Ops A La Carte Statistical Process Control (SPC) Seminar

&

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Ops A La Carte / (408) 654-0499 / askops@opsalacarte.com / www.opsalacarte.com

Statistical
Process
Control
Greg Swartz // (650) 274-6001 // gregs@opsalacarte.com

Ops A La Carte LLC // www.opsalacarte.com

The following presentation materials are
copyright protected property of
Ops A La Carte LLC.
These materials may not be distributed
outside of your company.

Presenter’s Biographical Sketch – Greg Swartz

◈Greg Swartz has worked successfully for over twenty years in the fields
of statistics and process improvement, as a Consultant and Trainer.
His consulting experience includes working with a number of Biotech,
high tech companies, Aerospace, and Defense. His expertise includes
analysis of technical data, a hands-on approach towards design of
experiments, and Failure Analysis, e.g. with Ops A La Carte for
Semiconductor Equipment. Greg was a Sr. Quality Program Manager
at Sun Microsystems for 6 years.

◈Mr. Swartz has worked in the fields of Applied Data Analysis (ADA)
techniques, yield improvement, quality assessments, and reliability
studies. Additionally, Greg has a background in software reliability,
CMM, and Software Product Life Cycle (SPLC).

© 2008 Ops A La Carte

Metrics and
Statistical Process Control

Level II for Operations,
Engineering and Research

Developed and Presented by:

Greg Swartz, CQE
Ops A La Carte

April 11, 2008

Gregory Swartz, © 2008 (650) 274-6001 Page 1

Metrics and
Statistical Process Control

Learning Objectives

Overview:
This Level II Statistical Process Control (SPC) course presents
a number of valuable tools to assist you in evaluating process
variation and to make sound decisions based on your data.
Topics covered included the following:

♦ Pareto Charts and Check sheets for Attribute and Visual Data
♦ Histograms for understanding variation in measurable data
♦ Variables and Attribute Control Charts including p Charts for
varying sample sizes
♦ Process Capability (Cp & Cpk) and Sample Size Determination
♦ Interpretation and Corrective Action including
Out-Of-Control guidelines
♦ Correlation and Regression Studies with
guard-banding techniques.

Learning Objectives:
Upon completion of this Metrics/SPC Level II course,
participants will be able to do the following:

♦ Construct p, NP, and C Charts for attribute process control
♦ Be able to construct Ave. and Range control charts for
variables data
♦ Construct 90 and 95% Confidence Intervals for process data.
♦ Distinguish between Process Control and Process Capability.
♦ Perform a Correlation Studies and interpret results.


Metrics/Statistical Process Control
Level II Content Outline
Chapter 1: INTRODUCTION TO SPC
• Benefits of Metrics and SPC
• SPC Tools — Overview
• SPC Implementation Strategy
Chapter 2: PROBLEM SOLVING TOOLS
• Cause and Effect Diagrams (Fishbone)
• Check Sheets
• Pareto Analysis using Excel with ExcelTM
Chapter 3: DESCRIPTIVE STATISTICS
• Measures of Central Tendency and Variation
• Histograms and Specification Limits
• SPC vs. Process Capability
Chapter 4: PROCESS CAPABILITY AND YIELD STUDIES
• "Central Limit Theorem"
• Cp and Cpk Indices — A practical approach
• “t” Test and Confidence Intervals in Excel
* Sample Size Determination
Chapter5: PROCESS CONTROL TOOLS FOR VARIABLES DATA
• X Bar & R Chart
• X Bar & S Charts (n>10) (for reference)
• Short Run Charting Techniques
Chapter 6: PROCESS CONTROL TOOLS FOR ATTRIBUTE DATA
• NP Charts and • P Charts (fraction defective)
• C Charts
Chapter 7: INTERPRETATION AND CORRECTIVE ACTION
• Interpreting Trends and Shifts in Data
• Planning Corrective Action
• Implementing Continuous Process Improvement
Chapter 8: CORRELATION AND REGRESSION

Appendix Terms and Definitions
Formula Summary


Metrics and Statistical Process Control

Chapter One: Introduction to SPC

• SPC is a tool that uses analytical techniques to:

Investigating Monitoring Improving

• SPC measures quality during the production
process, using statistics to determine and
maintain a state of process control in your area.

• SPC ensures that quality is built into the product
at each step, as shown in the overview process
flow of Genotyping.

Sample Combine Auto-
Receiving (VI) QC Assays caller Bioinformatics



Key Features of Implementing
Metrics and SPC

• Baseline Data

1st thing to do: catching abnormal variations
where special causes to problems can be
identified and corrected!

• SPC responds to trends by making changes
before reaching an out of control condition.
Emphasis is on prevention, versus after the fact.

• "Corrective action guidelines" are determined
statistically, and are commonly known as
Control Limits.

• Corrective Action Planning can be performed by
cross-functional metric improvement teams.



Benefits of Measuring your
process with SPC

Improved customer
satisfaction, both
internal and external

%
%
% %
Increased product yield
Failure rate = 1-yield %

Reduced operating costs

Improved product flow

Increased profits



Value Add of Statistical
Process Control

Process decisions are made based on
”Fact versus Opinion.”

Increases knowledge base regarding analysis of
your process data, in-process inspection, and
improves your out-going quality!

Improves long-term relationships between your
company, suppliers, and your customers.

Targets critical process, for product optimization
and Capability, for example, in meeting Six
Sigma criteria.

Allows sound decision making, using empirical
methods, versus opinions, or whims.



Types of Data – Flowchart

Raw Fabri-
Assembly Test
Materials cation

TYPES OF DATA

Variable Attribute

Process
Improvement
with
SPC!



SPC Implementation - Overview

Initially, Flowchart
Your Process

Variable Data i.e. measurable -
Identify Critical costs, cycle time, or response time
Product or Service or
Process Parameters Attribute Data e.g. categorical -
error types, PPM, defects by type

Independent
Causes
Dependent
Effect
Use Cause & Effect
Diagrams to Brainstorm
all Cause Variables

Use Pareto Charts
to Prioritize $
Key Problem Areas

Key Problem Areas

Continue
To Page 7



Continued Tools of Quality - (Con’t.)
from
Page 6

Do you
2
have one or Build a Scatter Y
two variables Diagram
?

