# SPC by jianglifang

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```									      BUSA 3110
Statistical Process Control
Kim I. Melton, Ph.D.
Transition
 Population                  Process
 We have a fixed             We have an on-going
population (a group of       process that produces
interest)                    output
 We want to use              We want to assess if
information from a           the observed output
sample from the              from the process can
existing group to draw       be considered as a
a conclusion about the       sample from a
entire group.                population and
 Goal                          We want to be able to
 Describe— “What”             “infer” to future output
from the process
 Goal
 Explain— “Why?”
Population and Sample
Population:
All of the items/measurements of         Inference is back
interest                                 to the population
that produced the
sample

Sample:
The items/measurements
actually obtained
Process and “Sample”
 Two issues arise:
 Changes can occur in an on-
going process while you are
collecting data—i.e., you don’t
know if all of your data is
coming from the same
population
 Although describing past output
may be useful, this is
descriptive (history). You really
want to be able to know what
to expect in the future—i.e.,
you aren’t trying to make an
it existed while you were
collecting data.
Classical Statistical Inference?
Suppose you could obtain data on the
process of delivering shipments to a
customer (at much cost in terms of dollars
and time). You would like to say something
about the expected delivery time for the near
future. How can you use the data?

  9.222

  3.772
1                   17
  9. 22

  3. 77
Looking at the SAME Data as
Output from a Process…

?
17

?
?
?
?
?
1
Process Based Studies
 We don’t know if there is a
population. Conditions may not
be the same for repeated
?     observations from the process.
?
?    The goal is to make decisions
that will relate to the future
output from the process; to do
time ==>         this we must understand the
underlying cause system.
 If the process is stable, the
history of the process (combined
with subject matter knowledge)
can be used to predict the future
output. No numerical measure
of confidence can be calculated!
Aims:
Monitoring and Improvement
 Monitoring
 Determine if conditions were similar enough over the
entire period to make some kind of descriptive statement
 Determine if the variation in the process comes from the
design of the process or from a combination of the design
of the process and some other sources of variation
 Improvement
 Determine what type of action is called for to influence
future outcomes
 Improvement comes from
 Changes to the process (not just to the numbers)
   To change the average
   To reduce variation
   To make the process more robust to variation
   Innovation (change the problem!)
SIPOC Model
 Suppliers provide
 Inputs to the
 Process which produces
 Output for
 Customers
Voices of the Process
 Voice of the Customer
 Voice of the Product/Service
 Voice of the Process
 Voice of the System
Example: Bottle Closures
 Voice of the Customer –What the customer wants
(may be the end customer or the bottler)
 Fits
 e.g., The bottler may provide specs to the closure
manufacturer:
E dimension - target: 1.003; LSL = .995; USL 1.011
 Clean
 Safe
 e.g., The FDA says that no more than 3 in 1000 TE bands
can fail
 Reusable (can be taken off and put back on)
 Easy to open
 Doesn’t leak
 Stays on
Bottle Closure
 Voice of the Product – Data from output from the
process about the product that relates to what
the customer wants:
 Fits  actual results on various diameters and
other dimensions
 Clean color and lack of debris
 Safe actual number of TE bands that fail
 Reusable (can be taken off and put back on) 
 Easy to open  actual torque required to open
 Doesn’t leak  data on liners or clearances
 Stays on  results of heat tests
Bottle Closure
 Voice of the Process
 The factors that can be controlled during the
production process that might impact the
product/service characteristic
 Fit/Reusable/Doesn’t leak/Stays on dimensions 
 Machine settings: temperatures, pressure, time
 Resin: type, color
 Clean free from contamination
 Screening of resin prior to injection into the mold
 Safe TE Band
 Amount of resin forced into the mold (to control the size
of the “bridges”)
 Doesn’t leak Liner
Voice of the System
 Research and Design (design of closure and
molds)
 Hiring practices
 Maintenance practices
 Training
Seven
Basic
Tools                                                                             time ==>

Process Flow Diagram   Cause and Effect Diagram                     Run Chart

variable 2
R C P DO               measurement
time ==>            Product
variable 1

