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Chapter 7 SupplementChapter 7 Supplement

Statistical Quality ControlStatistical Quality Control ChaseChaseAquilanAquilan

ooJacobsJacobs

©The McGraw-Hill Companies, Inc., 1998 and (c) Stephen A. DeLurgio, 2000Irwin/McGraw-Hill

2

Chapter 7 Supplement - 1

Statistical Quality Control

• Process Control Procedures - 1– Variable data– Attribute data

• Process Capability - 2

• Acceptance Sampling - 3– Operating Characteristic Curve

3

Basic Forms of Statistical Sampling for Quality Control

• Sampling to accept or reject the immediate lot of product at hand (Acceptance Sampling). Trying to Inspect Quality Into Product!

• Sampling to determine if the process is within acceptable limits (Statistical Process Control). Building Quality Into Product and Process!

IMPORTANT UNDERLYING PRINCIPLE

IT IS POSSIBLE TO DESIGN A PROCESS SO THAT EVEN WHEN WE DETECT IT AS BEING OUT OF

CONTROL, NO DEFECTS ARE PRODUCED.

OUR GOAL REDUCE PROCESS VARIATION SO MUCH THAT

DEFECTS ARE NOT PRODUCED. WE DO THAT BY CREATING CONTROL DEVICES, ELIMINATING

THE CAUSES OF LARGE, ASSIGNABLE PROCESS VARIATIONS, AND COORDINATING PRODUCT

DESIGN AND PROCESS CAPABILITY.

PRODUCTIVITY/QUALITY GAINS FROM SPC ARE TRULY EXTRAORDINARY !

WE STUDY SCIENTIFIC METHODS OF SPC TO

Eliminate Causes of DefectsIdentify Assignable Variations Adjust the Process Reduce Risks of Defective Products

ACHIEVE VALUE FOR EVERYONE!

UNDERSTANDING VARIABILITY

To understand variability, we need to understand some basic statistics and random behavior.

These concepts apply to industrial processes, how we perform at sports, how physical and biological systems

behave, and many other occurrences.

Well designed processes yield output that is Normally Distributed. Your understanding of the Normal

Distribution(ND) is Essential -WHAT IS AND WHAT CAUSES NORMALLY DISTRIBUED VALUES?

WHY IS THIS IMPORTANT?

NORMALLY DISTRIBUTED

MEAN +/- ONE STANDARD DEVIATION 68%

MEAN +/- 1.96 STANDAR DEVIATRIONS 95%

MEAN +/- 3.00 STANDARD DEVIATIONS 99.73%

MEAN +/- 4.00 STANDARD DEVIATIONS 99.994%

ND CHARACTERISTICS

• SYMMETRICAL - BELL SHAPED• DISCOVERED BY K. F. GAUSS• DEFINED COMPLETELY BY MEAN AND

STANDARD DEVIATION• GENERATED BY IN CONTROL

RANDOM PROCESS• CONTINUOUS DISTRIBUTION FROM -

INFINITY TO + INFINITY

WHAT GENERATES ND OUTPUT?

“IF AN EVENT IS THE RESULT OF A RELATIVELY LARGE NUMBER OF SMALL, CHANCE, INDEPENDENT INFLUENCES, THEN ITS OUTPUT WILL BE ND.”

MANY PROCESSES ARE ND BECAUSE:WE HAVE WORKED HARD TO ELIMINATE THE VERY LARGE INFLUENCES, THUS ONLY A RELATIVELY LARGE NUMBER OF SMALL, INDEPENDENT INFLUENCES REMAIN.

FOR EXAMPLE:

THINK ABOUT THE PROCESS OF PRODUCING GOLD COINS,

IT IS IMPORTANT THAT EACH WEIGHS 1.0 OZ.

TO ACHIEVE A 1 OZ. WEIGHT WE CONTROL:

THE SIZE OF GOLD STRIPS GOING INTO THE PRESS.THE ADJUSTMENTS ON THE MACHINE.THE TEMPERATURE OF THE MACHINE.THE HUMIDITY OF THE ROOM.THE CLEANLINESS OF THE SET UP.THE CONDITION OF THE TOOLS (DIES) USED.ALL OTHER FACTORS THAT INFLUENCE WEIGHT.