1
X
Snapshot
Y
Do you
No
wish to display Construct a
your data Histogram #
over time
?

Yes Measurement Scale X

UCL
Plot data over time
on the chart, then Ave.
calculate controls.
LCL

Time / Date

Assign causes to
out- of - control points Monitor Charts for
with corrective action Improvement



Symbols Summary
Σ To Sum

X Individual Score

X Mean, or average

R Range (max-min)

σ or S Standard Deviation

UCL Upper Control Limit

LCL Lower Control Limit

K # of groups

n Subgroup sample size

p Proportion Defective

NP Number of Defects per sample

C Number of defects per unit or area

Cp Basic Capability Index

Cpk Capability Index (including process shifts)

t Used to determine yield with n < 30

Z Used to determine yield with large samples



SPC Tools Overview
Continuous Quality
Improvement

Need for Feedback
Data based on data

Type of
Data

Attribute Variables
Data Data

Check
Histogram Process Yield
Sheets Capability Improvement

Pie
Charts Two Yes Scatter-
Variables grams

Pareto
Charts No
Correlation

Run
Chart
Attribute
Charts

Control
Chart



Chapter Two:

Problem Solving Techniques

• Process Flow Analysis

• Cause and Effect Diagrams (Fishbone)

• Check Sheets

• Pareto Analysis with Excel Example



Problem Solving Tools Flow Chart

Start

Big Picture
Detailed Flowchart

Check Sheet

Pareto
Analysis

Determine NO Fishbone
root cause?? Diagram

YES

Take Corrective
Action

pp. 9-13



Process Flow Chart Exercise:

PROCESS FLOW STEPS A/V TYPES OF DATA



Cause and Effect Diagrams
(Ishikawa Diagram)

• Cause and Effect Diagrams can be used
for any service or product problem

• Serve as the basis for group discussion
and brainstorming

• Effect could be a quality, yield or
productivity problem

• Provide guidance for concrete
corrective action

People Equipment Methods

Effect
(Problem)
Causes (Independent Variables)

Materials Measurement Environment

24-29



How to Create a Cause and
Effect Diagram
1. Identify the problem (effect).

2. Brainstorm several causes — include all
ideas generated without evaluating causes.

3. Identify and circle a branch for corrective
action.

CAUSES EFFECT

pg. 24



Process Improvement Flow
“Plan, Do, Check, Act” PDCA Method.

Ca uses Effect

Ne ed No Take
More Data? Corrective
Action

Yes

Ca use Tally



Pareto Analysis (The 80-20 effect)
Errors in a process are categorical (attribute) in
nature where defects can easily be tallied with a
check sheet. Pareto charts display the 80-20
effect.

Key Advantages:

• When you identify the “vital few” you
improve your ability to identify the
root causes to the majority of the
problems.

• By solving the largest problem decreases
the overall percent defective product.

• Cost benefit of product can be
determined with the assistance of
Pareto Analysis.

• Solving major problems often
reduces or eliminates the minor
problems.

17-23



Procedure for Using Check Sheets for Pareto Charts

1. Rank causes by frequency of occurrence.

2. Calculate both percentage and Cum %.

3. Draw Pareto Diagram.

4. Concentrate corrective action on the "vital few."

Pareto Analysis Worksheet

Causes Tally Mark Freq. Rank %

Smear II 2

Color IIII I 6

IIII IIII IIII IIII IIII
Contaminatio IIII I 36

IIII IIII IIII IIII
M isc. 24
IIII

Misc.2 IIII
4
IIII IIII IIII IIII IIII IIII
IIII IIII IIII IIII IIII IIII 78
empty well IIII III
Totals 150



Procedure for Creating a Pareto Diagram in ExcelTM
1. Sort your cause categories so they are ranked (highest to
lowest).
2. Create an ordered Check Sheet as in page 14.
3. Tabulate both % and Cum. % as in the table below:

R a n k in g C auses Count F re q . (% ) C u m . (% )
1 S c r a tc h e s 77 5 1 .3 0 % 5 1 .3 0 %
2 M is a lig n e d 36 2 4 .0 0 % 7 5 .3 0 %
3 M is c . 25
4 W ro n g # 11
5
6

4. Use the mouse to block off the causes, Frequency in %,
and Cumulative %.

5. Use the Chart Wizard to create your Pareto
Chart (see below).

Partially Completed Pareto Exercise

0.6 100.00%
0.5 80.00%
0.4
60.00%
0.3
40.00% Freq. (%)
0.2
20.00% Cum. (%)
0.1
0 0.00%
Scratches Misaligned Misc. Wrong #



Procedure for Performing a
Cost Pareto Analysis
1. List all possible causes to problem.

2. Tally frequency for each cause.

3. Assign option $ cost value to causes (unit cost).

4. Rank causes by total cost for each category.

5. Derive the cumulative cost.

6. Concentrate corrective action on
the most costly cause(s).

p.20



Cost ($) Pareto Procedure in Excel:
1. Create a new table with your Unit Cost per defect type

2. Determine your total cost by multiplying Unit $ x Freq.

3. Rank your whole table by (total) Cost, and then Create a Cumulative Cost
Column

4. Swipe your mouse over the Causes, Cost, and Cumulative Cost.

5. Create a bar and line chart with the Chart Wizard

6. Comment on Leading Cost Issue and compare it with the Freq. Pareto

Freq. Unit Cost (total) Cost
2 $ 0.015 $ 0.030
6 $ 0.010 $ 0.060
36 $ 0.030 $ 1.080
24 $ 0.005 $ 0.120
4 $ 0.005 $ 0.020
78 $ 0.010 $ 0.780
150



SPC Tools Integration

1. Use Pareto Diagram to determine the
major causes of rejects.

1

1

2. Brainstorm possible causes for the biggest
problem.

3. Plan Corrective Action



Chapter Three: DESCRIPTIVE
STATISTICS

Concept Variation

Measures of Central Tendency

Measures of Variation

Histograms and Specification Limits

Coefficient of Variation (CV)



Variation defined by Cause
There are two types of causes of variation:

• Normal causes of variation result
from the problems in the system as
a whole.

