Statistical Quality Control
by R.B. Clough – UNH
M. E. Henrie - UAA
Statistical Quality Control
The Right Statistical Tools
Statistical Quality Control
Basic Statistics

Descriptive Statistics


A straightforward presentation of facts. A
survey or summary of a population in
which all data are known.
Inferential Statistics

Drawing conclusions about a population
from a random sample
Statistical Quality Control
Inferential Statistics

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Inferential statistics is a valuable tool because it allows us to look at a
small sample size and make statements on the whole population.
Samples must be pulled RANDOMLY from a population so that the
sample truly represents the population. Every unit in a population
must have a equal chance of being selected for the sample to be truly
random.
The distribution or shape of the data is important to know for analytical
purposes.
The most common distribution is the bell shaped or normal distribution.
Parameters can be estimated from sample statistics. Two of the most
common parameters are the mean and standard deviation.
The mean (or average, denoted by μ) measures central tendency
This is estimated by the sample mean or xbar.
The standard deviation (σ ) measures the spread of the data and is
estimated by the sample standard deviation
Three SQC Categories


Statistical quality control (SQC) is the term used to describe
the set of statistical tools used by quality professionals
SQC encompasses three broad categories of;

Descriptive statistics
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e.g. the mean, standard deviation, and range
Statistical process control (SPC)
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Involves inspecting the output from a process

Quality characteristics are measured and charted

Helpful in identifying in-process variation
Acceptance sampling used to randomly inspect a batch of goods to
determine acceptance/rejection
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Does not help to catch in-process problems
Statistical Quality Control
Sources of Variation
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Variation exists in all processes.

Variation can be categorized as either;
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Common or Random causes of variation, or

Random causes that we cannot identify

Unavoidable

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e.g. slight differences in process variables like diameter, weight, service
time, temperature
Assignable causes of variation

Causes can be identified and eliminated
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e.g. poor employee training, worn tool, machine needing repair
Statistical Quality Control
Traditional Statistical Tools
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Descriptive Statistics
include
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n
The Mean- measure of central
tendency
x
The Range- difference
between largest/smallest
observations in a set of data
Standard Deviation
measures the amount of data
dispersion around mean
Distribution of Data shape


x
i 1
n
 x
n
σ
Normal or bell shaped or
Skewed
Statistical Quality Control
i
i 1
i
X
n 1

2
Distribution of Data

Normal distributions

Skewed distribution
Statistical Quality Control
SPC Methods-Control Charts
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Control Charts show sample data plotted on a graph with CL,
UCL, and LCL
Control chart for variables are used to monitor characteristics
that can be measured, e.g. length, weight, diameter, time
Control charts for attributes are used to monitor characteristics
that have discrete values and can be counted, e.g. % defective,
number of flaws in a shirt, number of broken eggs in a box
© Wiley 2010
Analysis of Patterns on Control Charts
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When do you have a problem with your process?
One or more points outside of the control limits
A run of at least seven points (up, down or above or
below center line)
Two or three consecutive points outside the 2-sigma
warning limits, but still inside the control limits
Four or five consecutive points beyond the 1-sigma
limits
An unusual or nonrandom pattern in the data
From Douglas C. Montgomery “Introduction to Statistical Quality Control”
Setting Control Limits
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Percentage of values
under normal curve
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Control limits balance
risks like Type I error
Statistical Quality Control
Hypothesis Tests
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Results of hypothesis tests fall into one of
four scenarios:
Type I Error
OK
OK
Type II Error
Statistical Quality Control
Type I and Type II Error
ART and BAF
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Type I - ART (Alpha, Reject Ho
when true)
Type II - BAF (Beta, Accept Ho
when false)
Statistical Quality Control
Jury Trial vs. Hypothesis Test
Jury Trial
Hypothesis
Test
Assumption
Defendant is
Innocent
Null hypothesis
is true
Standard of Proof
Beyond a
reasonable doubt
Determined by

