Control charts

2WS02 Industrial Statistics

A. Di Bucchianico

Goals of this lecture

Further discussion of control charts:

– variable charts

• Shewhart charts

– rational subgrouping

– runs rules

– performance

• CUSUM charts

• EWMA charts

– attribute charts (c, p and np charts)

– special charts (tool wear charts, short-run charts)

Statistically versus technically in control

“Statistically in

control”

• stable over time /

• predictable

“Technically in control”

• within specifications

Statistically in control vs technically in control

statistically controlled process:

– inhibits only natural random fluctuations (common causes)

– is stable

– is predictable

– may yield products out of specification

technically controlled process:

– presently yields products within specification

– need not be stable nor predictable

Shewhart control chart

graphical display of product characteristic which is important for

product quality

X-bar Chart for yield

Subgroup

X-b

ar

0 4 8 12 16 2013,6

13,8

14

14,2

14,4

UpperControl Limit

Centre Line

Lower Control

Limit

Control charts

Basic principles

take samples and compute statistic

if statistic falls above UCL or below LCL, then out-of-control signal:

X-bar Chart for yield

Subgroup

X-b

ar

0 4 8 12 16 2013,6

13,8

14

14,2

14,4

how to choose control limits?

Meaning of control limits

limits at 3 x standard deviation of plotted statistic

basic example:

9973.0)33(

)33(

)33(

)(

ZP

XP

XP

UCLXLCLP

XX

XX

UCL

LCL

Example

diameters of piston rings

process mean: 74 mm

process standard deviation: 0.01 mm

measurements via repeated samples of 5 rings yields:

mmLCL

mmUCL

mmn

x

9865.73)0045.0(374

0135.74)0045.0(374

0045.05

01.0

Individual versus mean

Centre line

1 1073,97

74,03

group means

individualobservations

Range chart• need to monitor both mean and variance

• traditionally use range to monitor variance

• chart may also be based on S or S2

• for normal distribution:

– E R = d2 E S (Hartley’s constant)

– tables exist

• preferred practice:

– first check range chart for violations of control limits

– then check mean chart

Design control chart

• sample size

– larger sample size leads to faster detection

• setting control limits

• time between samples

– sample frequently few items or

– sample infrequently many items?

• choice of measurement

Rational subgroups

how must samples be chosen?

choose sample size frequency such that if a special cause

occurs

– between-subgroup variation is maximal

– within-subgroup variation is minimal.

between subgroup variation

within subgroup variation

Strategy 1

• leads to accurate estimate of

• maximises between-subgroup variation

• minimises within-subgroup variation

process mean

Strategy 2

•detects contrary to strategy 1 also temporary changes of process

mean

process mean

Phase I (Initial study): in control (1)

Phase I (Initial study): in control (2)

Phase I (Initial Study): not in-control

Trial versus control

•if process needs to be started and no relevant historic data is

available, then estimate µ and or R from data (trial or initial study)

•if points fall outside the control limits, then possibly revise control

limits after inspection. Look for patterns!

•if relevant historical data on µ and or R are available, then use

these data (control to standard)

Control chart patterns (1)

Cyclic pattern,

three arrows with different weight

Control chart of height

Observation

Heig

ht

CTR = 0.00

UCL = 10.00

LCL = -10.00

0 3 6 9 12 15 18-10

-6

-2

2

6

10

Control chart patterns (2)

Trend,

course of pin

Control chart of height

Observation

Heig

ht CTR = 0.00

UCL = 10.00

LCL = -10.00

0 4 8 12 16 20-10

-6

-2

2

6

10

Control chart patterns (3)

Shifted mean,

Adjusted height Dartec

Control chart of height

Observation

Heig

ht CTR = 0.00

UCL = 10.00

LCL = -10.00

0 4 8 12 16 20-10

-6

-2

2

6

10

Control chart patterns (4)

A pattern can give explanation of the cause

Cyclic different arrows, different weight

Trend course of pin

Shifted mean adjusted height Dartec

Assumption: a cause can be verified by a pattern

The feather of one arrow is damaged outliers below

Phase II: Control to standard (1)

Phase II: Control to standard (2)

Runs and zone rules

•if observations fall within control limits, then process may still be

statistically out-of-control:

– patterns (runs, cyclic behaviour) may indicate special causes

– observations do not fill up space between control limits

•extra rules to speed up detection of special causes

•Western Electric Handbook rules:

– 1 point outside 3-limits

– 2 out of 3 consecutive points outside 2 -limits

– 4 out of 5 consecutive points outside 1 -limits

– 8 consecutive points on one side of centre line

•too many rules leads to too high false alarm rate

Warning limits

•crossing 3 -limits yields alarm

•sometimes warning limits by adding 2 -limits; no alarm but

collecting extra information by:

– adjustment time between taking samples and/or

– adjustment sample size

•warning limits increase detection performance of control chart

Detection: meter stick production

• mean 1000 mm, standard deviation 0.2 mm

• mean shifts from 1000 mm to 0.3 mm?