• Abnormal causes of variation
result from special problems within
a system.

Normal cause Abnormal cause
common special
random non-random
systematic local
expected irregular
unidentifiable identifiable

One recommended method is to identify
abnormal causes of variation, first. And
then, to continually reduce variation by
effecting both abnormal and normal
causes.



Measures of Variation

Variation: Spread, dispersion or
scatter around the Central
Tendency

Range: Difference between the largest
and smallest value (Max. –
Min.)

Standard Deviation ( σ or S ):
A measure of the differences
around the average.

mean

-3 o -2 o -1o X +1 o +2 o +3 o
Normal Distribution Curve



The Standard Deviation (Sigma)

Sigma % of Distribution

X +/- 1σ 68 %

X +/- 2 σ (1.96) 95 %

X +/- 3σ 99.97 %

mean

-3 o -2 o -1o X +1 o +2 o +3 o



Histograms
A histogram graphically represents the
frequency of an attribute or variable, and
displays its distribution of data as a snap shot
representation. This visual format shows the
variability of your data and can be use for
further analysis, e.g. capability analysis.

160

140
N 120
u 100
m
80
b
60
e
r 40

20

0
60 65 70 75 80 85 90 95 100

Yield Percentages

pp. 36-43



Histograms vs. Specs

LS US LS US

1 5

2 6

3 7

4 8



Standard Deviation Exercise

Pick a sequence of seven numbers and list
them below in the left column. Units.
Determine the standard deviation.

2
X (X - X) (X - X)

Σ 0

Σ (X-X)2
Formula: σ= n



Standard Deviation for Attribute Data
Binomial Standard Deviation or σp =
p x (1 – p )
n
Where, p is the average fraction defective ∑ np
--------------
n is the average sample size Total N

Standard Deviation Attribute Exercise:

Given the following set of data, determine the standard
deviation of the fraction defective, and then create a 95%
confidence interval.

Sample Sample
# Size n np p = np/n
1 45 2
2 50 1
3 60 3
4 40 0
5 35 1
6 70 2
7 30 5
8 65 3
9 55 4
10 50 3
Totals 500 ∑ = 24 ∑=



Coefficient of Variation
The standard deviation depends on units as a
measure of variation. A comparison of relative
variation cannot be made using the standard
deviation, so a unitless (dimensionless) measure
called the coefficient of variation (CV) is used.

population standard deviation = σ

the population mean = μ

sample standard deviation = S
__
sample mean = X

Sometimes the CV is normally expressed as a
percentage. Then, the equation becomes:

S
CV % = 100 • X

The Coefficient of Variation (CV) can be used to
signal changes in the same group of data, or to
compare the relative variability to two or more
different sets of data. The larger the CV, the
greater its relative variability.



Coefficient of Variation for Variables Data

Procedure:

a. Determine the (Grand) Average of ROX at 2 Ul
b. Determine the Std. Dev. of ROX
c. Divide the Std. Dev. by the Average to obtain CV.
d. Now repeat procedure for the .5 ul group
e. Label each CV value
f. Compare the relative variability between the two groups.

2 ul .5 ul

Avg ROX % Genotyped Avg ROX % Genotyped
857.2 100 196.74 0
609.54 100 193.11 0 .5 ul
775.25 100 235.59 0 Ave. 471.18
742.79 100 174.13 0 Std. Dev. 52.61506
834.79 100 300.52 12.5
721.02 100 300.2 12.5 2ul
791.38 87.5 340.55 12.5 Ave. 731.602
834.03 37.5 192.79 0 Std. Dev. 69.75184
791.5 100 260.26 12.5
796.88 100 147.19 0 CV .5ul 0.111667
744.67 100 305.19 12.5 CV 2ul 0.095341
679.79 87.5 226.12 50
722.99 100 266.5 50 CV .5ul%
622.92 100 236.47 12.5 CV 2ul %
654.22 87.5 322.33 12.5
821.46 50 253.65 0
758.24 100 309.33 12.5
781.04 25 202.03 12.5
726.51 100 269.09 25
670.2 100 236.88 25
697.83 100 214.98 12.5
630.47 100 196.84 0
704.11 100 364.15 12.5
815.79 87.5 220.58 0
715.18 87.5 242.1 37.5
755.32 87.5 311.26 0
721.65 87.5 294.34 25
650.83 100 248.83 25
699.65 100 284.14 37.5
620.81 100 225.05 0



Attribute Coefficient of Variation Example

Now, let’s try applying this concept
to attribute data:
__
Group one: Sp1 = .00142, P1 = .007
__
Group two: Sp2 = .00178, P2 = .01

Which group has the larger variability? Hint: At
first you might be led to thinking that group two
has the larger relative variation…

CV1 = .00142/.007 x (100) =

CV2 = .00178/.01 x (100) =



Chapter Four: Process Capability,
Yield and Attribute Proportion Tests

This chapter includes the following key areas for
effectively performing process capability studies,
accurate yield determination, and testing for
single and two sample proportions:

• Central Limit Theorem

σx
• = σx
n

• X = X
i

• Cp and Cpk Indices

• Yield Determination

• Attribute Proportional Testing



Central Limit Theorem (CLT)
The CLT states that the average of individual
values (X) tends to be normally distributed
regardless of the individual (x) distribution.
Individual Data:
If you randomly selected 100 four-digit numbers
and charted them, the distribution will be
somewhat uniform. Each digit shows
approximately the same frequency.
10

8

6

4

2

0
0 1 2 3 4 5 6 7 8 9
Phone Digit

Chart the number 4186 by using one each of 4, 1, 8, and 6.
Mean = 4.5225 Standard Deviation = 2.8266



Averages Data

This histogram of average values shows the normal
distribution of the averages. For example, each digit
in the number 4286 would be averaged (4 + 2 + 8 +
6) = 20/4 = 5. Averages are distributed below:
Averages Bar Chart

27

24

21

18

(#) 15
12

9

6

3

0
1 2 3 4 5 6 7 8 9
Phone Digit Midpoint
Mean = 4.5225 Standard Deviation = 1.4133

Question: Can you now validate the Central
Limit Theorem with the above example?