1 OZ.

MEAN= 0.999952STDEV= 0.000984LOWER UPPER Frequency0.996645 0.996764 00.996892 0.997033 00.997139 0.997302 00.997386 0.997571 10.997633 0.99784 20.99788 0.998109 7

0.998127 0.998378 110.998374 0.998648 250.998621 0.998917 380.998868 0.999186 380.999115 0.999455 410.999362 0.999724 630.999609 0.999993 610.999856 1.000262 641.000103 1.000531 621.00035 1.000801 57

1.000597 1.00107 451.000844 1.001339 311.001091 1.001608 231.001338 1.001877 111.001585 1.002146 111.001832 1.002415 41.002079 1.002684 31.002326 1.002954 1

More More 0

COINING OUTPUT FOR n = 600

NOTE SYMMETRY AND BELL SHAPE

Histogram

0

10

20

30

40

50

60

70

Bin

Frequency

Frequency

HISTOGRAM OF COINING OUTPUT, n=600

NOTE SYMMETRY AND BELL SHAPE

IN CONTROL PROCESS VARIATION

BY ELIMINATING ALL OF THE LARGE INFLUENCES WE ARE LEFT WITH MANY SMALL INFLUENCES ACTING SEPARETLY.

THIS YIELDS A PROCESS WITH:

MEAN = 1 OZ.STD. DEV. = .001 OZS.

AND IMPORTANTLY, THE OUTPUT IS

NORMALLY DISTRIBUTEDCONSIDER THE INTERVALS:

MEAN = 1 OZ., STD DEV=.001

1 +/- .001 68% 6,800 OF 10,000 IN THIS RANGE

1 +/- .00196 95% 9,500 OF 10,000 IN THIS RANGE

1 +/- .003 99.73% 9,973 OF 10,000 IN THIS RANGE

1 +/- .004 99.994% 9,999.4 OF 10,000 IN THIS RANGE

DESCRIPTIVE STATISTICS

• MEAN = CENTER OF DEVIATIONS

• POPULATION MEAN, = X / N

• MEDIAN VALUE HAVING 50% ABOVE, 50% BELOW

• MODE MOST FREQUENT VALUE

• FOR SYMMETRICAL DISTRIBUTION

• MEAN = MEDIAN = MODE

MEAN= 0.999952STDEV= 0.000984LOWER UPPER Frequency0.996645 0.996764 00.996892 0.997033 00.997139 0.997302 00.997386 0.997571 10.997633 0.99784 20.99788 0.998109 7

0.998127 0.998378 110.998374 0.998648 250.998621 0.998917 380.998868 0.999186 380.999115 0.999455 410.999362 0.999724 630.999609 0.999993 610.999856 1.000262 641.000103 1.000531 621.00035 1.000801 57

1.000597 1.00107 451.000844 1.001339 311.001091 1.001608 231.001338 1.001877 111.001585 1.002146 111.001832 1.002415 41.002079 1.002684 31.002326 1.002954 1

More More 0

COINING OUTPUT FOR n = 600

NOTE SYMMETRY AND BELL SHAPE

STANDARD DEVIATION

• MEASURES VARIATION OR SCATTER• SQUARE ROOT OF THE MEAN

SQUARED ERROR•

x = (X - )2 /N Population std. deviation of X with census.

• Sx = (X -X)2/(n-1) Sample standard deviation of X.

• Formulas may not yield much information, not as meaningful unless for known distribution.

MEAN = 1 OZ., STD DEV=.001

1 +/- .001 68% 6,800 OF 10,000 IN THIS RANGE

1 +/- .00196 95% 9,500 OF 10,000 IN THIS RANGE

1 +/- .003 99.73% 9,973 OF 10,000 IN THIS RANGE

1 +/- .004 99.994% 9,999.4 OF 10,000 IN THIS RANGE

MEAN +/- ONE STANDARD DEVIATION 68%

MEAN +/- 1.96 STANDAR DEVIATRIONS 95%

MEAN +/- 3.00 STANDARD DEVIATIONS 99.73%

MEAN +/- 4.00 STANDARD DEVIATIONS 99.994%

MEAN +/- 5.00 STANDARD DEVIATIONS 99.99994%

MEAN +/- 6.00 STANDARD DEVIATIONS 99.99999%

OTHER ND INTERVALS

THE CENTRAL LIMIT THEOREM

NOTE THAT SAMPLE MEANS

ARE ND!