• how long does it take before control chart signals?

Performance of control charts

expressed in terms of time to alarm (run length)

two types:

– in-control run length

– out-of-control run length

X-bar Chart for yield

Subgroup

X-b

ar

0 4 8 12 16 2013,6

13,8

14

14,2

14,4

Statistical control and control charts

•statistical control: observations

– are normally distributed with mean and variance 2

– are independent

•out of (statistical) control:

– change in probability distribution

•observation within control limits:

– process is considered to be in control

•observation beyond control limits:

– process is considered to be out-of-control

In-control run length

•process is in statistical control

•small probability that process will go beyond 3 limits (in spite of

being in control) -> false alarm!

•run length is first time that process goes beyond 3 limits

•compare with type I error

Out-of-control run length

•process is not in statistical control

•increased probability that process will go beyond 3 limits (in spite

of being in control) -> true alarm!

•run length is first time that process goes beyond 3 sigma limits

•until control charts signals, we make type II errors

Metrics for run lengths

•run lengths are random variables

– ARL = Average Run Length

– SRL = Standard Deviation of Run Length

Run lengths for Shewhart Xbar- chart

in-control: p = 0.0027

UCL

LCL0.99730.99730.99730.0027

• time to alarm follows geometric distribution:– mean 1/p = 370.4– standard deviation: ((1-p))/p = 369.9

Geometric distribution

Event prob.0.0027

Geometric Distribution

probab

ility

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1(X 1000)

00.5

11.5

22.5

3(X 0.001)

Numerical values

Shewhart chart for mean (n=1)

single shift of mean: P(|X|>3) ARL SRL

0 0.0027 370.4 369.9

1 0.022 43.9 43.4

2 0.15 6.3 5.3

3 0.5 2 1.4

Scale in Statgraphics

Are our calculations wrong???

ARL Curve for X-bar

Process mean

Avera

ge ru

n len

gth

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

50100

150

200

250

300350

400

Sample size and run lengths

increase of sample size + corresponding control limits:

– same in-control run length

– decrease of out-of-control run length

Numerical values

Shewhart chart for mean (n=5)

single change of standard deviation ( -> c)

c P(|Xbar|>3 ARL SRL

1 0.0027 370.4 369.9

1.1 0.0064 156.6 156.1

1.2 0.012 80.5 80.0

1.3 0.021 47.6 47.0

1.4 0.032 31.1 30.6

1.5 0.046 22.0 21.4

Runs rules and run lengths• in-control run length: decreases (why?)

• out-of-control run length: decreases (why?)

Performance Shewhart chart

•in-control run length OK

•out-of-control run length

– OK for shifts > 2 standard deviation group average

– Bad for shifts < 2 standard deviation group average

•extra run tests

– decrease in-control length

– decrease out-of-control length

CUSUM Chart

plot cumulative sums of observation

change point

CUSUM tabular form

assume

– data normally distributed with known

– individual observations

HCCCC

CXKC

CKXC

ii

iii

iii

,max if alarm ;0

,0max

,0max

00

10

10

Choice K and H

•K is reference value (allowance, slack value)

•C+ measures cumulative upward deviations of µ0+K

•C- measures cumulative downward deviations of µ0-K

•for fast detection of change process mean µ1 :

– K=½ |µ0- µ1|

•H=5 is good choice

CUSUM V-mask form

UCL

LCL

CL

change point

Drawbacks V-mask

• only for two-sided schemes

•headstart cannot be implemented

•range of arms V-mask unclear

• interpretation parameters (angle, ...) not well determined

Rational subgroups and CUSUM

• extension to samples:

– replace by /n

• contrary to Shewhart chart , CUSUM works best with individuals

Combination•CUSUM charts appropriate for small shifts (<1.5)

•CUSUM charts are inferior to Shewhart charts for large

shifts(>1.5)

•use both charts simultaneously with ±3.5 control limits

for Shewhart chart

Headstart (Fast Initial Response)

•increase detection power by restart process

•esp. useful when process mean at restart is not equal at target

value

•set C+0 and C-

0 to non-zero value (often H/2 )