Cp & Cpk: The Inherent
Capability of a Process
The Cp index relates the allowable spread of the
specification limits (USL - LSL) to the actual
variation of the process. The variation is
represented by 6 sigma.

USL − LSL
Cp =
6σ

If the tolerance width is exactly the same as the
6 standard deviations width, then you have a
Cp = 1.

1.333
LSL X USL
mean

-3σ -2σ -1σ X +1σ +2σ +3σ
12.5% 75% 12.5%



Cpk Defined
Cpk expresses the worst case capability index —
a process that is off-center.

Cpk also takes into account the location of the
process average.

Cpk = the smaller result of the following two
formulas:

USL − X X − LSL
C pu = or C pl =
3s 3s

Where:
Cpu = Upper Capability Index
and
Cpl = Lower Capability Index

pp. 64-66



Process Capability Indices
Example:
Your boss attended this statistical seminar and is
familiar with process capability indices. He or she
threatens to take away your new sports car if the
process capability indices (Cp) from the new oxide
manufacturing process is not greater than 1.0.

Tolerances = 250 to 400 μ

Average = 300 μ

sample S = 35 μ

Question: Will you be driving to work in
your old car tomorrow?



400 − 250
Cp = =.714
6(35)

6(35)
Cratio = =1.4
400 − 250

400 − 300
Cpu = =.95
3(35)

300 − 250
Cpl = =.476 = Cpk
3(35)

Your boss has taken away your new sports car.



Process Capability Index Exercise
Given the following specifications,
determine Cp and Cpk.

• Upper Spec. (US) = 100.0

• Lower Spec. (LS) = 24.0

• Mean (X) =

• Sigma =

Cp = _________

Cpk = _________



MEAS MIN MAX
26.81 24 100
26.67 24 100
26.9 24 100
27.04 24 100
26.63 24 100
26.92 24 100
26.73 24 100
26.8 24 100
26.94 24 100
26.85 24 100
27.54 24 100
27.22 24 100
25.84 24 100
25.77 24 100
26.93 24 100
26 24 100
26.96 24 100
25.79 24 100
27.04 24 100
25.98 24 100
27.48 24 100
27.03 24 100
26.92 24 100
27.69 24 100
26.83 24 100
25.38 24 100
25.87 24 100
26.81 24 100
27.36 24 100
27.3 24 100
26.73



Interpretation of Cp and Cpk Indices

Cpk < 1.00 Not Capable

Cp < 1

Cpk = 1.00 Barely Capable

Cp = 1

Cpk > 1.33 Very Capable

Cp = 1.33

Is the process truly capable of meeting the customer
requirements? _________

Why or why not?



Creating Confidence Intervals for Variables Data
The fish and game commission have been feeding robin yearlings a special bird seed.
Sample weights of 13 robins are listed below. What are the 95% and 99%
confidence intervals?

Procedure:

1. Determine the Mean and Std. Deviation of the data set.

2. Create Lower and Upper Confidence Intervals
based on the “t” values provided.

Data Set
95% Confidence Interval (Mean) (in Grams)
Mean = 12.5
Std Dev.= Tinv(95)= 2.178813 12.3
n= 13 Tinv(99)= 3.05454 12.7
Confidencence Limits 12.5
Lower= 0 Upper= 0 12.4
12.1
12.6
12.7
99% Confidence Interval (Mean) 12.2
Mean = 12.1
StdDev= 11.9
n= 13 12.3
Confidence Limits 12.6
Lower 0 Upper 0 Sum = 160.9

Question: Why are we using “t” scores versus
standardized “Z” scores?



Sample Size Determination for Means and Proportions
Determining a sample size for means.

The formula for determining a sample size for a mean is

Ζ 2σ 2
η=
(χ − μ ) 2

The Ζ -value depends on the level of confidence required. Remembering that:

A 99 percent confidence results in a Ζ -value of 2.58.



σ is the standard deviation or variance.

χ −μ is the difference between the sample mean and the population mean referred
to as the error.

Sample Size Determination

Z-Value = 2.576321008
Std. Dev.= 0.75
error= 0.15
sample size= 165.9357483



Determining a sample size for proportion:

The formula for determining a sample size for a proportion is

n =
Ζ 2
(
p 1− p )
(ρ − p )2

The Ζ -value depends on the level of confidence required.

p is the population proportion if known. If the proportion is not
known, π is assigned a value of .5

ρ − p is the difference between the sample proportion and the
population proportion referred to as the error.

The Easy technique for determining sample “n”:

np > 5



Scenario: You have been selected as the “improvement Expert”
in your lab to determine the appropriate “n” size per sample after
your team has determined an average failure rate of .035. Due
to new equipment in the lab an initial confidence level of 95% is
selected, and degree of precision (error) @ .02.

Procedure:

1. On a worksheet, key in the following information:

Sample Size Determination - Proportion

Z-Value=
Pop.Prop.=
error=
sample size= #DIV/0!

2. In cell B3, input the Z value for 95% Confidence

3. In cell B4, input the failure rate

4. In cell B5, input the error.

5. In cell B6, key in =B3^2*B4*(1-B4)/B5^2

Questions:



1. What is the required random sample size for a degree of
precision of .05?
2. What sample size is required for the same precision, with
99% confidence?

Chapter Five: Process Control Tools
for Variables Data

♦ X Bar and R Charts

♦ X Bar and S Charts (n>10) for reference

♦ Short Run SPC Charting Technique



Control Limits
• Help define acceptable variations
of the process.

• Are calculated and represent true capability
of the target process, or where baseline
metrics have been implemented.

• Can change in time as the process improves.

UCL

X
LCL
1 2 3 4 5 6 7 8 9 10 11 12
Time or Sample Number

General Rule: Don’t apply specification
limits on control charts.