THE CENTRAL LIMIT THEOREM

DISTRIBUTION OF SAMPLE MEANS IS ND FOR LARGE SAMPLES FROM ANY GENERAL

POPULATION!

MEAN OF MEANS ARE ND

__ __ X = Z / n

MEAN OF MEAN = POP MEAN

STD. DEV. OF MEANS = POP STD.DEV /n^.5

__ ___

X = 1.0 Z .001/ 100

15

Control Limits

Let’s establish control limits at +/- 3 standard deviations, then

We expect 99.7% of observations to fall within these limits

xLCL UCL

CONTROL CHARTS BASED ON ND

TIME TO THE CONTROL CHART ADDS POWERFUL INFERENCES!

ALL POINTS IN CONTROL

A B“A” IS OUT OF CONTROL, TWO PTS. IN B

ARE OUT OF CONTROL, TREND OF 7 = OUT OF CONTROL

7=TREND

X-BAR CHART FORMULAS

When using known mean and standard deviation :

_ __ X = Z / n

When and are unknown, they are estimated:

_ = _ __ X = X Z S/ n

When using measured Ranges:

_ = _ X = X A2 R

THE RELATIONSHIP BETWEEN COOKBOOK FORMULAS AND THEORY

A2R = 3

n

S-Charts and R-Charts

The S-chart uses the following formula:

S = Z /2n

The R-Chart uses the following formulas:

D4R (UCL)R = {

D3R (LCL)

The results of both will be the same in use, however, numerical values using S and R will be

different, the plots will look nearly identical.

A LITTLE MORE THEORY

When small samples (n<30) are used, the assumption is that the sample comes from a ND. When this is not true, then the above formulas MAY NOT BE valid.

If the process is NOT ND, then large samples are necessary, or other statistical tests called Nonparametric methods must be used.

23

Example: x-Bar and R Charts2 10.787 10.86 10.601 10.746 10.7793 10.78 10.667 10.838 10.785 10.7234 10.591 10.727 10.812 10.775 10.735 10.693 10.708 10.79 10.758 10.6716 10.749 10.714 10.738 10.719 10.6067 10.791 10.713 10.689 10.877 10.6038 10.744 10.779 10.11 10.737 10.759 10.769 10.773 10.641 10.644 10.72510 10.718 10.671 10.708 10.85 10.71211 10.787 10.821 10.764 10.658 10.70812 10.622 10.802 10.818 10.872 10.72713 10.657 10.822 10.893 10.544 10.7514 10.806 10.749 10.859 10.801 10.70115 10.66 10.681 10.644 10.747 10.728

Averages

24

Calculate sample means, sample ranges, mean of means, and mean of ranges.

Sample Obs 1 Obs 2 Obs 3 Obs 4 Obs 5 Avg Range1 10.682 10.689 10.776 10.798 10.714 10.732 0.1162 10.787 10.86 10.601 10.746 10.779 10.755 0.2593 10.78 10.667 10.838 10.785 10.723 10.759 0.1714 10.591 10.727 10.812 10.775 10.73 10.727 0.2215 10.693 10.708 10.79 10.758 10.671 10.724 0.1196 10.749 10.714 10.738 10.719 10.606 10.705 0.1437 10.791 10.713 10.689 10.877 10.603 10.735 0.2748 10.744 10.779 10.11 10.737 10.75 10.624 0.6699 10.769 10.773 10.641 10.644 10.725 10.710 0.13210 10.718 10.671 10.708 10.85 10.712 10.732 0.17911 10.787 10.821 10.764 10.658 10.708 10.748 0.16312 10.622 10.802 10.818 10.872 10.727 10.768 0.25013 10.657 10.822 10.893 10.544 10.75 10.733 0.34914 10.806 10.749 10.859 10.801 10.701 10.783 0.15815 10.66 10.681 10.644 10.747 10.728 10.692 0.103

