Which Control Charts to Use Where

NATIONAL PRODUCTIVITY COUNCIL

WELCOMES YOU TO A PRESENTATION

ON

CONTROL CHARTSBy B.Girish

Dy. Director

Three SQC Categories Traditional descriptive statistics

e.g. the mean, standard deviation, and range

Acceptance sampling used to randomly inspect a batch of goods to determine acceptance/rejection

Does not help to catch in-process problems

Statistical process control (SPC) Involves inspecting the output from a process Quality characteristics are measured and charted Helpful in identifying in-process variations

Statistical Process Control (SPC)

A methodology for monitoring a process to identify special causes of variation and signal the need to take corrective action when appropriate

SPC relies on control charts

SPC Implementation Requirements Top management commitment Project champion Initial workable project Employee education and

training Accurate measurement system

Traditional Statistical Tools 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

n

xx

n

1ii

1n

Xxσ

n

1i

2

i

Distribution of Data Normal distributions Skewed distribution

Sources of Variation Common causes of variation

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

Common Causes

Special Causes

Histograms do not take into account changes over time.

Control charts can tell us when a process changes

Introduction to Control Charts

Important uses of the control chart

1. Most processes do not operate in a state of statistical control

2. Consequently, the routine and attentive use of control charts will identify assignable causes. If these causes can be eliminated from the process, variability will be reduced and the process will be improved

3. The control chart only detects assignable causes. Management, operator, and engineering action will be necessary to eliminate the assignable causes.

Monitor Variation in Data Exhibit trend - make correction before

process is out of control A Process - A Repeatable Series of

Steps Leading to a Specific Goal


Show When Changes in Data are Due to: Special or assignable causes

Fluctuations not inherent to a process Represent problems to be corrected Data outside control limits or trend

Chance or common causes Inherent random variations Consist of numerous small causes of random

variability

(continued)Introduction to Control Charts

Graph of sample data plotted over time

020406080

1 3 5 7 9 11

X

Time

Special Cause Variation

Common Cause Variation

Process Average

Mean

UCL

LCL


Commonly Used Control Charts

Variables data x-bar and R-charts x-bar and s-charts Charts for individuals (x-charts)

Attribute data For “defectives” (p-chart, np-chart) For “defects” (c-chart, u-chart)


Popularity of control charts

1) Control charts are a proven technique for improving productivity.

2) Control charts are effective in defect prevention.

3) Control charts prevent unnecessary process adjustment.

4) Control charts provide diagnostic information.

5) Control charts provide information about process capability.

Control Chart Selection

Quality Characteristicvariable attribute

n>1?

n>=10 or computer?

x and MRno

yes

x and s

x and Rno

yes

defective defect

constant sample size?

p-chart withvariable samplesize

no

p ornp

yes constantsampling unit?

c u

yes no

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

Control Charts for Attributes –P-Charts & C-Charts

Use P-Charts for quality characteristics that are discrete and involve yes/no or good/bad decisions Number of leaking caulking tubes in a box of 48 Number of broken eggs in a carton

Use C-Charts for discrete defects when there can be more than one defect per unit Number of flaws or stains in a carpet sample

cut from a production run Number of complaints per customer at a hotel

Control Chart Design Issues Basis for sampling Sample size Frequency of sampling Location of control limits

Developing Control Charts

1. Prepare Choose measurement Determine how to collect data,

sample size, and frequency of sampling

Set up an initial control chart

2. Collect Data Record data Calculate appropriate statistics Plot statistics on chart

Pre-Control

nominal value

Green Zone

Yellow Zones

RedZone

RedZone

LTL UTL

Control LimitsUCL = Process Average + 3 Standard

DeviationsLCL = Process Average - 3 Standard

DeviationsProcess Average

UCL

LCL

X

+ 3

- 3

TIME

Next Steps3. Determine trial control limits

Center line (process average) Compute UCL, LCL

4. Analyze and interpret results Determine if in control Eliminate out-of-control points Recompute control limits as

necessary

Setting Control Limits Percentage of values

under normal curve

Control limits balance

risks like Type I error

Comparing Control Chart Patterns

X XX

Common Cause Variation: No Points

Outside Control Limits

Special Cause Variation: 2 Points

Outside Control Limits

Downward Pattern: No Points Outside Control Limits but

Trend Exists

Typical Out-of-Control Patterns Point outside control limits Sudden shift in process average Cycles Trends Hugging the center line Hugging the control limits Instability

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?

