29
6.39
6.34
6.47
6.41
6.35
0.1
0.20
0.0876
0
17
6.41
6.4
6.29
6.34
6.36
6.47
6.41
6.35
0.12
0.20
0.0876
0
18
6.38
6.44
6.28
6.58
6.42
6.47
6.41
6.35
0.3
0.20
0.0876
0
19
6.35
6.41
6.37
6.38
6.38
6.47
6.41
6.35
0.06
0.20
0.0876
0
20
6.56
6.55
6.45
6.48
6.51
6.47
6.41
6.35
0.11
0.20
0.0876
0
21
6.38
6.4
6.45
6.37
6.4
6.47
6.41
6.35
0.08
0.20
0.0876
0
22
6.39
6.42
6.35
6.4
6.39
6.47
6.41
6.35
0.07
0.20
0.0876
0
23
6.42
6.39
6.39
6.36
6.39
6.47
6.41
6.35
0.06
0.20
0.0876
0
24
6.43
6.36
6.35
6.38
6.38
6.47
6.41
6.35
0.08
0.20
0.0876
0
25
6.39
6.38
6.43
6.44
6.41
6.47
6.41
6.35
0.06
0.20
0.0876
0
Calculation
From Table 5.2:
 Sigma X-bar = 160.25
 Sigma R = 2.19
 m = 25
Thus;
 X-double bar = 160.25/29 = 6.41 mm
 R-bar = 2.19/25 = 0.0876 mm
Trial control limit




UCLx-bar = X-double bar + A2R-bar = 6.41 +
(0.729)(0.0876) = 6.47 mm
LCLx-bar = X-double bar - A2R-bar = 6.41 –
(0.729)(0.0876) = 6.35 mm
UCLR = D4R-bar = (2.282)(0.0876) = 0.20
mm
LCLR = D3R-bar = (0)(0.0876) = 0 mm
For A2, D3, D4: see Table B Appendix, n = 4.
X-bar Chart
R Chart
Revised CL & Control Limits



Calculation based on discarding subgroup 4 & 20 (Xbar chart) and subgroup 18 for R chart:
X new
X X


Rnew
RR


d
m  md
d
m  md
= (160.25 - 6.65 - 6.51)/(25-2)
= 6.40 mm
= (2.19 - 0.30)/25 - 1
= 0.079 = 0.08 mm
New Control Limits
New value:
X o  X new ,

RO
Ro  Rnew ,  o 
d2
Using standard value, CL & 3 control limit obtained
using formula:
UCLX  X o  A o ,
LCLX  X o  A o
UCLR  D2 o ,
LCLR  D1 o
From Table B:
 A = 1.500 for a subgroup size of 4,
 d2 = 2.059, D1 = 0, and D2 = 4.698
Calculation results:
X o  X new  6.40mm
Ro  Rnew  0.079,
o 
Ro 0.079

 0.038mm
d 2 2.059
UCLX  X o  A o  6.40  (1.500)(0.038)  6.46mm
LCLX  X o  A o  6.40  (1.500)(0.038)  6.34mm
UCLR  D2 o  (4.698)(0.038)  0.18 mm
LCLR  D1 o  (0)(0.038)  0 mm
Trial Control Limits & Revised Control Limit
6.65
Revised control limits
Mean, X-bar
6.60
6.55
UCL = 6.46
6.50
6.45
CL = 6.40
6.40
6.35
LCL = 6.34
6.30
0
2
4
6
8
Subgroup
UCL = 0.18
Range, R
0.20
0.15
CL = 0.08
0.10
0.05
0.00
0
2
4
Subgroup
6
8
LCL = 0
Revise the charts

In certain cases, control limits are revised
because:
 out-of-control points were included in the
calculation of the control limits.
 the process is in-control but the within
subgroup variation significantly
improves.
Revising the charts




Interpret the original charts
Isolate the causes
Take corrective action
Revise the chart

Only remove points for which you can determine an
assignable cause
Process in Control


When a process is in control, there occurs a
natural pattern of variation.
Natural pattern has:



About 34% of the plotted point in an imaginary
band between 1 on both side CL.
About 13.5% in an imaginary band between 1
and 2 on both side CL.
About 2.5% of the plotted point in an imaginary
band between 2 and 3 on both side CL.
The Normal
Distribution
 = Standard deviation
Mean
-3 -2 -1
+1 +2 +3
68.26%
95.44%
USL 99.74%
LSL
-3
CL
+3





34.13% of data lie between  and 1 above the mean ().
34.13% between  and 1 below the mean.
Approximately two-thirds (68.28 %) within 1 of the mean.
13.59% of the data lie between one and two standard deviations
Finally, almost all of the data (99.74%) are within 3 of the mean.
Normal Distribution Review

Define the 3-sigma limits for sample means as follows:
3
3(0.05)
Upper Limit   
 5.01 
 5.077
n
5
3
3(0.05)
Low erLimit   
 5.01 
 4.943
n
5


What is the probability that the sample means will lie
outside 3-sigma limits?
Note that the 3-sigma limits for sample means are
different from natural tolerances which are at
  3
Common Causes
Process Out of Control


The term out of control is a change in the
process due to an assignable cause.
When a point (subgroup value) falls outside
its control limits, the process is out of control.
Assignable Causes
Average
Grams
(a) Mean
Assignable Causes
Average
(b) Spread
Grams
Assignable Causes
Average
Grams
(c) Shape
Control Charts
Assignable
causes
likely
UCL
Nominal
LCL
1
2
Samples
3
Control Chart Examples
Variations
UCL
Nominal
LCL
Sample number
Control Limits and Errors
Type I error:
Probability of searching for
a cause when none exists
(a) Three-sigma limits
UCL
Process
average
LCL
Control Limits and Errors
Type I error:
Probability of searching for
a cause when none exists
(b) Two-sigma limits
UCL
Process
average
LCL
Control Limits and Errors
(a) Three-sigma limits
Type II error:
Probability of concluding
that nothing has changed
UCL
Shift in process
average
Process
average
LCL
Control Limits and Errors
(b) Two-sigma limits
Type II error:
Probability of concluding
that nothing has changed
UCL
Shift in process
average
Process
average
LCL
Achieve the purpose



Our goal is to decrease the variation inherent
in a process over time.
As we improve the process, the spread of the
data will continue to decrease.
Quality improves!!
Improvement
Examine the process

A process is considered to be stable and
in a state of control, or under control,
when the performance of the process
falls within the statistically calculated
control limits and exhibits only chance, or
common causes.
Consequences of misinterpreting the
process





Blaming people for problems that they cannot
control
Spending time and money looking for problems that
do not exist
Spending time and money on unnecessary process
adjustments
Taking action where no action is warranted
Asking for worker-related improvements when
process improvements are needed first
Process variation

When a system is subject to only
chance causes of variation, 99.74% of
the measurements will fall within 6
standard deviations
 If 1000 subgroups are measured,
997 will fall within the six sigma
limits.
Mean
-3 -2 -1
+1 +2 +3
68.26%
95.44%
99.74%
Chart zones

Based on our knowledge of the normal curve, a
control chart exhibits a state of control when:
♥ Two thirds of all points are near the center
value.
♥ The points appear to float back and forth
across the centerline.
♥ The points are balanced on both sides of the
centerline.
♥ No points beyond the control limits.
♥ No patterns or trends.