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Technical Note 8Process Capability and

Statistical Quality Control

- Process Variation
- Process Capability
- Process Control Procedures
- Variable data
- Attribute data
- Acceptance Sampling
- Operating Characteristic Curve

Basic Causes of Variation

- Assignable causes
- Common causes
- Key

Types of Control Charts

- Attribute (Go or no-go information)
- Defectives
- p-chart application
- Variable (Continuous)
- Usually measured by the mean and the standard

deviation. - X-bar and R chart applications

Types of Statistical Quality Control

Statistical

Statistical

Quality Control

Quality Control

Process

Acceptance

Process

Acceptance

Control

Sampling

Control

Sampling

Variables

Attributes

Variables

Attributes

Variables

Attributes

Variables

Attributes

Charts

Charts

Charts

Charts

Statistical Process Control (SPC) Charts

Control Limits

- We establish the Upper Control Limits (UCL) and

the Lower Control Limits (LCL) with plus or minus

3 standard deviations. Based on this we can

expect 99.7 of our sample observations to fall

within these limits.

99.7

Example of Constructing a p-Chart Required Data

Statistical Process Control FormulasAttribute

Measurements (p-Chart)

Given

Compute control limits

Example of Constructing a p-chart Step 1

1. Calculate the sample proportions, p (these

are what can be plotted on the p-chart) for each

sample.

Example of Constructing a p-chart Steps 23

2. Calculate the average of the sample

proportions.

3. Calculate the standard deviation of the sample

proportion

Example of Constructing a p-chart Step 4

4. Calculate the control limits.

Example of Constructing a p-Chart Step 5

5. Plot the individual sample proportions, the

average of the proportions, and the control

limits

R Chart

- Type of variables control chart
- Interval or ratio scaled numerical data
- Shows sample ranges over time
- Monitors variability in process
- Example Weigh samples of coffee compute ranges

of samples Plot

R Chart Control Limits

From Table (function of sample size)

Sample Range in sample i

Samples

R Chart Example

- Youre manager of a 500-room hotel. You want to

analyze the time it takes to deliver luggage to

the room. For 7 days, you collect data on 5

deliveries per day. Is the process in control?

R Chart Hotel Data

- Sample
- Day Delivery Time Mean Range
- 1 7.30 4.20 6.10 3.45 5.55 5.32

R Chart Hotel Data

- Sample
- Day Delivery Time Mean Range
- 1 7.30 4.20 6.10 3.45 5.55 5.32 3.85

R Chart Hotel Data

- Sample
- Day Delivery Time Mean Range
- 1 7.30 4.20 6.10 3.45 5.55 5.32 3.85 2 4.60 8.7

0 7.60 4.43 7.62 6.59 4.27 3 5.98 2.92 6.20 4.20

5.10 4.88 3.28 4 7.20 5.10 5.19 6.80 4.21 5.70

2.99 5 4.00 4.50 5.50 1.89 4.46 4.07 3.61 6 10

.10 8.10 6.50 5.06 6.94 7.34 5.04 7 6.77 5.08 5.

90 6.90 9.30 6.79 4.22

R Chart Control Limits Solution

From Table (n 5)

R Chart Control Chart Solution

UCL

R-bar

?X Chart

- Type of variables control chart
- Interval or ratio scaled numerical data
- Shows sample means over time
- Monitors process average
- Example Weigh samples of coffee compute means

of samples Plot

?X Chart Control Limits

From Table

Mean of sample i

Range of sample i

Samples

?X Chart Example

- Youre manager of a 500-room hotel. You want to

analyze the time it takes to deliver luggage to

the room. For 7 days, you collect data on 5

deliveries per day. Is the process in control?

X Chart Hotel Data

- Sample
- Day Delivery Time Mean Range
- 1 7.30 4.20 6.10 3.45 5.55 5.32 3.85 2 4.60 8.7

0 7.60 4.43 7.62 6.59 4.27 3 5.98 2.92 6.20 4.20

5.10 4.88 3.28 4 7.20 5.10 5.19 6.80 4.21 5.70

2.99 5 4.00 4.50 5.50 1.89 4.46 4.07 3.61 6 10

.10 8.10 6.50 5.06 6.94 7.34 5.04 7 6.77 5.08 5.