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Processes In Control

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Process Out of Control

Shift in Process Average

Identifying Potential Shifts

Cycles

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Sudden stability

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Mixtures

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UCL

Center

line

LCL

PROCESS STREAMS

Chart

ProcessIn

Control

ProcessOut ofControl

Run

Chart points donot form a parti-cular pattern andlie within theupper and lowerchart limits

Chart points form a particular pattern OR oneor more pointslie beyond theuppor or lowerchart limits

Chart points areon one side of thecenter line. The number of pointsin a run is calledthe “length of therun”

UCL 10

x 19

lcl 30

UCL 10

x 19

lcl 30

UCL 10

x 19

lcl 30

UCL 10

x 19

lcl 30

UCL 10

x 19

lcl 30

UCL 10

x 19

lcl 30

The process is stable, notchanging. Doesn’t necesarily mean to leavethe process alone. Maybe opportunities to improvethe process and enjoy substantial benefits

Alerts us that the processis changing. Doesn’t mean you need to take a corrective action. May berelate to a change you havemade. Be sureto identifythe reason\(s) before takingany constructive actions(w)

Suggest the process hasundergone a permanentchange (positive ornegative) and is nowbecoming stable. Often requires tha t you recompute the controllines for future interpre-tation efforts.

Description Example # 1 Example # 2 Interpretation

Interpreting Control Charts

Chart

Trend

Cycle

Hugging

A continued riseor fall in a seriesof points (7 ormore consecutivepoints direction)

Chart ponts showthe same patternchanges (e.g.riseor fall) over equalperiods of time

Chart points areclose to the centerline or to a control limit line(2 out of 3, 3 out of4, or 4 out of 10.)

UCL 10

x 19

lcl 30

UCL 10

x 19

lcl 30

UCL 10

x 19

lcl 30

UCL 10

x 19

lcl 30

UCL 10

x 19

lcl 30

CL 10

x 19

lcl 30

Description Example # 1 Example # 2 Interpretation

12 3

45

67

1/2

1/2

12

34

56

1/2

1/2

Often seen after some changehas been made. Helps tellyou if the change(s) had apositive or negative effect.may also be part of a learning curve associatedwith some form of training

often relates to factors thatinfluence the process in apredictable manner. Factorsoccur over a set time periodand a positive/negative effectHelps determine future workload and staffing levels

Suggests a different type ofdata has been mixed into thesub-group being sampled.Often need to change thesub-group, reassemble thedata, redraw the controlchart

Interpreting Control Charts

When to Take Corrective Action Corrective Action Should Be Taken

When Observing Points Outside the Control Limits or when a Trend Has Been Detected Eight consecutive points above the

center line (or eight below) Eight consecutive points that are

increasing (decreasing)

Out-of-Control Processes If the Control Chart Indicates an Out-of-

Control Condition (a Point Outside the Control Limits or Exhibiting Trend) Contains both common causes of variation

and assignable causes of variation The assignable causes of variation must be

identified If detrimental to quality, assignable causes of

variation must be removed If increases quality, assignable causes must be

incorporated into the process design

In-Control Process If the Control Chart is Not Indicating

Any Out-of-Control Condition, then Only common causes of variation exist It is sometimes said to be in a state of

statistical control If the common-cause variation is small,

then control chart can be used to monitor the process

If the common-cause variation is too large, the process needs to be altered

Types of Error First Type:

Belief that observed value represents special cause when, in fact, it is due to common cause

Second Type: Treating special cause variation as if

it is common cause variation

Remember Control does not mean that the

product or service will meet the needs. It only means that the process is consistent (may be consistently bad).

Capability of meeting the specification.

How to use the results By eliminating the special causes

first and then reducing the common causes, quality can be improved.