X and R Control Chart

UCL x
M
E
A
S
U
R
E
x
M
E
N
T LCL x

UCL R

R



Control Limits vs. Spec. Limits

Control limits monitor the performance of the
process.
y
UCL
X
Measure

X

LCL
X
X
1 2 3 4 5 6 7 8 9 10
time or sample number -->

Spec. limits monitor the quality of the product as to
the individual distribution below:

X

LS US



Short Run SPC

The Short Run Individual X and Moving Range
Charts can be applied to the following:

• Low production volume

• Temperature, humidity, concentration of
solutions

• When data must be obtained at the end of a
reporting period (per quarter, month, day)

• When the testing is costly or time consuming



X & R Charts

Control Chart Plotting Procedure:

1. Accurately measure the required number of
readings for the lot.

2. Calculate the mean. (Add readings together and
divide by the number of readings.)

3.Calculate the range. (Subtract lowest reading
from the highest reading.)

4. Plot both the mean and range on the SPC chart.
Log the lot number and date.

pg. 59



Example Data /Analysis for Control

Date MEAS 1 Meas. 2 Meas. 3 Ave. Grand AveUCL LCL
1/18/2007 0:00 0.3637 0.3663 0.2118
1/19/2007 0:00 0.1322 0.426 0.2178
1/20/2007 0:00 0.09442 -0.02428 0.02284
1/21/2007 0:00 0.3333 0.1105 0.2807
1/22/2007 0:00 0.04403 0.2663 0.02492
1/23/2007 0:00 0.4842 0.1715 0.0816
1/24/2007 0:00 0.07829 0.1304 0.1919
1/25/2007 0:00 -0.04909 -0.09284 -0.2375
1/26/2007 0:00 0.1948 0.4446 -0.02368
1/27/2007 0:00 0.1614 -0.1326 0.2387
1/28/2007 0:00 -0.206 0.0127 0.2065
1/29/2007 0:00 0.0201 0.1632 0.2199
1/30/2007 0:00 0.04176 0.1323 0.2523
1/31/2007 0:00 0.338 0.09527 0.9097
2/1/2007 0:00 0.2842 -0.05588 -8.97
2/2/2007 0:00 -0.1014 0.04255 0.07366
2/3/2007 0:00 -0.2253 0.3117 0.2042
2/4/2007 0:00 6.543 0.2073 0.000886
2/5/2007 0:00 10.03 -0.1436 9.883
2/6/2007 0:00 0.2127 0.1612 0.4555
2/7/2007 0:00 0.4352 0.1162 0.1387
2/8/2007 0:00 0.744 0.2604 0.5681
2/9/2007 0:00 0.1054 0.2471 0.04124
2/10/2007 0:00 -0.2962 0.05815 0.6354
2/11/2007 0:00 0.4714 9.732 0.2281
2/12/2007 0:00 0.2151 0.0752 0.2977
2/13/2007 0:00 0.2146 0.6519 0.6632
2/14/2007 0:00 0.3294 0.7231 0.1349
2/15/2007 0:00 0.7159 0.2251 0.3108
2/16/2007 0:00 0.5853 0.4141 0.2791



Average Control Chart using 2 Sigma Limits

Below is an Average Control Chart using the
data from the previous page. Limits were
generated in Excel at the 95% confidence
interval using 1.96 Sigma + Grand Average.

Control Chart of Plate Data w ith 2 Sigma Limits

180.0
175.0
170.0
165.0 Average
160.0 UCL
155.0 LCL
150.0 Grand Ave.
145.0
140.0
135.0
4

4

4

4

4

4

4
04

04

04

00

00

00

00

00

00

00
20

20

20

/2

/2

/2

/2

/2

/2

/2
4/

6/

8/

10

12

14

16

18

20

22
1/

1/

1/

1/

1/

1/

1/

1/

1/

1/

Interpretation: There is good reason with the
above data set to consider implementing 2 Sigma
Control Limits as shown. In this case, data point on
1/09/04 fell just outside the Upper Control Limit.

Do you think 3 Sigma Limits would have caught the
abnormal cause?



Factors and Control Limits

Shewhart Factors

n 2 3 4 5 6

D4 3.268 2.574 2.282 2.115 2.004
D3 0 0 0 0 0
A2 1.880 1.023 0.729 0.577 0.483
d2 1.128 1.693 2.059 2.326 2.534

Control Limit Formulas

UCL X = X + (A2• R)

LCL X = X − (A2• R)

UCL R = RD4

LCL = RD 3
R



Exercise - Short-Run Control Charts

Key Points for Plotting the X (individual)
Control Charts:

• X is the individual measurement to be plotted.

• X is the average of the individual plot points.
This becomes the center line for the control
chart.

• UCL is the Upper Control Limit and is
calculated by: UCL = [ Average + (2 x σ) ]

• LCL is the Lower Control Limit and is
calculated by: LCL = [ Average - (2 x σ) ]



Creating a Short-Run Control Chart in ExcelTM

1. Arrange your data from left to right as seen in table below.

2. Assign a # or date for the individual data being
collected (see table below).

3. Calculate the average of your data with the function wizard and
create separate rows repeating the average across all data points.

4. Determine the Standard Deviation (σ ) with the function wizard.

5. Calculate the Upper & Lower Control Limits (UCL & LCL) by
multiplying the Standard Deviation times 2, and then both add and
subtract the product from the [X ± (2 xσ )]

6. Repeat the Control Limits across all data points.

7. Use the mouse to block off date, data, average, & control limits.

8. Use the Chart Wizard to create your Control Chart (see below).

9. Interpret Control Chart for shifts, trends, or out-of-control points.

pp. 51-63



Variables Data Excel Exercise

1. Determine Averages across date or assay type
2. Create Upper Control Limit = Ave. plus 1.96 Std. Dev.
3. Create Lower Control Limit = Ave. minus 1.96 Std. Dev.
4. Create 3 additional columns for UCL, LCL, and Ave.
5. Swipe Mouse over Dates, Averages, UCL, LCL, and Average
6. Use chart wizard to create a multiple line chart
7. Include Interpretation Section for Out-Of-Control points
Date Phred 20 Ave. UCLx LCLx
5/1/2007 351
5/2/2007 375
5/3/2007 368
5/4/2007 364
5/5/2007 321
5/6/2007 289
5/7/2007 325
5/8/2007 366
5/9/2007 378
5/10/2007 347
5/11/2007 339
5/12/2007 335
5/13/2007 389
5/14/2007 348
5/15/2007 354
5/16/2007 368
5/17/2007 356
5/18/2007 392
5/19/2007 373
5/20/2007 352
Sum=
Ave. =
Std. Dev.=

Questions:
1. Since the above Phred scores are individual readings, what
might be a realistic lower specification limit?
2. What degree of confidence in % have you created with your
control limits?