Control Chart        Pareto Chart            Histogram                         Scatter Diagram
Stability
(aka: Statistical Control)

 Stability depends on the consistency of output
(in terms of the range of variation seen); a
stable process could consistently produce
product that is too small, too large, or within
the acceptable range.
 Variation is considered stable when there are
only common causes of variation present; if
there are both common and special causes of
variation present the process is not considered
stable (i.e., it is not in a state of statistical
control)
Stability, Acceptability, and
Desirability
 Stability: What is consistently happening
 A process is called stable if the variation in the
outputs is predictable (may be predictably small
or predictably large—the issue is that the range
can be predicted)
 Acceptability: What is considered “OK” from
the process
 Output that meets the requirements set by the
customer is considered “acceptable.”
 Desirability: The ideal output from the process
 A process that is producing acceptable output can
still be improved by reducing variation around a
target.
Stability, Accuracy, Precision
Stability, Accuracy,   Acceptable vs.
Precision              Desirable
Bottle Filling Process
 All three process
could be stable
 Only the third
process might be
1
considered
acceptable
 Even the third
process could be
desirable (by
filling every
bottle to the
reported fill
volume)             3
Zero Defects vs.
Continual Improvement
 E Dimension on a closure
 Operational Definition identifies how “deep” in the closure to
measure, how to orient the closure on the measurement
machine, how to obtain the “diameter,” and where to start and
stop measuring (the threads spiral down the closure)
 Acceptable (Specifications):
 Closure: LSL .995     USL 1.011
 Bottle:  LSL .972     USL .997
 Desirable (Nominal):
 Closure: 1.003
 Bottle: .984
 If all product were made to nominal, clearance would be
.019
 If “meeting specs” is good enough
 Clearances could range from -.002 to .039    [Some closures
won’t fit on the bottles!!!]
____ Limits
 Spec Limits (specifications)
 Measurements that are “acceptable” to the
customer
 Control limits
 Statistically calculated limits used to assess
stability and estimate the range of values for
the characteristic plotted on a control chart
 Natural Process Limits
 Range of measurements for individual items
expected to be seen from a process (depends
on stability)
Run Charts
 Measurements for a
process characteristic
are plotted in time
order
 Patterns in the data
indicate special causes
of variability
 Trends
time ==>
 Clusters
 A repeating pattern
 No variability in plotted
points
Run Charts
If the process is stable all of the
following will be true:
• Most points will plot near some
central value
• Some variation will exist
• Individual points will not be
predictable, but the overall
clustering of points will be
predictable
• No patterns will show up
Control Charts
 A Run Chart with statistically
calculated limits
   Limits are based on data
collected from the process
   If points plot within the limits
and show no patterns, the
process is said to be "in
control" or stable
   If the process is considered
stable, limits represent the
time ==>
expected range of variation
for the value plotted
   A process that is in control is
predictable (it may or may
not be producing desirable
output)
Setting Up Control Charts
Step    Determine question(s) to be answered
1:
Step    Design data collection plan and collect data
2:
Step    Plot run chart and look for obvious patterns
3:
Step    If no patterns, calculate control limits (using
4:
formulas for the appropriate type of chart)
Step 5: Conduct runs tests
Step 6: Interpret the chart
Step 7: Determine appropriate type of action and
take steps to accomplish this
Runs Tests-Melton
A control chart fails to show
stability
if any of the following occur:
• at least one point plots       A
outside the control limits     B
• two of three consecutive       C
points in the same A zone      C
• fifteen consecutive points     B
plot in the C zones            A
• more than seven
consecutive points on the
same side of the center line
• seven or more consecutive
increases (or decreases)
Note: Other runs tests
• fifteen consecutive points
alternating up and down        are available in other
books.
Obs. # Measure   Obs. # Measure
Runs Tests
1     11        16     17
2     14        17     14
3     17        18     12                    25
4     15        19     17
23
5     17        20     19
6     21        21     12                    21
7     17        22     12
8     16        23     14                    19
9     17        24     19
17