Averages 10.728 0.220400

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Control Limit Formulas

x Chart Control Limits

UCL = x + A R

LCL = x - A R

2

2

R Chart Control Limits

UCL = D R

LCL = D R

4

3

n A2 D3 D42 1.88 0 3.273 1.02 0 2.574 0.73 0 2.285 0.58 0 2.116 0.48 0 2.007 0.42 0.08 1.928 0.37 0.14 1.869 0.34 0.18 1.8210 0.31 0.22 1.7811 0.29 0.26 1.74

26

x-Bar Chart

10.601

10.856

=).58(0.2204-10.728RA - x = LCL

=).58(0.2204-10.728RA + x = UCL

2

2

10.550

10.600

10.650

10.700

10.750

10.800

10.850

10.900

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Sample

Mea

ns

UCL

LCL

27

R-Chart

0

0.46504

)2204.0)(0(R D= LCL

)2204.0)(11.2(R D= UCL

3

4

UCL

LCL0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Sample

R

6

Statistical Sampling--Data

• Attribute (Go no-go information)– Defectives--refers to the acceptability of product

across a range of characteristics.– Defects--refers to the number of defects per unit--

may be higher than the number of defectives.

• Variable (Continuous)– Usually measured by the mean and the standard

deviation.

DISTRIBUTION OF SAMPLE PROPORTIONS

POP IS NOT ND

= .98

SAMPLE LOOKS LIKE POP,

P = .99

DIST. OF SAMPLE P’S ARE ND

P-CHARTS

Require large samples n30.

When population proportion is known:

————— P = Z (1 - )/n

When population proportion is unknown:

_ _ _ P = P Z P(1 - P)/n

Where P-Bar is an estimate of

17

Constructing a p-Chart

Sample n Defectives1 100 42 100 23 100 54 100 35 100 66 100 47 100 38 100 89 100 1

10 100 211 100 312 100 213 100 214 100 815 100 3

18

Statistical Process Control--Attribute Measurements (P-Charts)

p =Total Number of Defectives

Total Number of Observations

= p (1- p)

npS

UCL = p + Z

LCL = p - Z p

p

s

s

1. Calculate the sample proportion, p, for each

sample.Defectives p UCL LCL

4 0.04 ? ?2 0.02 ? ?5 0.05 ? ?3 0.03 ? ?6 0.06 ? ?4 0.04 ? ?3 0.03 ? ?7 0.07 ? ?1 0.01 ? ?2 0.02 ? ?3 0.03 ? ?2 0.02 ? ?2 0.02 ? ?8 0.08 ? ?3 0.03 ? ?

19©The McGraw-Hill Companies, Inc., 1998Irwin/McGraw-Hill

2. Calculate the average of the sample proportions.

0.037=1500

55 = p

3. Calculate the standard deviation of the sample proportion

.0188= 100

.037)-.037(1=

n

)p-(1 p = ps

20©The McGraw-Hill Companies, Inc., 1998

Irwin/McGraw-Hill

4. Calculate the control limits.

3(.0188) .037 UCL = 0.093

LCL = -0.0197 (or 0)

UCL = p + Z

LCL = p - Z p

p

s

s

21©The McGraw-Hill Companies, Inc., 1998

Irwin/McGraw-Hill

22

p-Chart (Continued)5. Plot the individual sample proportions, the average

of the proportions, and the control limits

1. Calculate the sample proportion, p, for each

sample.Defectives p UCL LCL

4 0.04 0.093049 02 0.02 0.093049 05 0.05 0.093049 03 0.03 0.093049 06 0.06 0.093049 04 0.04 0.093049 03 0.03 0.093049 07 0.07 0.093049 01 0.01 0.093049 02 0.02 0.093049 03 0.03 0.093049 02 0.02 0.093049 02 0.02 0.093049 08 0.08 0.093049 03 0.03 0.093049 0

19©The McGraw-Hill Companies, Inc., 1998Irwin/McGraw-Hill

Control Chart: VAR00001

Sigma level: 3

151413121110987654321

Pro

port

ion N

onconfo

rmin

g

.10

.08

.06

.04

.02

0.00

VAR00001

UCL = .0930

Center = .0367

LCL = .0000