Now: Let’s try this with another example with min and
max specifications:

Date MEAS MIN MAX
7/23/2007 26.81 24 100
7/24/2007 26.67 24 100
7/25/2007 26.9 24 100
7/26/2007 27.04 24 100
7/27/2007 26.63 24 100
7/28/2007 26.92 24 100
7/29/2007 26.73 24 100
7/30/2007 26.8 24 100
7/31/2007 26.94 24 100
8/1/2007 26.85 24 100
8/2/2007 27.54 24 100
8/3/2007 27.22 24 100
8/4/2007 25.84 24 100
8/5/2007 25.77 24 100
8/6/2007 26.93 24 100
8/7/2007 26 24 100
8/8/2007 26.96 24 100
8/9/2007 25.79 24 100
8/10/2007 27.04 24 100



Control Chart Tools Overview

Data

Yes/No
Measurable Good/Bad
Pass/ Fail

Variable Data Attribute Data

Defects Defects
Unlimited Limited

X/MR
X/R Chart X/S Chart c Chart u Chart p Chart np Chart
Chart

Sample Sample Fixed Variable Variable Fixed
size less size more Individuals Sample Sample Sample Sample
than 7 than 6 Size Size Size Size



Chapter Six: Process Control Tools
For Attribute Data

NP Charts - # of defective in a sample
(sample size is constant

P Charts - fraction defective
(sample size can vary

C Charts - # of defects per unit

SPC Charting Guidelines



Attribute Control Charts
Attribute Control Charts consist of primarily three
basic types of charts following the binomial and
poisson distributions:

• np Charts - used for monitoring the # of
defects per sample when the sample size is
constant, for example, n = 50.

• p Charts - can be used either with a constant
sample size or variable sample (n) size.
(variable control limits or average control
limits may be imposed)

• c Charts – is applicable for the number on
defects per sample unit, e.g. # of defects on a
car. Sample unit size is constant.

• u Charts – is used in the same way as a
c Chart, but the sample unit size may vary.



p Chart Formulas NP Chart Formulas

(
p 1− p )
UCL p = p + 3.
n (
UCLnp = np + 3. np 1 − p )
(
p 1− p )
LCL p = p − 3.
n (
LCLnp = np − 3. np 1 − p )
C Chart Formulas

UCLc = c + 3. c

LCLc = c − 3. c



Benefits of an “Attribute P Chart”

Allows for accurate monitoring of fraction defective.

Control Limits act as guidelines when your process is
producing bad product.

The average fraction defective is a good indicator of
“Failure Rate.”

Attribute P Chart Procedure
1. Determine fraction defective for each sample in adjacent
column

2. Calculate the average fraction defective (Ave. p) into
additional column

3. Determine the Std. Dev. Of the proportion defective.

4. Create Upper and Lower Control Limits based on 1.96 Sigma

5. Drag mouse over p, Ave. p, UCLp, and LCLp

6. Create multiple line chart in Chart Wizard

7. Interpret Results and comment on Outliers



P Chart Exercise with variable sample sizes in Excel

Instructions: Using the data set below with varying sample n, construct a P Chart in
Excel, using +/- 2.58 standard deviation limits.

Question: What confidence Interval am I generating?

sample n np (defects np/n =p Ave. p UCLp LCLp
1 50 2 0.040
2 35 4 0.114
3 45 3 0.067
4 65 5 0.077
5 75 1 0.013
6 35 3 0.086
7 45 2 0.044
8 75 3 0.040
9 50 2 0.040
10 45 5 0.111
11 58 8 0.138
12 25 5 0.200
13 40 3 0.075
14 60 1 0.017
15 80 0 0.000
16 65 1 0.015
17 46 4 0.087
18 50 3 0.060
19 25 4 0.160
20 85 5 0.059
Totals 1054 64

Average P 0.060721

Questions:

1. Is the average fraction defective a good indicator of the failure rate?

2. What processes would lend themselves to p charts in your lab areas?



Process Control Tools
Overview Flowchart
Data

Attribute Variable

Display Display
Data Over Data Over
Time? Time?

No Yes No Yes

Check Data X and MR
P, NP, or
Sheet Collection Run Chart
C Charts Sheet

_
Pareto X and R
Chart Histogram
Control
Chart

Pie Process
Chart Capability
Tools



Chapter
Seven: INTERPRETATION &
CORRECTIVE ACTION

• Interpreting Trends and Shifts in Data

• Planning Corrective Action

• Implementing Continuous Process
Improvement



Control Chart Interpretation

• Detecting "Out-of-Control" Conditions

• Assigning Causes to Problems

• Guidelines for Control and Stability

Corrective Action

• Assigning Causes to Problems

• Selecting SPC Tools

• Corrective Action Plan

• SPC Report Form



Detecting Out of Control Conditions

Bonnie Small's guidelines for interpreting control
chart data

• Points beyond the control limits usually
indicate:
- The process performance is sporadic
- Measurement has changed (inspector, shift,
gage, etc.)

• Runs indicate a shift or trend. Runs include:
- 7 points in a row on one side of the average
- 7 points in a row that are consistently
increasing or decreasing

• Non-random patterns may indicate:
- The plot points have been miscalculated or
misplotted.
- Subgroups may have data from two or more
processes



Determine whether Bonnie Small rules were
broken:

• One average (mean) above or below control
limit.

