1. Statistical Quality Control
by R.B. Clough – UNH
M. E. Henrie - UAA
Statistical Quality Control

2. The Right Statistical Tools
Statistical Quality Control

3. 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

4. Inferential Statistics
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

5. Statistical Quality Control
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
e.g. the mean, standard deviation, and range
Statistical process control (SPC)
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
Does not help to catch in-process problems
Statistical Quality Control

6. Statistical Quality Control
Sources of Variation
Variation exists in all processes.
Variation can be categorized as either;
Common or Random causes of variation, or
Random causes that we cannot identify
Unavoidable
e.g. slight differences in process variables like diameter, weight, service time, temperature
Assignable causes of variation
Causes can be identified and eliminated
e.g. poor employee training, worn tool, machine needing repair
Statistical Quality Control

7. Traditional Statistical Tools
Descriptive Statistics include
The Mean- measure of central tendency
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
Normal or bell shaped or
Skewed
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8. Statistical Quality Control
Distribution of Data
Normal distributions
Skewed distribution
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9. SPC Methods-Control Charts
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
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10. Analysis of Patterns on Control Charts
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”

11. Setting Control Limits
Percentage of values under normal curve
Control limits balance
risks like Type I error
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12. Statistical Quality Control
Hypothesis Tests
Results of hypothesis tests fall into one of four scenarios:
Type I Error OK OK
Type II Error
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13. Type I and Type II Error ART and BAF
Type I - ART (Alpha, Reject Ho when true)
Type II - BAF (Beta, Accept Ho when false)
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14. Jury Trial vs. 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
(not guilty) or
reject (guilty)
Fail to reject H0 or
Reject H0 in favor
of Ha
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15. Context? 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?

16. Control Charts for Variables
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

17. Control Charts for Variables
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

18. Graphical Analysis “A picture is worth a thousand words.”
Graphical analysis is the first step in analyzing your data. Examples:
Distribution (histogram, dotplot, boxplot)
Time Series plot for trending
I-chart (for Individual data points)
Normality
Cpk (when applicable) graph (Minitab)
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19. Dotplot of Tensile Test Data
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20. Statistical Quality Control
Time Series Plot
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21. Statistical Quality Control
Individuals (I) Chart
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22. Normal Probability Plot
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23. Statistical Quality Control
Cpk Graph (Minitab)
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24. Confidence Statements
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

25. Confidence Statements
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

26. Confidence Statements
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

27. 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

28. Confidence Statements
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29. Confidence Statements
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:
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30. Statistical Quality Control
Sample Size
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.
Statistical Quality Control

31. Statistical Quality Control
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:
Statistical Quality Control

32. Statistical Quality Control
Sample Size
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) = or rounded up, 97
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33. 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.
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.
Typically process capability studies occur in five stages;
(1) process characterization,
(2) metrology characterization,
(3) capability determination,
(4) optimization or reduction of variability, and
(5) preventive control.
The two standard methods for measuring process capability are Cp and CpK.
Statistical Quality Control

34. 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:
Where:
USL=
The Upper Specification Limit
LSL =
The Lower Specification Limit
σ =
The population standard deviation
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35. 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
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36. Statistical Quality Control
Process Capability
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

37. 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 tolerance window. Cpk is the smallest resulting ratio of the following two (2) equations:
USL =
The upper specification limit
LSL =
The lower specification limit =
The product related process mean.
s =
The product related standard deviation
Statistical Quality Control

38. Statistical Quality Control
Process Capability
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

39. Statistical Quality Control
CpK
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, % 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 % of the product is within specification.
Statistical Quality Control

40. Statistical Quality Control
Acceptance Sampling
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
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

41. Acceptance Sampling Plans
Goal of Acceptance Sampling plans is to determine the criteria for acceptance or rejection based on:
Size of the lot (N)
Size of the sample (n)
Number of defects above which a lot will be rejected (c)
Level of confidence we wish to attain
There are single, double, and multiple sampling plans
Which one to use is based on cost involved, time consumed, and cost of passing on a defective item
Can be used on either variable or attribute measures, but more commonly used for attributes
Statistical Quality Control

42. Acceptance Sampling Plans
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

43. Sample Size Calculation –Z 1.4
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44. Sampling Plan for Normal Inspection
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45. AQL Inspector’s Rule
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46. AQL Inspector’s Rule
Accept/Reject
Sample Size
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47. 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

48. Acceptance Sampling
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49. Implications for Managers
How much and how often to inspect?
Consider product cost and product volume
Consider process stability
Consider lot size
Where to inspect?
Inbound materials
Finished products
Prior to costly processing
Which tools to use?
Control charts are best used for in-process production
Acceptance sampling is best used for inbound/outbound
Statistical Quality Control

50. SQC Across the Organization
SQC requires input from other organizational functions, influences their success, and are actually used in designing and evaluating their tasks
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

51. Quality Control
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