Evidence
Facts presented
at trial
Summary
statistics
Decision
Fail to reject
assumption
Fail to reject H0
(not guilty)
or
or
Reject H0 in favor
reject (guilty)
of Ha
© Wiley 2007
Context?
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What does it mean to make a type I error here?
 Convict an innocent person of a crime.
What does it mean to make a type II error?
 Fail to convict a guilty person.
What do we usually say about type I and type II
error rates in this context?
Control Charts for Variables
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Use x-bar and R-bar
charts together
Used to monitor
different variables
X-bar & R-bar Charts
reveal different
problems
In statistical control on
one chart, out of control
on the other chart? OK?
Statistical Quality Control
Control Charts for Variables
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Use x-bar charts to monitor the
changes in the mean of a process
(central tendencies)
Use R-bar charts to monitor the
dispersion or variability of the process
System can show acceptable central
tendencies but unacceptable variability or
System can show acceptable variability
but unacceptable central tendencies
Statistical Quality Control
Graphical Analysis
“A picture is worth a thousand words.”
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Graphical analysis is the first step in
analyzing your data. Examples:
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Distribution (histogram, dotplot,
boxplot)
Time Series plot for trending
I-chart (for Individual data points)
Normality
Cpk (when applicable) graph (Minitab)
Statistical Quality Control
Dotplot of Tensile Test Data
Dotplot of CONTROL
50
55
60
65
CONTROL
70
Statistical Quality Control
75
80
Time Series Plot
Time Series Plot of CONTROL
80
75
CONTROL
70
65
60
55
50
1
7
14
21
28
35
42
Index
49
Statistical Quality Control
56
63
70
Individuals (I) Chart
I Chart of CONTROL
85
UCL=82.74
80
Individual Value
75
_
X=71.72
70
65
LCL=60.69
60
1
1
55
1
1
1
50
1
8
15
1
1
22
29
36
43
Observation
50
57
Statistical Quality Control
64
71
Normal Probability Plot
Probability Plot of CONTROL
Normal
99.9
Mean
71.72
StDev
6.485
N
73
AD
7.366
P-Value <0.005
99
Percent
95
90
80
70
60
50
40
30
20
10
5
1
0.1
50
60
70
CONTROL
80
Statistical Quality Control
90
Cpk Graph
(Minitab)
Process Capability of CONTROL
LSL
USL
Process Data
LSL
65
Target
*
USL
85
Sample Mean
71.7151
Sample N
73
StDev(Within) 3.67538
StDev(Overall) 6.4853
Within
Overall
Potential (Within) Capability
Cp
0.91
CPL 0.61
CPU 1.20
Cpk 0.61
Overall Capability
Pp
PPL
PPU
Ppk
Cpm
54
Observed Performance
PPM < LSL 123287.67
PPM > USL
0.00
PPM Total 123287.67
60
Exp. Within Performance
PPM < LSL 33846.99
PPM > USL
150.42
PPM Total 33997.41
66
72
78
Exp. Overall Performance
PPM < LSL 150234.16
PPM > USL 20257.02
PPM Total 170491.18
Statistical Quality Control
84
0.51
0.35
0.68
0.35
*
Confidence Statements
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A confidence statement is used to state the level of
quality of manufactured product. Whether it is
dimensional or pass/fail data, confidence statements
can help to state the quality level achieved by a
process in relation to the specification.
When the true means and standard deviations are
not known, estimates of these parameters such as
sample standard deviations and sample means are
used to make confidence statements based on
tolerance limits using either binomial probabilities or
k-factors.
There are three types of confidence statements that
are primarily used.
Statistical Quality Control
Confidence Statements
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1. Attribute data confidence statements are used to
state the quality level when data is of a pass/fail
type. A binomial probability is used to calculate a
95% confidence statement that at least x% of the
population will pass the required specification.
2. Two sided confidence statements are used to
describe the quality level of data that has an upper
and lower specification limit. The data is assumed to
come from a normally distributed population. A two
sided tolerance limit table is used for determining
probability levels for percent of population. This
probability is stated as a 95% confidence that at
least x% of the population will be within the
specification.
Statistical Quality Control
Confidence Statements
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3. One sided confidence statements are used to
describe the quality level of data that has either a
maximum or minimum specification limit. As with the
two sided confidence statement the data is assumed
to be from a normally distributed population. A one
sided tolerance limit table is used for the probability
levels for percent of population in this case. This
probability is stated as a 95% confidence that at
least x% of the population will be either above the
minimum specification or below the maximum
specification.
Statistical Quality Control
Confidence Statements
For confidence that data is greater than min spec
Sample mean – K*(sample sd) = min spec
xbar- Ks = min
- Ks = min - xbar
K = (xbar - min)/s
For confidence that data is less than max spec
Sample mean + K*(sample sd) = max spec
xbar + Ks = max
Ks = max - xbar
K = (max - xbar)/s
For two sided tolerance limit both calculations should be made and lowest k-factor
compared with table value.
Statistical Quality Control
Confidence Statements
Statistical Quality Control
Confidence Statements
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In the case of attribute data sample size will determine the level that is
reached with a confidence statement. The higher the sample size used
(with zero or minimal failures), the higher the percent of population is
when stating the confidence.
Below is a chart showing how sample sizes can effect the 95%
confidence statements:
Percent of Population
90
95
99
99.9
99.99
99.999
Defects
0
0
0
0
0
0
Statistical Quality Control
Sample Size
30
60
300
3,000
30,000
300,000
Sample Size
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The following simple formula may be used to estimate sample
size (for any distribution) to determine a sample mean, or
average, when estimates of the standard deviation are known.
2 2
z s
n 2
B
n represents the sample size to be calculated
z represents the table value for the specified confidence desired
(i.e., z ( 90%) = 1.65, z (95%) = 1.96, z ( 99%) = 2.58)
s represents the estimated standard deviation
B represents the bound of the error of estimation, or ½ of the
desired range of accuracy, e.g., if you desire accuracy of
then B = 3 psi.
Statistical Quality Control
x ± 3 psi,
Sample Size
The following simple formula may be used to estimate sample
size to determine a proportion (fraction) defective.
n= p (1-p) (z / B)
Where:

n represents the sample size to be calculated.
p represents the estimate of the population fraction defective. If no
estimate of p is available, assume worst case of p = 0.5.
z represents the table value for the specified confidence desired
(i.e., z (90%) ) = 1.65, z (95%) = 1.96, z (99%) - 2.58).
B represents the bound of the error of estimation, or ½ of the
desired range of accuracy, e.g., if you desire accuracy of p ± 0.002.
Statistical Quality Control
Sample Size
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Example: An engineer wants to estimate a sample size to determine the
proportion of unacceptable attributes that may be present in a
manufacturing process, e.g., the number of molded components with
flash present on the parting line.
If a known history of scrap is already present in a similar product, then
that proportion can be used.
If the expected proportion is unknown, then you should use the worst
case, or 0.5 as your estimated proportion.
Let’s say the engineer does not know the proportion and uses 0.5 as the
estimate.
He/she wants to know at 95% confidence what the sample size should
be and is willing to be accurate within ± 0.1.
n = 0.5 (0.5) (1.96/0.1) = 96.04 or rounded up, 97
Statistical Quality Control
Process Capability
Process Capability Study is an approach to determine the
inherent variability of each process, sub-process, and piece of
equipment.
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This study provides a method to compare the relationship
between the variability of the process and the tolerance range
to assure that the process is capable of achieving the
tolerance window.
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Typically process capability studies occur in five stages;
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(1) process characterization,
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(2) metrology characterization,
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(3) capability determination,
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(4) optimization or reduction of variability, and
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(5) preventive control.
The two standard methods for measuring process capability are Cp
and CpK.
Statistical Quality Control
Process Capability

Cp: Process Cp is a numeric index that represents the inherent
capability of a process to meet the requirements of the
tolerance range without respect to centeredness. It represents
precision, and is calculated as follows:
USL  LSL
Cp 
6
Where:
USL=
The Upper Specification Limit
LSL =
The Lower Specification Limit
σ
The population standard deviation
=
Statistical Quality Control
Process Capability

Cp represents the precision, but not the
accuracy of the process in respect to the
tolerance window.
High Accuracy but low
precision
High Precision but low
Accuracy
Statistical Quality Control
Process Capability
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The 6  is estimated from the process, and is
more accurate as the sample size gets larger.
Decisions about process capability may not be
valid with data from a single run, and when
possible, should be based on data from 2 or
more runs.
Cp is only valid when the distribution of the
data is statistically normal.
Outliers, bimodal tendencies and skewness may
lower the Cp value.
Statistical Quality Control
Process Capability