Measurement
10     15        25     17
11     13        26     19
15
12     15        27     10
13     19        28     19
13
14     11        29     14
15     15        30     16                    11

LCL = 6.387                                 9

CL = 15.467                                 7
UCL = 24.547
5

11

13

15

17

19

21

23

25

27

29
1

3

5

7

9
Width = 3.02666
Observation #
Insurance Quotes

An insurance company staffs quote lines
so that independent agents can call in for
quotes on insurance. The following data
represent the time (in seconds) for one
operator to respond to five consecutive calls
from the same state. One subgroup (of five
observations) is collected each day.
The Data
Day Obs. 1   Obs. 2 Obs. 3   Obs. 4   Obs. 5
1   197      190     162     159      194
2   200      192     177     227      180
3   186      178     209     197      190
4   206      168     209     208      182
5   182      175     158     207      226
6   195      179     216     213      193
7   197      195     213     198      217
8   208      248     193     158      177
9   184      166     224     186      180
10   203      185     212     214      161
11   189      183     207     176      207
12   223      175     196     213      200
13   200      168     193     233      164
14   186      161     179     155      203
15   199      218     211     217      230
X-bar and R charts
Collect Data                 Plot Data on
Start          Calculate X-bar and R for                                   Patterns?
appropriate chart
each subgroup

yes                        no

Place limits on R chart
Look for cause
Conduct Runs Tests

Look for cause of
inconsistent                Look for cause of
variation between            inconsistent variation     yes            Runs?
subgroups                   within subgroups

yes                                           no

Appears                                              Place limits on X-bar chart
Stable             no              Runs?               Conduct Runs Tests
X-Bar and R Charts

Result
Result

Time              Time

Result

Result

Time            Time
Control Limits
R Chart             X Chart

LCL R  D3 R     LCL X  X  A 2 R
CL R  R         CL X  X
UCLR  D 4 R     UCL X  X  A 2 R

If the process                R
appears to be     X
ˆ        
ˆ
stable, then:                 d2
Using Excel to Calculate X-Bar and R

Assume that the data for the first subgroup are in
cells B2, C2, D2, E2, and F2

If you want to put X-Bar for this subgroup in cell
I2, then click on cell I2 and type:
=average(B2:F2)

If you want to put R for this subgroup in cell J2,
then click on cell J2 and type:
=max(B2:F2)-min(B2:F2)

Copy the formulas down the column to find the
corresponding values for each subgroup
X-Bars and Rs
X-Bar    R
180.4   38
195.2   50
192    31
194.6
189.6
41
68   X  X  2912  194.133
199.2   37       k     15
204
196.8
22
90   R  R  715  47.667
188    58       k    15
195    53
192.4   31
201.4   48
191.6   69
176.8   48
215    31
Control Chart Constants

n   D3     D4      A2      d2
2   None 3.267     1.880   1.128
3   None 2.574     1.023   1.693
4   None 2.282     .729    2.059
5   None 2.114     .577    2.326
6   None 2.004     .483    2.534
7   .076   1.924   .419    2.704
R Chart
 Calculating Control Limits
 CL: 47.667
 UCL: 2.114*47.667 = 100.767
 LCL: none (NOTE: this is different from 0)
 Conducting Runs Tests
 Width of zones: (100.767 – 47.667)/3 = 17.7
 Heights:
 100.767
 83.067
 65.367
 47.667
 29.967
 12.266
R Chart
R Chart                                        Runs:

100

80
R (n=5)

60

40

20

0
1   2   3   4   5   6   7   8   9    10   11   12   13   14   15
Subgroup #
R Chart
(What does this tell us?)
 Remember that R looks at within subgroup
variation.
 R does not address the magnitude of the
observations—just the difference between the
highest and lowest in a subgroup.
 For example, the following two subgroups would
have the same range:
 5, 10, 15, 20
 105, 110, 115, 120
within subgroup variation)
 Lack of stability relates to changes within a
subgroup.
X-Bar Chart
 Calculating Control Limits
 CL: 194.133
 LCL: 194.133 - .577(47.667) = 166.63
 UCL: 194.133 + .577(47.667) = 221.637
 Conducting Runs Tests
 Width of zones: (221.637 – 194.133)/3 = 9.168
 Heights:
 221.637
 212.469
 203.301
 194.133
 184.965
 175.798
 166.63
X-Bar Chart
X-Bar Chart                                        Runs:
230