• Seven consecutive averages (means) above or
below the center line.

• A trend of seven consecutive points in an
upward or downward trend.

Now, take corrective action as follows:

1. Circle the point or group of points

2. Comment on the cause(s) of the unstable
point(s).

3. Detail Corrective Action Plan.



Taking Corrective Action

• Implementing change in the process

• Identify key problem area(s)

• Determine root cause(s)

• Document causes and Corrective Action

• Implement SPC Team Action Plan



SPC Report Form

Name: Date:

Department: Extension:

Statement of the Problem:

Corrective Action Objective:

Method:

Results: (attach charts, data analysis to form)

Corrective Action/Recommendation:



Chapter
Eight: Correlation and Regression

Procedure for Creating a Scatter Diagram in ExcelTM

Arrange your paired (X and Y) data in table format.

Assign a # for each pair of data being collected (see table below).

Conc. % Genotype
1.50 72
1.00 65
2.50 87
1.00 63
3.00 92
4.00 95
1.00 60
2.00 80
1.50 68
3.00 90

Use the mouse to block off the X and Y data columns.

Use the Chart Wizard to create your Scatter Diagram.



Appendix: Terms & Definitions:
Acceptance Criteria -the amount of acceptable rejects before a lot
will be rejected based on the sample. Used in sampling plans as the
criteria for passing or failing a lot of items inferred from the sample.

Acceptable Quality Level (AQL) - a coordinate point for the fraction
defective on the x axis of the Operating Characteristic Curve of an
attribute sampling plan. This point is the region of good quality and
reasonably low rejection probability - 5% alpha error.

Accuracy - how close a measurement comes to its actual value. In a
particular process, accuracy could be a function of calibration. See
Precision.

Alpha Error - the probability of error in making an assumption
incorrectly. In sampling plans, it is the probability of rejecting a lot
which is truly good. In Control Charts, it is the assumption that a
process point is out-of-control, when in fact it is not, and is due to
statistical chance alone. Therefore, the smaller the alpha error in any
case, the more confidence there is in the result(s) we‘ve obtained.

Analysis - implies some conclusion based on statistical results in
order to interpret some meaning from the statistical test(s) performed.
Interpretation.

Ambient -certain intervening variables in a environment that have
some effect on the result being measured. Generally, ambient
variables or factors in an industrial environment are those which are
not wanted, such as dust particles, temperatures, or sources of light.

Arithmetic Average - the mean of the distribution. It is a measure of
Central Tendency indicating the center weight of a distribution of
scores.

Assignable Causes - those causes to problems which are sporadic in
nature and not due to statistical chance alone. Assignable causes can
be assigned a reason as to why that problem point exists. Usually,
points outside of control chart limits are associated with an
assignable cause and this cause can be identified.

Attribute Data - qualitative data based on the absence or presence of
a characteristic, usually determined by a specification. Common
types of attribute data would include: go no-go data, pass-fail,


accept/reject, yield/reject. Attribute data is based on binomial
population of mutually exclusive events designated by P and Q= (1-P).

Average Outgoing Quality (A.O.Q.) - based on the fraction defective
(P) and the probability of acceptance (PA) for that fraction defective.
Also takes into account the characteristics of an attribute sampling
plan, that is, its sample size and decision criteria. A.O.Q. = P.A. x P.

Average Outgoing Quality Limit (A.O.Q.L.) - the threshold point on
the A.O.Q. curve. It is the worst possible case outgoing quality, and is
generally derived from the area of indifference off the Operating
Characteristic Curve.

Awareness - attention to the relationships between quality and
productivity. Directing this attention to the requirement for
management commitment and statistical thinking leads toward
improvement.

Beta Error - In sampling plans, beta error is associated with the
L.T.P.D. point and implies a 10% risk in accepting a lot which is truly
rejectable. In hypothesis testing, it is the error made in rejecting an
alternative hypothesis when in fact, it is true. In control charts, beta
is the error made in assuming the process is in control when in fact, it
is not.

Bimodal Distribution - a distribution having two modes depicted by
two distinctive humps in the curve. The presence of two frequently
occurring scores, or groups of scores is noticeable.

Binomial Distribution - A discrete probability distribution for
attributes data that applies to the conformance and non conformance
of units. This distribution also is the basis for attribute control charts
such as p and np charts.

Capability - whether or not product is truly capable of conforming to
specifications. This capability can only be determined after the
process is in statistical control. A process may be defined as being
truly capable when the aim of the process is well centered and the
variance or spread of the process on an individual unit basis does not
exceed the specification limits.

Cause and Effect Diagram - a simple tool for individual or group
problem-solving that uses a graphic description of the various process
elements to analyze potential sources of process variation. Also called



a fishbone diagram (because of its appearance) and developed by
Ishikawa.

Capricious Data - the natural occurring chaos in all things, or the
unexpected results one derives from attempting to sort out dirty data,
like sudden shifts or abnormal changes.

Central Limit Theorem (C.L.T.) - when collecting a distribution of
averages or subgroup scores, the distribution will tend to centralize
around the center value. The distribution will be evenly distributed
about the mean or average. This is true if the averages are sampled
from an abnormal distribution (skewed, bimodal, etc.).

Control - in Statistical Quality Control, control means to get a
handle on the process and be able to manipulate it in a desirable
fashion.

Control Charts - a tool one uses to visualize a particular process
over time and/or across units. It is a way to graphically represent a
parameter in an unbiased manner. The various types of control charts
are as follows:

C Charts - used to depict the number of defects per unit. For
example, the number of defects per automobile. An average number
of defects per automobile can also be obtained - (C bar).

P Charts - used when the Percent or Fraction Defective is
graphically desired. It depicts the fraction defection per sample, and
an average can be obtained.

NP Charts - used to the depict the number of defects per
sample. Similar to a C Chart, NP easily counts the number of defects
which makes charting fairly simple. The main requirement for a NP
Chart is the sample size must remain constant.