CpK: Process Cpk is a numeric index that represents the ability
of the process to manufacture parts that are within
specification. It represents accuracy. Cpk provides a numeric
index that focuses on the centeredness of the process on the
xx
tolerance window. Cpk is the
smallest resulting ratio of the
following two (2) equations:
x  LSL
Cpk 
3S
USL  x
Cpk 
3S
USL =
The upper specification limit
LSL =
The lower specification limit
x=
The product related process mean.
s =
The product related standard deviation
Statistical Quality Control
Process Capability
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A machine or process is sometimes referred to as being
capable when its Cpk has a minimum value of one (1.00) and
when process stability has been proven.
A Cpk equal to one (1.00) implies that 99.73% of the product
is within specification limits, provided that the process is
stable. However, it should be noted that if the machine
capability is only 1.0, it will be impossible to maintain a Cpk of
1.0 or higher.
The goal should be a Cp as high as possible.
It is possible for a process to have a high Cp, but a low Cpk, if
the process is not centered in the tolerance window. A Cpk of
1.33 or higher should be targeted.
Statistical Quality Control
CpK
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A CpK of 1.33 means that the difference between the
mean and specification limit is 4σ (since 1.33 is 4/3).
With a CpK of 1.33, 99.994% of the product is
within the within specification.
Similarly a CpK of 2.0 is 6σ between the mean and
specification limit (since 2.0 is 6/3).
With a CpK of 2.0 99.9999998% of the product is
within specification.
Statistical Quality Control
Acceptance Sampling
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Definition: the third branch of SQC refers to the
process of randomly inspecting a certain number of
items from a lot or batch in order to decide whether to
accept or reject the entire batch
Different from SPC because acceptance sampling is
performed either before or after the process rather
than during
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Sampling before typically is done to supplier material
Sampling after involves sampling finished items before shipment
or finished components prior to assembly
Used where inspection is expensive, volume is high, or
inspection is destructive
Statistical Quality Control
Acceptance Sampling Plans
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Goal of Acceptance Sampling plans is to determine the criteria
for acceptance or rejection based on:
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Size of the lot (N)
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Size of the sample (n)
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Number of defects above which a lot will be rejected (c)
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Level of confidence we wish to attain
There are single, double, and multiple sampling plans
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Which one to use is based on cost involved, time consumed, and cost of
passing on a defective item
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Can be used on either variable or attribute measures, but more
commonly used for attributes
Statistical Quality Control
Acceptance Sampling Plans
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ANSI/ASQC Z1.4 (Attribute or P/F Data)
ANSI/ASQC Z1.9 (Variable Data)
C=0 (Attribute, reject on 1)
MIL STD 1235C (Continuous
Production)
Statistical Quality Control
Sample Size Calculation –Z 1.4
© Wiley 2007
Sampling Plan for Normal Inspection
© Wiley 2007
AQL Inspector’s Rule
© Wiley 2007
AQL Inspector’s Rule
Accept/Reject
Sample Size
© Wiley 2007
Acceptance Sampling Plans
• As mentioned acceptance sampling can reject “good” lots and accept “bad”
lots. More formally:
Producers risk refers to the probability of rejecting a good lot. In order to
calculate this probability there must be a numerical definition as to what
constitutes “good”
– AQL (Acceptable Quality Limit) - the numerical definition of a good lot. The
ANSI/ASQC standard describes AQL as “the maximum percentage or proportion
of nonconforming items or number of nonconformities in a batch that can be
considered satisfactory as a process average”
• Consumers Risk refers to the probability of accepting a bad lot where:
– LTPD (Lot Tolerance Percent Defective) - the numerical definition of a bad lot
described by the ANSI/ASQC standard as “the percentage or proportion of
nonconforming items or noncomformities in a batch for which the customer
wishes the probability of acceptance to be a specified low value.
Statistical Quality Control
Acceptance Sampling
OC Curve
1.2
Producers Risk
Probability of Acceptance
1
0.8
0.6
Consumers Risk
0.4
0.2
0
AQL
LTPD
Percent Defective
© Wiley 2007
Implications for Managers
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How much and how often to inspect?
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Where to inspect?
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Consider product cost and product volume
Consider process stability
Consider lot size
Inbound materials
Finished products
Prior to costly processing
Which tools to use?
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Control charts are best used for in-process production
Acceptance sampling is best used for inbound/outbound
Statistical Quality Control
SQC Across the Organization

SQC requires input from other organizational
functions, influences their success, and are actually
used in designing and evaluating their tasks

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Marketing – provides information on current and future
quality standards
Finance – responsible for placing financial values on SQC
efforts
Human resources – the role of workers change with SQC
implementation. Requires workers with right skills
Information systems – makes SQC information accessible for
all.
Statistical Quality Control
Quality Control
© Wiley 2007