Control Charts – Part 1

Data collection, chart ‘anatomy’, control

charts for variables

2

Data Collection

Recall the story about the drunk who loses his keys

and is seen near a street light looking for them.

Someone comes along and offers to help.

Helper – “You sure you dropped them near here?”

Drunk – “Nope…dropped them over there in the bushes.”

Helper – “Then why are we looking here???”

Drunk – “The light is better here.”

3

Data Collection FAQs

Q: What should I measure?

A: Depends on what you want to do with the

information and on the process itself

Q: How many pieces should I measure (i.e.,

sample size)?

A: Depends on how accurate you want to be

and the amount of variability.

4

Rational Subgrouping

Choices:

Subgroup size

Frequency

Goal:

minimize variability within the subgroup and

maximize (opportunity for variability) between

subgroups

5

Rational Subgrouping

Which approach would best meet the goal:

A. Measure 140 pieces at noon each day

B. Measure 5 pieces every 20 minutes between 8

am and 5 pm

C. Measure 70 pieces at 8 am, 70 pieces at 5 pm

D. All of the above

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Anatomy of a Control Chart

Variables (X-bar/R) chart - one chart for

process centering, one chart for spread

Points are average (or range) for each subgroup

Centerline (the ‘grand average’)

Upper and lower control limits (UCL and LCL)

7

Calculating Control Limits

To calculate the centerline and limits,

remember what we’re monitoring

For X-bar chart we’re monitoring the average and

the variability of the average

For R chart we’re monitoring the variability and

the variability of the variability (???)

8

Setting Up an X Chart

Assumed – Estimates of µ and have

already been made from a large sample.

Draw centerline at X

Draw trial upper limit at X + 3 X

Draw trial lower limit at X - 3 X

But what is this?This we know. It’s the grand

average of all the X from the

samples

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1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

The distribution of X

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Sample 1

Sample 2

Sample 3

Sample 4

“Spread” of the X’s

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1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

The spread of the X’s

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

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1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

The spread of the X’s

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

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The Distribution of X

The spread of X ( X) is much less than

the spread of the individual data points.

In fact, statistical theory tells us that:

nX

This we know

This we have to estimate

13

Q How can we estimate ?

A From our sample data, either s or R

Q Are these good estimates of ?

A Unfortunately, NO

How well s or R estimate depends on

the sample size. So we use correction

factors:

42

.ˆ

c

sor

d

Rest

14

X Chart Limits

Therefore, the limits for the X chart are:

RAXn

dR

XXUCLXX 2

233

RAXn

dR

XXLCLXX 2

233

15

Setting up R Charts

Draw centerline at R

Draw trial upper limit at D4R

Draw trial lower limit at D3R

16

Control Charts – Let’s build one

Back to XYZ Forest Products

One recommendation based on DOE results was

to control/monitor MC at 6%

Easier said than done!

What does it mean to ‘control?’

How are things now?

How much variability is acceptable?

17

Control ChartHands-On

XYZ decides to collect 5 handles from

production every 20 minutes.

They cut 2 inches from the tip of each handle

and measure MC using the ovendry test

Data are in X bar R data.xlsx

Let’s calculate the centerline and control

limits for both the X-bar and the R chart

18

Interpreting the Chart

Start with R chart

If variability not ‘in control’, then limits are

meaningless

What are indicators of ‘out of control’?

Next, look at X-bar chart

Histogram

Capability – can we meet specs.?

19

Process Capability AnalysisCan we meet the specifications?

Process capability - Measuring process potential by comparing the specification width to the variation of the process

or

How “wide” are the specs relative to the natural spread of the process?

20

Capability Indices:

6

LSLUSLC p

ˆ3

ˆ LSLC pl

ˆ3

ˆ

USLC pu

pupl CC

pkC ,min

21

A visual look at process variation

µLSL USL(Target) 22

It’s up, it’s… good???

23 ft. 4 inches

B

E

A

V

S

23

What are the “specs?”

23.3011.65

B

E

A

V

S

24

Kicker “Capability”

Kicker 1 – After 100 kicks, side-to-side

variation is 18 feet (± 9).

Kicker 2 – …variation is 8 feet (± 4).

25

Kicker “Capability”

23.30 11.65

Kicker 1

Kicker 2

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Cp

A Simple Ratio

The Specifications - Each kicker has 23.3 total feet of “wiggle room”

“Process” Spread Kicker 1 - 18 feet

Kicker 2 - 8 feet

Their Cp is spec ÷ process Kicker 1 – 23.3/18 = 1.29

Kicker 2 – 23.3/8 = 2.91

Who is the better kicker?