220

210
X-Bar (n=5)

200

190

180

170

160
1   2   3   4   5   6   7   8   9    10   11   12   13   14   15
Subgroup #
X-Bar Chart
(What does this tell us?)
 Remember that the X-Bar Chart plots
subgroup averages.
 For the limits on the X-Bar Chart to have
meaning, you need stability with respect to
spread—the average R needs to be
meaningful.
 The X-Bar chart assesses if there are
subgroups (above and beyond the variation
captured within the subgroups).
Interpreting Results
When the Process is Judged to be Stable

 Recall that control limits deal with the expected
range of variation for the characteristic plotted on
the control chart.
 Therefore, the limits on the X-bar chart deal with
the expected range of variation for subgroup
averages.
 Natural process limits, expected ranges for the
individual values, can be calculated as:
R
  3 where   X and  
ˆ    ˆ       ˆ         ˆ
d2
*based on Empirical Rule (and assumes mound
shaped distribution for individual measurements)
Can we use the Empirical Rule?
 Stability allows                                          Calculating Natural
us to consider                                 Process Limits appears
the observations                               reasonable.
as coming from a
single                                           194.133
ˆ
distribution.
  R / d 2  47.667 / 2.326  20.493
ˆ
Histogram of
Individual Obs.
 LNPL=194.133-3(20.493)
20

15
= 132.654
Frequency

10

5                                               UNPL=194.133+3(20.493)
0                                                    = 255.612
159 169 179 189 199 209 219 229 239 249
Time in Seconds
(max in bin given)
Limits
 Control Limits (LCL and UCL)
 calculated from data collected from the process
 used to assess stability of the process
 related to the characteristic plotted (e.g., X-Bar or R)
 Natural Process Limits (LNPL and UNPL)
 only calculated if the process appears to be stable
 related to measurements for individual items
 provides information about range of measurements for
individual items that can be expected
 Based on the Empirical Rule (so assumes mound shaped
distribution of individual measurements
 Specification Limits - “Specs” (LSL and USL)
 determined by the user based on desired or needed
measurements for an item
 related to desired (acceptable) range of measurements for
individual items
Call Center Goals
 Suppose that management of the insurance
company wants quotes times to be
between two minutes and four minutes.
(They believe any times below two minutes
will be rushed and unfriendly, and times
above four minutes would discourage
 With the current process, what proportion
of the calls would you expect to take more
than four minutes?
How are we doing?

 The manager’s expectation that calls will be
between 2 and 4 minutes provides “specs” that
we will write in terms of seconds:
LSL = 120; USL = 240
 We just found Natural Process Limits of:
LNPL = 132.654; UNPL = 255.612

120                    240

132.654                   255.612

PROBLEM: The process is expected to
produce some calls that are too long.
How many calls will be too long?
 Based on a comparison of our Natural
Process Limits and the Specs, we would
expect to find a few calls that are too long.
 Calculating, we see
 P(X > 240) =
 =1-NORMDIST(240,194.133,20.493,1) =
.0126
 About one and a quarter percent (1.26%) of
the calls will be too long.
How many calls will be too short?
 We can compare our Natural Process Limits and
the specs to estimate that very few calls will be
too short.
 To find the proportion of calls that would be
expected to be too short, we find:
 P(X < 120) =
 =NORMDIST(120,194.133,20.493,1) = .00015
 Less than .02% would be expected to be too
short.

Suppose the manager wants to create an
ad telling people how quickly they can
provide a quote. The manager wants to
include a statement along the lines of:

“If you call us, our agent will provide you
with a quote in less than ________
minutes.”