Final Steps

5. Use as a problem-solving tool

Continue to collect and plot data

Take corrective action when necessary

6. Compute process capability

Process Capability Product Specifications

Preset product or service dimensions, tolerances e.g. bottle fill might be 16 oz. ±.2 oz. (15.8oz.-16.2oz.) Based on how product is to be used or what the customer expects

Process Capability – Cp and Cpk Assessing capability involves evaluating process variability relative

to preset product or service specifications Cp assumes that the process is centered in the specification range

Cpk helps to address a possible lack of centering of the process

6σ

LSLUSL

width process

width ionspecificatCp

3σ

LSLμ,

3σ

μUSLminCpk

Relationship between Process Variability and Specification Width

Three possible ranges for Cp

Cp = 1, as in Fig. (a), process variability just meets

specifications

Cp ≤ 1, as in Fig. (b), process not capable of producing within specifications

Cp ≥ 1, as in Fig. (c), process exceeds minimal

specifications

One shortcoming, Cp assumes that the process is centered on the specification range

Cp=Cpk when process is centered

Computing the Cp Value at Cocoa Fizz: three bottling machines are being evaluated for possible use at the Fizz plant. The machines must be capable of meeting the design specification of 15.8-16.2 oz. with at least a process capability index of 1.0 (Cp≥1)

The table below shows the information gathered from production runs on each machine. Are they all acceptable?

Solution: Machine A

Machine B

Cp=

Machine C

Cp=

Machine

σ USL-LSL

6σ

A .05 .4 .3

B .1 .4 .6

C .2 .4 1.2

1.336(.05)

.4

6σ

LSLUSLCp

Computing the Cpk Value at Cocoa Fizz

Design specifications call for a target value of 16.0 ±0.2 OZ.

(USL = 16.2 & LSL = 15.8) Observed process output has

now shifted and has a µ of 15.9 and a

σ of 0.1 oz.

Cpk is less than 1, revealing that the process is not capable

.33.3

.1Cpk

3(.1)

15.815.9,

3(.1)

15.916.2minCpk

Target theis T

12

2 T

CC ppm

10.7171mmat centered is process assumebut above, as same :Example

2 where1

,min3

3

Tolerance

TKKCC

CCC

LTLC

UTLC

ppk

puplpk

pl

pu

Process Capability (2)

086.10868.03

7171.100.11

puC

834.00868.03

5.107171.10

plC

8977.0

868.075.107171.10

1

960.0

2

2

pmC

Capability Versus Control

Control

Capability

Capable

Not Capable

In Control Out of Control

IDEAL

Excel Template

Special Variables Control Charts

x-bar and s charts x-chart for individuals

Charts for Attributes Fraction nonconforming (p-chart)

Fixed sample size Variable sample size

np-chart for number nonconforming

Charts for defects c-chart u-chart

p Chart Control Chart for Proportions

Is an attribute chartattribute chart Shows Proportion of Nonconforming (Success Success

) Items E.g., Count # of nonconforming chairs & divide by

total chairs inspected Chair is either conforming or nonconforming

Used with Equal or Unequal Sample Sizes Over Time Unequal sizes should not differ by more than

±25% from average sample size

p Chart Control Limits

(1 )max 0, 3p

p pLCL p

n

(1 )3p

p pUCL p

n

1

k

ii

nn

k

Average Group Size

1

1

k

iik

ii

Xp

n

Average Proportion of Nonconforming Items

# Defective Items in Sample i

Size of Sample i

# of Samples

p Chart Example

You’re manager of a 500-room hotel. You want to achieve the highest level of service. For 7 days, you collect data on the readiness of 200 rooms. Is the process in control?

p Chart Hotel Data

# NotDay # Rooms Ready Proportion

1 200 16 0.0802 200 7 0.0353 200 21 0.1054 200 17 0.0855 200 25 0.1256 200 19 0.0957 200 16 0.080

1

1

121.0864

1400

k

iik

ii

Xp

n

p Chart Control Limits Solution

16 + 7 +...+ 16

1 1400200

7

k

ii

nn

k

1 .0864 1 .08643 .0864 3

200

.0864 .0596 or .0268,.1460

p pp

n

Mean

p Chart Control Chart Solution

UCL

LCL

0.00

0.05

0.10

0.15

1 2 3 4 5 6 7

P

Day

Individual points are distributed around without any pattern. Any improvement in the process must come from reduction of common-cause variation, which is the responsibility of the management.

p

p

p Chart in PHStat PHStat | Control Charts | p Chart …

Excel Spreadsheet for the Hotel Room Example

Microsoft Excel Worksheet

Worker Day 1 Day 2 Day 3 All Days

A 9 (18%) 11 (12%) 6 (12%) 26 (17.33%)

B 12 (24%) 12 (24%) 8 (16%) 32 (21.33%)

C 13 (26%) 6 (12%) 12 (24%) 31(20.67%)

D 7 (14%) 9 (18%) 8 (16%) 24 (16.0%)

Totals 41 38 34 113

Understanding Process Variability:Red Bead Example

Four workers (A, B, C, D) spend 3 days to collect beads, at 50 beads per day. The expected number of red beads to be collected per day per worker is 10 or 20%.