•if process equals target value µ0 is, then CUSUMs quickly return

to 0

•if process mean does not equal target value µ0, then faster alarm

CUSUM for variability

•define Yi = (Xi-µ0)/ (standardise)•define Vi = (|Yi|-0.822)/0.349

•CUSUMs for variability are:

HSSSS

SVKS

SKVS

ii

iii

iii

,max if alarm ;0

/,0max

/,0max

00

1

1

Exponentially Weighted Moving Average chart

•good alternative for Shewhart charts in case of small shifts of mean

•performs almost as good as CUSUM

•mostly used for individual observations (like CUSUM)

•is rather insensitive to non-normality

EWMA Chart for Col_1

Observation

EW

MA

CTR = 10.00

UCL = 11.00

LCL = 9.00

0 3 6 9 12 159

9.4

9.8

10.2

10.6

11

11.4

Why control charts for attribute data

•to view process/product across several characteristics

•for characteristics that are logically defined on a classification

scale of measure

N.B. Use variable charts whenever possible!

Control charts for attributes

Three widely used control charts for attributes:

• p-chart: fraction non-conforming items

• c-chart: number of non-conforming items

• u-chart: number of non-conforming items per unit

For attributes one chart only suffices (why?).

Attributes are characteristics which have a countable number of possible outcomes.

p-chart

xnx ppx

nxDP

1}{ nx ,...,1,0

Number of nonconforming products is binomially distributed

n

Dp ˆsample fraction of nonconforming:

n

ppp

)1(ˆ 2

ˆ

mean: p variance

p-chart

m

p

mn

Dp

m

ii

m

ii

1 1

ˆ

average of sample fractions:

n

pppLCL

pCLn

pppUCL

13

13

Fraction Nonconforming Control Chart:

Assumptions for p chart

• item is defect or not defect (conforming or non-conforming)

• each experiment consists of n repeated trials/units

• probability p of non-conformance is constant

• trials are independent of each other

•Counts the number of non-conformities in sample.

•Each non-conforming item contains at least one non-

conformity (cf. p chart).

•Each sample must have comparable opportunities for non-

conformities

•Based on Poisson distribution:

Prob(# nonconf. = k) =

c-chart

!k

ce kc

c-chart

Poisson distribution: mean=c and variance=c

ccLCL

cCL

ccUCL

3

3

Control Limits for Nonconformities:

is average number of nonconformities in samplec

u-chart

monitors number of non-conformities per unit.

n

cu

•n is number of inspected units per sample• c is total number of non-conformities

n

uuLCL

uCLn

uuUCL

3

3

Control Chart for Average Number of Non-conformities Per Unit:

Moving Range Chartuse when sample size is 1indication of spread: moving range

Situations:automated inspection of all unitslow production rateexpensive measurementsrepeated measurements differ only because of laboratory error

Moving Range Chart

calculation of moving range:

d2, D3 and D4 are constants depending number of observations

1 iii xxMR

2

2

3

3

d

MRxLCL

xCL

d

MRxUCL

MRDLCL

MRCL

MRDUCL

3

4

individualmeasurements

moving range

Example: Viscosity of Aircraft Primer Paint

Batch Viscosity MR

9 33.49 0.22

10 33.20 0.29

11 33.62 0.42

12 33.00 0.62

13 33.54 0.54

14 33.12 0.42

15 33.82 0.72

Batch Viscosity MR

1 33.75

2 33.05 0.70

3 34.00 0.95

4 33.81 0.19

5 33.46 0.35

6 34.02 0.56

7 33.68 0.34

8 33.27 0.41 52.33x 48.0MR

Viscosity of Aircraft Primer Paint

since a moving range is calculated of n=2 observations, d2=1.128,

D3=0 and D4=3.267

24.32128.1

48.0352.33

52.33

80.34128.1

48.0352.33

LCL

CL

UCL

CC for individuals CC for moving range

048.00

48.0

57.148.0267.3

LCL

CL

UCL

Viscosity of Aircraft Primer Paint

X

0 3 6 9 12 1532

32.5

33

33.5

34

34.5

35

CTR = 0.48

UCL = 1.57

LCL = 0.00

0 3 6 9 12 150

0.4

0.8

1.2

1.6

X

MR

Tool wear chart

known trend is removed (regression)

trend is allowed until maximum

slanted control limits

LSL

USL

LCL

UCL reset

Pitfalls

bad measurement system

bad subgrouping

autocorrelation

wrong quality characteristic

pattern analysis on individuals/moving range

too many run tests

too low detection power (ARL)

control chart is not appropriate tool (small ppms, incidents, ...)

confuse standard deviation of mean with individual