What number should be placed in the
blank?
Graphically…
 Logically, the
be high enough that
“most” of the calls will
be handled in the
represents the calls
that are completed
time.                                               z
ˆ     ˆ
will depend on the
proportion of calls that   Where z is the value of z that
we want to complete        has an area  to the left and is
within the claim           found with
number of seconds.         =normsinv()
Process Capability
 To talk about capability of a process, we must have
stability
   Capability refers to the “voice of the process”
   The capability of the process tells the range of
values that can be expected for the measurements
of some process characteristic
   Specifications (specs) provide a “voice of the
customer.”
   Capability indexes are a fairly common way of
communicating the relationship between
specifications and process performance.
   Capability indexes attempt to compare the “voice
of the process” with the “voice of the customer.”
Two Approaches

 Approach 1:
 Comparison of Engineering Tolerances to
Natural Tolerances
 Engineering Tolerances refer to the specifications
for the characteristic
ET = USL - LSL
 Natural Tolerances refer to the natural process
limits for the characteristic
NT = UNPL - LNPL (where natural process limits
are calculated as 3 and  is estimated by
Rbar/d2 from a stable process)
 If NT < ET, we say the process is capable
 If NT < ET and the natural process limits are
within the specification limits, we say the
process is capable and meeting spec.
Capable but Not Meeting Specs???

When capability is described in terms of the amount
of variation (without looking at location), a process
with very little variation could consistently produce
unacceptable product.

Example:
Nails are sold by weight, but builders need to know how
many nails are contained in boxes of a given weight.
Suppose a builder specifies that each box of nails should
contain 990 to 1010 nails
(i.e., 1000 ± 10).

The producer has reduced the variation in the weight of
nails to the point where there is only a difference of 1 to 4
nails from one box to another—but, boxes actually contain
983 to 987 nails.
Capability (cont.)
 Approach 2:
 Capability Indexes
 Cp tells us if the natural variation is smaller than the
allowed variation. Cp does not look at process location;
therefore it is possible to have a ‘good’ Cp and be
making large amounts of unacceptable product.

 Cpk tells us if the natural variation is ‘small enough’ and
‘far enough’ from the specifications for most product to
meet specs for the characteristic. Cpk cannot be larger
than Cp.
Capability Formulas
USL - LSL              ET
Cp =                      =
6              NT

Cpk = min   {       USL - X
3     ,   X - LSL
3      }
If the process is centered between the specs:

Cpk =        X  nearest spec
3
Selecting the Appropriate Type
of Control Chart

 Since control limits are
calculated from data
collected from the process,
we need to know which
formulas to use!
Control Charts–Variables Data

 X-bar and R charts (used together)
 X-bar (the average of n observations)
attempts to
assess location
 R (the range of n observations) attempts to

 X and Moving Range charts (used together)
 When there is no logical grouping, individual
values are plotted on the X chart
 A moving range is used to assess spread
Control Charts–Attributes Data
 p or np charts
 "n" items are studied
 each item is classified in one of two categories
 we are counting the number in one of the
categories
 a p chart plots the proportion in one category
 an np chart plots the number in one category
 c or u charts
 an inspection unit (IU) is defined
 the number of occurrences are counted and
plotted
Determine
characteristic
to study
Counting or
counting          Measuring      measuring

Attributes Data                                   Variables Data

Classifying                                       How many
no
into two                                         items per
yes    categories?                                       subgroup?

one                    2 to 8

Constant                      Constant
subgroup                       area of
size?                      opportunity?          Consider       Consider
X/mR            X/R
yes                no
yes          no

Consider     Consider       Consider          Consider
np or p        p              c                 u
Examples
 Potholes per mile
 Time of delivery of the meal cart
at lunch at a nursing home
   Complaints per day (lunch meal) at a
nursing home
   1:1s in a mental health facility
   Confiscated items per day at Atlanta
Hartsfield Jackson Airport
   Late arriving flights per day by Delta at ATL
   Weight of contents of cans of tomato soup
Food Service