Average Day 1 Day 2 Day 3 All Days

X 10.25 9.5 8.5 9.42

p 20.5% 19% 17% 18.83%

Understanding Process Variability:Example Calculations

113.1883

50(12)p

(1 ) .1883(1 .1883)3 .1883 3

50 .1883 .1659

p pp

n

_

.1883 .1659 .0224

.1883 +.1659 .3542

LCL

UCL

0 A1 B1 C1 D1 A2 B2 C2 D2 A3 B3 C3 D3

Understanding Process Variability:Example Control Chart

.30

.20

.10

p

UCL

LCL

_

Morals of the Example

Variation is an inherent part of any process. The system is primarily responsible for worker performance. Only management can change the system. Some workers will always be above average, and some will be below.

The c Chart Control Chart for Number of

Nonconformities (Occurrences) in a Unit (an Area of Opportunity) Is an attribute chartattribute chart

Shows Total Number of Nonconforming Items in a Unit E.g., Count # of defective chairs

manufactured per day Assume that the Size of Each Subgroup

Unit Remains Constant

c Chart Control Limits

3cLCL c c 3cUCL c c

1

k

ii

cc

k

Average Number of Occurrences

# of Samples

# of Occurrences in Sample i

c Chart: Example

You’re manager of a 500-room hotel. You want to achieve the highest level of service. For 7 days, you collect data on the readiness of 200 rooms. Is the process in control?

c Chart: Hotel Data# Not

Day # Rooms Ready1 200 162 200 73 200 214 200 175 200 256 200 197 200 16

c Chart: Control Limits Solution

1 16 7 19 1617.286

7

3 17.286 3 17.285 4.813

3 29.759

k

ii

c

c

cc

k

LCL c c

UCL c c

c Chart: Control Chart Solution

UCL

LCL0

10

20

30

1 2 3 4 5 6 7

c

Day

c

Individual points are distributed around without any pattern. Any improvement in the process must come from reduction of common-cause variation, which is the responsibility of the management.

c

Variables Control Charts: R Chart Monitors Variability in Process

Characteristic of interest is measured on numerical scale

Is a variables control chartvariables control chart Shows Sample Range Over Time

Difference between smallest & largest values in inspection sample

E.g., Amount of time required for luggage to be delivered to hotel room

R Chart Control Limits

Sample Range at Time i or Subgroup i

# Samples

From Table4RUCL D R

3RLCL D R

1

k

ii

RR

k

R Chart Example

You’re 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 and Mean Chart Hotel Data

Sample SampleDay Average Range

1 5.32 3.852 6.59 4.273 4.88 3.284 5.70 2.995 4.07 3.616 7.34 5.047 6.79 4.22

R Chart Control Limits Solution

From Table (n = 5)

1 3.85 4.27 4.223.894

7

k

ii

RR

k

4

3

2.114 3.894 8.232

0 3.894 0

R

R

UCL D R

LCL D R

R Chart Control Chart Solution

UCL

02468

1 2 3 4 5 6 7

Minutes

Day

LCL

R_

Variables Control Charts: Mean Chart (The Chart) Shows Sample Means Over Time

Compute mean of inspection sample over time

E.g., Average luggage delivery time in hotel

Monitors Process Average Must be preceded by examination of

the R chart to make sure that the process is in control

X

Mean Chart

Sample Range at Time i

# Samples

Sample Mean at Time i

Computed From Table

2XUCL X A R

2XLCL X A R

1 1 and

k k

i ii i

X RX R

k k

Mean Chart ExampleYou’re 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 and Mean Chart Hotel Data


