Statistical Quality Control (SQC)
- Why control the process?
• Products and services require uniform quality.
• Process variation is inevitable
• reduce variation for better quality (conformance to
specs)
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Some Generic Tools for QC
- Process flow chart
- Pareto analysis
- Run chart
- Histogram
- Checksheet
- Causes and effect diagram (Fishbone diagram)
- Control charts
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SQC
-
Sources of variation in process:
• random variations (can not be controlled)
•
-
non-random variations (can be controlled)
What should we do?
• We can not eliminate the variation, but we can identify the
sources of non-random variations and hence improve the
process.
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Statistical Sampling
-
Identify the problems
• 100% inspection
• sampling
º DO NOT TEMPER WITH THE SYSTEM!!!
-
Sample measurement:
• variable (continuous): such as weight, length, ….
• Attribute (good, no-good)
º defectives: the acceptability of products across a range
of characteristics
º defects: a product that has at least one defective
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Control Charts
-
Purpose:
• Using the samples collected from the process, and
calculating the required statistics to construct a time
sequence chart to detect the process variation.
-
Process control with variable measurements:
• For each sample, we measure the weight, volume, length, or
other variable measurements.
• Process control charts (variable measures):
º X-bar chart
º R chart
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Control Charts (variable measures)
-
The sample mean follows a normal distribution.
• If the process does not change over time, every sample
mean should follow the same normal distribution.
-
X-bar chart:
• A plot of the sample means taken from the process. It is
used to detect the change of the process mean (mean for
the normal distribution).
-
R chart:
• A plot of the sample variation range. It is used to detect the
change of process variation (variance for the normal
distribution).
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X-bar and R charts
- Change of process mean
- Change of variance
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Developing Control Charts
1. For each sample,
- calculate the sample mean.
Xi 
X 1  X 2  X 3 ... X n
n
(n is the number of observations in a sample (sample size))
- calculate the range.
2. For all the samples,
- calculate the overall mean.
- calculate the average range.
Ri  (max. in sample i) - (min. in sample i)
X
X 1  X 2  X 3 ... X k
k
R
R1  R2  R3 ... Rk
k
(k is the number of samples)
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Developing Control Charts (continued)
3. For the X-bar chart:
- Upper control limit (UCL)= X  A2 R
- center line = X
- Lower control limit (LCL)= X  A2 R
4 For the R chart:
- Upper control limit (UCL)= D4 R
- center line = R
- Lower control limit (LCL)= D3 R
5. Look for A2, D3, D4 in Exh. S5.4.
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Control Charts-Example
The following measures of shaft diameters were taken from a
company producing transmission boxes. Construct the X-bar
and R charts.
sample
1
2
3
4
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diameter (in inches)
2.10
1.99
2.00
2.09
2.08
1.98
2.01
2.10
1.97
2.05
1.98
2.05
Example (continued)
- X-bar chart
- R chart
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Example (continued)
Two new measures were taken this week. Using the
control charts developed, explain your findings.
sample
1
2
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diameter (in inches)
2.31
1.86
1.86
1.85
1.91
1.79
Interpreting the Control Charts
-
X-bar and R charts MUST BE USED TOGETHER!!
sample
measurements
1
10.0
10.1
10.2
2
13.0
9.5
7.8
- sample 1 and 2 have the same mean (10.1)
- sample 2 has greater variation than sample 1. (R1=0.2, R2=5.2)
sample
measurements
1
10.0
10.1
10.2
2
13.0
13.1
13.2
- sample 1 and 2 have the same variation (0.2)
- sample 2 has a higher mean than sample 1.
-
See Exh. S5.3.
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Control Chart for attribute measures: p chart
- Sample units are only classified into one of two
categories (good or bad, success or failure, etc.)
total # of defects from all samples
p
(# of samples) X (sample size)
Sp 
p (1- p )
n
UCL  p  zS p
LCL  p  zS p
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P Chart-Example
A visual inspection for scratches produced the following data for
last week. For each sample, 30 units were inspected.
Sample
1
2
3
4
5
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# of defects
5
4
4
5
7
sample
6
7
8
9
10
# of defects
4
5
6
4
5
Example (continued)
This week 30 units were inspected on each of 2 occasions. On
Monday, 6 units were found defective. On Tuesday, 9 were
found defective. Is this process in or out of control?
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Process Capability
-
Control charts are of little value if the process is not capable of
making products within design specification (or tolerance) limits.
-
Process capability = 6 s
(s: standard deviation of the process)
-
Process capability ratio =
Upper tolerance limit (UTL) - Lower tolerance limit (LTL)
6s
- higher ratio: the process is potentially more capable of
making the product within the design specification.
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Process Capacity vs. Design Specs (Tolerance)
-
process variability matches specs
-
process variability well within specs
-
process variability exceeds specs
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Capability Index (Cpk)
-
Indicates how well the process performs relative to the target
value.
-
Cpk is used to determine whether the process mean is closer to
the UTL or LTL.
 X  LTL UTL  X 
Cpk  min 
,

3s
3s


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Process Capability-Example
A metal fabricator produces connecting rods with an outer diameter that
has a specification of 1.000 +/- 0.010 inch. The process mean is
determined to be 1.002 inches with a standard deviation of 0.003
inches. What do you think about the process?
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Taguchi’s Cost Function
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