A nursing home serves meals to 100
patients each day. Meals are prepared in a
central kitchen facility, served onto trays,
placed in specially heat carts, and delivered
to the staff on the units for distribution to
patients. Lunch is scheduled to be served at
noon. In order to start to address
complaints about cold food being delivered
to patients, some data was collected.
The Data
Day of   Number of Time of Cart   Day of   Number of Time of
Cart
Week     Complaints   Delivery    Week     Complaints   Delivery
Mon.       39          11:30       Tues.     30          11:58
Tues.      31          11:35      Wed.       38          11:50
Wed.       30          12:00      Thurs.     24          12:20
Thurs.     27          12:15      Fri.       25          12:03
Fri.       38          11:45      Sat.       28          12:04
Sat.       33          11:32      Sun.       39          12:56
Sun.       35          12:00      Mon.       31          11:35
Mon.       26          12:15

Total number of complaints:
474
Complaints Data
(actually data on the # of “complainers”)

Day of   Number of            Day of    Number of
Week     Complaints           Week      Complaints
Mon.       39                 Tues.       30
Tues.      31                 Wed.        38
Wed.       30                 Thurs.      24
Thurs.     27                 Fri.        25
Fri.       38                 Sat.        28
Sat.       33                 Sun.        39
Sun.       35                 Mon.        31
Mon.       26

Total number of complaints: 474
Formulas
 Formulas:                    For this example:

CL np  np                           474
p         .316
1500
LCL np  np  3 np (1  p )     CL np  100(.316)  31.6
UCL np  np  3 np (1  p )     LCL np  31.6  3 31.6(1  .316)  17.65

total # non  confor min g   UCL np  31.6  3 31.6(1  .316)  45.55
p
total # inspected
np Chart for # of Complaints
# of Complaints
50

45

40
Complaints

35

30

25

20

15

Day of Week
Using Data to Generate Theories
Complaints vs. Delivery time

45
# of complaints

40

35

30

25

20
20   30     40         50          60        70   80

Number of minutes after 11:00
Revised Complaint Data
Day # Complaints# Meals Served   pi
1        39           95      .411
2        31          102      .304
3        30           87      .345
…         …            …      …
15        31          110      .282

Totals   474      1500
p chart for Complaints
 Formulas                   Point 1:
total # non  confor min g   CL p  .316
p
total # inspected                        .316(1  .316)
LCL p  .316  3                  .1729
95
CL p  p                                          .316(1  .316)
UCL p  .316  3                  .4591
95
p (1  p )
LCL p  p  3                     Point 2:
ni
CL p  .316
p (1  p )                         .316(1  .316)
UCL p  p  3                   LCL p  .316  3                  .1779
ni                                  102
.316(1  .316)
UCL p  .316  3                  .4541
102
p Chart for Complaints
Meal Complaints

0.5
0.45
0.4
Proportion

0.35
0.3
0.25
0.2
0.15

Day
1:1's in a Mental Hospital

When patients 'act-out' in ways that can be
hazardous to themselves or to others, they may
be placed on one-to-one (1:1). When a patient
is on 1:1 a staff member must remain within
arm's distance of the patient at all times. This
requires one staff member to have no other
duties other than the supervision of this one
patient. Obviously, this is very expensive.
In an attempt to improve service and
reduce costs, the number of 1:1's is being
studied. To start, data from the past quarter
were studied.
The Data (Day Shift)
Week   1:1’s          Week      1:1’s
1     23             10        18
2     18             11        19
3     27             12        17
4     35             13        16
5     10             14        14
6     19             15        21
7     24             16        11
8     20             17        14
9     21             18        15

Total number of 1:1's: 342
Formulas for c Chart
 Formulas:       For this example:

CL c  c          CL c  19
LCL c  19  3 19  5.923
LCL c  c  3 c
UCL c  19  3 19  32.077
UCL c  c  3 c
c Chart for 1:1s
1:1s for Day Shift
40

35

30

25
1:1's

20

15

10

5

0
1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18
Week
Improvement?
1:1s for Day Shift
40

35

30

25
1:1's

20

15

10

5

0
1   2   3   4   5   6   7   8   9   10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Week

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