R Charts - used to monitor the range variation when collecting
averaged or subgroup data. Usually seen in conjunction with an X
Bar Chart, the range chart gives information to the variance of a
process over time, across units, or across samples.
S Charts - similar to R charts and measure the process variation via
the sample standard deviations. The S Chart is especially applicable
with larger sample sizes.



X Bar Charts - used to monitor variables data (continuous variables)
over time. Generally, X Bar Charts, graphically represent averages or
groups of data over time. They serve as a good indication of any
process which has been identified as a problem area or for
monitoring purposes.

Control Limits - c the boundary lines set up on any control chart for
the purpose of determining whether a process is in or out of control.
Typically, the area between the control limits account for 99.7% of the
distribution of scores making up the control chart. When control
limits are set plus and minus three sigma (standard deviations), it
will accommodate again 99.7% of the distribution.

Control Limits for Averages - when taking average or subgroup
data, these limits are used for averages on an X Bar Chart. They also
serve as a boundary parameter for a majority of the scores being
marked on the chart (99.7%), but in this case it applies for averages
and not individual scores.

Control Limits for Individuals - also known as the natural process
limits help determine, with 99.7% confidence, where the expected
process will go. Because these limits are for individual scores, they
assist in determining the yield for a particular process.

Cost-Effectiveness - The reduction of quality costs, such as rework,
and waste, makes any operation more cost-effective. By being cost-
effective, savings and efficient operations will ensue. Quality is really
free, it only cost money when you don’t have it.

Fault-Tree Analysis - is a brainstorming and communication tool in
order to figure out all the possible causes to any particular yield,
productivity, or quality problem. This tool uses a fish-bone diagram
to analyze all the possible causes to an identified problem in the
categorized areas of People, Equipment, Specifications, Flow, Raw
Materials, and Measurement.

Kurtosis - Refers to the height of a distribution of scores. Platykurtic
means a flat and very dispersed distribution, whereas leptokurtic
means a tall and very tightened distribution.

L.T.P.D. - Lot Tolerance Percent Defective. Let Them Pay Dearly.
This particular defective level is guaranteed with 90% confidence of
meeting the plan, and a 10% Beta Error or probability of rejection. See
Beta Error.



Mean - arithmetic average.

Measure - the dictionary defines measure as the dimensions,
quantity, or capacity of anything ascertained by a scale or by the
variable condition. In S.Q.C., measure could be a reference standard
or sample used for the quantitative comparison of properties.

Median - is the middle score when the scores are ranked from highest
to lowest or lowest to highest. When the median is resolved half of the
scores will be on one side, and the other half will be on the other side.

Methodology - the systematic way in which an application is
addressed to a problem. S.Q.C. methodology involves a logical
approach with statistical tools to effectively solve problems.

Midpoint - in reference to cell intervals, it is the middle point of any
particular cell.

Modified Control Limits - are generally performed when the process
is well within the Specification Limits, and both the upper and lower
specification limits are outside the natural limits of the process.

Mode - a measure of central tendency indicating where the most
frequently occurring score or group of scores lies in a distribution.

Motivation - the impetus influencing the use of S.P.C. to its
maximum potential.
Participation.

Normal - a continuous, symmetrical, bell-shaped frequency
distribution for variables data which is the basis for control charts for
variables. The mean, median, and mode are approximately the same,
and a standard deviation (S) exists where plus and minus one S =
68%, plus and minus two S = 95%, and plus and minus three S =
99.7% which is a standard setting for control charts limits.

Pareto Chart - A simple tool for problem-solving that involves making
all potential problem areas or sources of variation. Pareto was an
Italian economist who resolved that a majority of the wealth resides in



a few elite or upper class. In relation to a process, this means a few
causes account for most of the cost (or variation).

Poisson Distribution - Another discrete probability distribution for
attributes data used as an approximation to the binomial. It can be
used when p<.1 and np<5. It is the basis for C charts using
attributes data.

Prevention - a strategy for maintenance of a process. This implies an
awareness of potential problems that can occur in the process and to
act on those problems before an “out-of-control” situation happens. A
preventative maintenance program (PM).

Process - a series of events leading to a desired result or product. A
process can involve any part of a business.

Process Control - having a process behave under an expected
frequency of occurrence or within the limits which have been
statistically derived. It is a state in which all the points fall in and
around the average in a random manner and very few of these
approach the limits of the distribution.

Quality - usually determined by the customer, quality is a current
issue today that challenges U.S. companies to surpass its
competition. Quality gives a product a characteristic of customer
satisfaction. If we care for good quality we should have the priority of
pleasing our customer.

Randomness - the state of collecting individual data values without
any expected frequency or basis. They may become defined once a
distribution is perceived.

Range - the difference between the minimum and maximum score.

Sample - a known quantity designated by (n) or the size of the
sample. It is randomly pulled from a population parameter in order to
provide statistical data.

Statistics - derived from a sampled population, the information is
arranged to make interpretation of the data easy and to infer
something about the population from the
sample which has been randomly drawn.



Special Cause - cause attributable to an assignable item off the x axis
of a control chart. Special Causes are People, Machine, Materials, etc.

Specification - These may be quality specs. or product specs. They
are set by engineering or determined by the demands of the customer,
keeping in mind Deming’s philosophy: “The customer is King”.

Spread - variability in a distribution of data. Can also be thought of
as the dispersion of data around the measures of Central Tendency
such as the mean.

Stable Process - a process which is under statistical control as well
as lacking in assignable or special causes of variation.

Standard Deviation - the main statistic to measure the spread or
dispersion of a distribution or of a process when applied with the use
of Control Charts.

Student’s t Distribution - used when the sample size is less than 50
or the variance of the distribution is unknown. This distribution
compensates for smaller sample sizes, and is used primarily for mean
comparisons or process capability studies.

Type I Error - see Alpha Error.

Type II Error - see Beta Error.

Variables Data - continuous data obtainable via measurable results
such as dimensional data (heights, widths), or electrical data
(resistance, current).

Variation - the degree of change in the spread of a distribution of
scores. Many things built by man and nature have some inherent
natural variability. This variation shows up graphically in a
distribution of scores.


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