1 5.32 3.852 6.59 4.273 4.88 3.284 5.70 2.995 4.07 3.616 7.34 5.047 6.79 4.22

Mean Chart Control Limits Solution

1

1

2

2

5.32 6.59 6.795.813

7

3.85 4.27 4.223.894

7

5.813 0.577 3.894 8.060

5.813 0.577 3.894 3.566

k

i

i

k

ii

X

X

XX

k

RR

k

UCL X A R

LCL X A R

From Table E.9 (n = 5)

Mean Chart Control Chart Solution

UCL

LCL

02468

1 2 3 4 5 6 7

Minutes

Day

X__

R Chart and Mean Chartin PHStat PHStat | Control Charts | R & Xbar

Charts …

Excel Spreadsheet for the Hotel Room Example

Microsoft Excel Worksheet

Process Capability Process Capability is the Ability of a

Process to Consistently Meet Specified Customer-Driven Requirements

Specification Limits are Set by Management in Response to Customer’s Expectations

The Upper Specification Limit (USL) is the Largest Value that Can Be Obtained and Still Conform to Customer’s Expectation

The Lower Specification Limit (LSL) is the Smallest Value that is Still Conforming

Estimating Process Capability Must Have an In-Control Process First Estimate the Percentage of Product or

Service Within Specification Assume the Population of X Values is

Approximately Normally Distributed with Mean Estimated by and Standard Deviation Estimated by

X

2/R d

Estimating Process Capability For a Characteristic with an LSL and a

USL

where Z is a standardized normal random variable

(continued)

2 2

P(an outcome will be within specification)

P( )

= P/ /

LSL X USL

LSL X USL XZ

R d R d

Estimating Process Capability For a Characteristic with Only a LSL


(continued)

2


P( )

= P/

LSL X

LSL XZ

R d

Estimating Process Capability For a Characteristic with Only a USL


(continued)

2


P( )

= P/

X USL

USL XZ

R d

You’re manager of a 500-room hotel. You have instituted a policy that 99% of all luggage deliveries must be completed within 10 minutes or less. For 7 days, you collect dataon 5 deliveries per day. Is the process capable?

Process Capability Example

Process Capability:Hotel Data


1 5.32 3.852 6.59 4.273 4.88 3.284 5.70 2.995 4.07 3.616 7.34 5.047 6.79 4.22

Process Capability:Hotel Example Solution

5.813X 3.894R 2and 2.326d P(A delivery is made within specification)

= P( 10)

10 5.813= P

3.894 / 2.326

= P( 2.50) .9938

X

Z

Z

5n

Therefore, we estimate that 99.38% of the luggage deliveries will be made within the 10 minutes or less specification. The process is capable of meeting the 99% goal.

Capability Indices Aggregate Measures of a Process’ Ability

to Meet Specification Limits The larger (>1) the values, the more capable

a process is of meeting requirements Measure of Process Potential Performance

Cp>1 implies that a process has the potential of having more than 99.73% of outcomes within specifications

2

specification spread

process spread6 /p

USL LSLC

R d

Capability Indices Measures of Actual Process Performance

For one-sided specification limits

CPL (CPU) >1 implies that the process mean is more than 3 standard deviations away from the lower (upper) specification limit

(continued)

23 /

X LSLCPL

R d

23 /

USL XCPU

R d

Capability Indices For two-sided specification limits

Cpk = 1 indicates that the process average

is 3 standard deviations away from the closest specification limit

Larger Cpk indicates larger capability of meeting the requirements

(continued)

min ,pkC CPL CPU

You’re manager of a 500-room hotel. You have instituted a policy that all luggage deliveries must be completed within 10 minutes or less. For 7 days, you collect data on 5 deliveries per day. Compute an appropriate capability index for the delivery process.

Process Capability Example

Process Capability:Hotel Data


1 5.32 3.852 6.59 4.273 4.88 3.284 5.70 2.995 4.07 3.616 7.34 5.047 6.79 4.22

Process Capability:Hotel Example Solution

5.813X 3.894R 2and 2.326d 5n

Since there is only the upper specification limit, we need to only compute CPU. The capability index for the luggage delivery process is .8337, which is less than 1. The upper specification limit is less than 3 standard deviations above the mean.

2

10 5.8130.833672

3 3.894 / 2.3263 /

USL XCPU

R d

Which Control Charts to Use Where

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