27

That depends…

23.30 11.65

Kicker 1

Kicker 2

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Cpk Accounts for Centering

Let’s say:

Kicker 1 centered at 11.65 feet (±9 feet)

Kicker 2 centered at 2 feet (±4 feet)

Cpk

Kicker 1 = MIN[(11.65-0)/9, (23.3-11.65)/9] = 1.29

Kicker 2 = MIN[(2-0)/4, (23.3-2)/4] = 0.5

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How about XYZ’s Process

Can it meet the specifications?

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Does out of control = out of spec?

No!

Control limits and specification limits are not

related

Control limits are based on the ‘voice of the

process’

Specification limits are based on the ‘voice of the

customer’

A process can be in control but out of spec.

A process can be out of control but in spec.

31

Attributes Control Charts

So far, only discussed variables data –

measurements of length, width, density, etc.

What about other types of data:

Fraction nonconforming

Counts of nonconformities

32

Attributes Control Charts

p charts – counts of fraction nonconforming in a

sample (sample size may vary)

np charts – counts of number of items

nonconforming (requires uniform sample size)

c charts – counts of nonconformities per unit

(sample size may be 1)

u charts – counts of nonconformities per unit area

Discrete data are things we count – require a

different method than continuous data

33

34

Percent Nonconforming p charts

Advantages:

can be constructed

from existing

inspection data

may suggest areas for

use of variables control

charts

fairly simple arithmetic

(look-up tables

unnecessary)

Disadvantages:

don’t provide as much

information as

variables charts

may encourage

continued use of

product QC over

process QC

p charts

Data collection:

n = number of items inspected

D = number of nonconforming items

p = D÷n

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1. Select m preliminary samples, each of size n

2. pi = Di / n where i = 1, 2, …, m (m should = 20 to 25)

3. p = S pi

m

p estimates the unknown fraction nonconforming p

Constructing a p chart: estimating p

i=1

m

n

pppLCL

pCenterline

n

pppUCL

13

13

36

p chart example

Let’s use data from XYZ Forest Products’

DOE

Open file pchart.xlsx

Enter data from Table 2, DOE publication

37

Interpreting the chart

Is the p-chart in control?

Consider the process we’re analyzing

38

Review

Voice of the Customer What’s the most important/costly quality problem?

(Checksheets, Pareto Charts)

Translation Where in our process might these problems occur?

(Flowcharts, C&E Diagrams)

Voice of the Process What variables impact the problem and how? (DOE)

How is our process currently operating? (Control Charts)

Once we ‘fix’ the problem, how can we ensure it stays fixed? (Control Charts)

39

Continuous Improvement

40

A Model for Continuous ImprovementNot just problem-solving!

Voice of the Customer

What can we improve that would benefit our customers?

Translation

Where in the process should we make changes?

Voice of the Process

What specific changes should we make?

Once we ‘improve’ how can we ensure things don’t slip back?

Repeat!

41

Plant Optimization

42

Q&A/Wrap-Up

43

Addendum

44

SPCStep-by-Step

Do

Plan

Check

Act

What to measure?

Where?

How?

How often?

Collect measurements

Calculate stats

Plot on chart

Interpret the charts

Compare to past data

Fix problems

Standardize processes

Monitor

Continuous improvement

Where does d2 come from?

The value d2 is a correction factor to

estimate from R. It is the average of the

w distribution, where:

Rw

• What does the w distribution look like?

46

Example: The w distribution for n = 5

Source: Duncan, A.J. 1986. Quality Control and Industrial

Statistics: Fifth Edition

For n = 5,

d2 = 2.326

47

To summarize -

the table values are:

d2 is the average of the w distribution – a

distribution that relates the sample range

R, to the population standard deviation

and

ndA

2

2

3

48

Where do D4 and D3 come from?

Recap: To construct an R-chart, we need a

centerline (R) and a value that estimates R,

the variation of sample ranges (the “variation

of the variation”)

Again, there is no formula, however as with

estimating (using d2), using the w

distribution, we get a ratio - R/

49

Another table value - d3

Recall that w is R/. The standard deviation

of w (w) is R/, also known as d3 and

2

3

d

RdR

2

3 ˆd

Rd RR

^

^

50

And now, to get D4:

RR RUCL 3

2

33d

RdRUCLR

4

2

331 DRd

dRUCLR

51

To get D3:

RR RLCL 3

2

33d

RdRLCLR

3

2

331 DRd

dRLCLR

52

To summarize:

d3 is the standard deviation of the w

distribution – a distribution that relates the

sample range R, to the population standard

deviation and

2

343 31,

d

dDD

53

Acknowledgements

SUSTAINABLE