90 6.90 9.30 6.79 4.22

?X Chart Control Limits Solution

?X ChartControl Chart Solution

UCL

X-bar

LCL

X AND R CHART EXAMPLEIN-CLASS EXERCISE

- The following collection of data represents

samples of the amount of force applied in a

gluing process - Determine if the process is in control
- by calculating the appropriate upper and lower
- control limits of the X-bar and R charts.

X AND R CHART EXAMPLEIN-CLASS EXERCISE

Example of x-bar and R charts Step 1. Calculate

sample means, sample ranges, mean of means, and

mean of ranges.

Example of x-bar and R charts Step 2. Determine

Control Limit Formulas and Necessary Tabled Values

Example of x-bar and R charts Steps 34.

Calculate x-bar Chart and Plot Values

Example of x-bar and R charts Steps 56

Calculate R-chart and Plot Values

SOLUTIONExample of x-bar and R charts

- 1. Is the process in Control?
- 2. If not, what could be the cause for the

process being out of control?

Process Capability

- Process limits -
- Tolerance limits -

Process Capability

- How do the limits relate to one another?

Process Capability Measurement

Cp index Tolerance range / Process range What

value(s) would you like for Cp?

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UTL

LTL

- While the Cp index provides useful information on

process variability, it does not give information

on the process average relative to the tolerance

limits. Note

UTL

LTL

Cpk Index

s

Together, these process capability Indices show

how well parts being produced conform to design

specifications.

Since Cp and Cpk are different we can conclude

that the process is not centered, however the Cp

index tells us that the process variability is

very low

An example of the use of process capability

indices

The design specifications for a machined slot is

0.5 .003 inches. Samples have been taken and

the process mean is estimated to be .501. The

process standard deviation is estimated to be

.001. What can you say about the capability of

this process to produce this dimension?

Process capability

Machined slot (inches)

0.497 inches LTL

0.503 inches UTL

? 0.001 inches

Process mean 0.501 inches

Sampling Distributions(The Central Limit Theorem)

- Regardless of the underlying distribution, if the

sample is large enough (gt30), the distribution of

sample means will be normally distributed around

the population mean with a standard deviation of

Computing Process Capability Indexes Using

Control Chart Data

- Recall the following info from our in class

exercise - Since A2 is calculated on the assumption of three

sigma limits

- From the Central Limit Theorem
- So,
- Therefore,

- Suppose the Design Specs for the Gluing Process

were 10.7 ? .2, Calculate the Cp and Cpk

Indexes - Answer

Note, multiplying each component of the Cpk

calculation by 3 yields a Z value. You can use

this to predict the of items outside the

tolerance limits From Appendix E we would

expect non-conforming product from this process

.002 or .2 of the curve

.02 or 2 of the curve

Capability Index In Class Exercise

- You are a manufacturer of equipment. A drive

shaft is purchased from a supplier close by. The

blueprint for the shaft specs indicate a

tolerance of 5.5 inches .003 inches. Your

supplier is reporting a mean of 5.501 inches.

And a standard deviation of .0015 inches. - What is the Cpk index for the suppliers process?

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- Your engineering department is sent to the

suppliers site to help improve the capability on

the shaft machining process. The result is that

the process is now centered and the CP index is

now .75. On a percentage basis, what is the

improvement on the percentage of shafts which

will be unusable (outside the tolerance limits)?

To answer this question we must determine the

percentage of defective shafts before and after

the intervention from our engineering department

Before

After

Since the process is centered then Cpk Cp Cp

UTL-LTL / 6s, so the tolerance limits are .75

x 6s 4.5s apart each 2.25s from the mean

-4

So the percentage decrease in defective parts is

Basic Forms of Statistical Sampling for Quality

Control

- Sampling to accept or reject the immediate lot of

product at hand - Sampling to determine if the process is within

acceptable limits

Acceptance Sampling

- Purposes
- Advantages

Acceptance Sampling

- Disadvantages

Risk

- Acceptable Quality Level (AQL)
- a (Producers risk)
- Lot Tolerance Percent Defective (LTPD)
- ? (Consumers risk)