Using Control Charts to

Keep an Eye on Variability

Operations Management

Dr. Ron Lembke

Goal of Control Charts

See if process is “in control”

Process should show random values

No trends or unlikely patterns

Visual representation much easier to interpret

Tables of data – any patterns?

Spot trends, unlikely patterns easily

NFL Control Chart?

Control Charts

UCL

LCL

avg

Values

Sample Number

Control Charts

Normal Too Low Too high

5 above, or below Run of 5 Extreme variability

Control Charts

UCL

LCL

avg

1σ

2σ

2σ

1σ

Control Charts

2 out of 3 in the outer third

Out of Control Point?

Is there an “assignable cause?”

Or day-to-day variability?

If not usual variability, GET IT OUT

Remove data point from data set, and recalculate

control limits

If it is regular, day-to-day variability, LEAVE

IT IN

Include it when calculating control limits

Attributes vs. Variables

Attributes:

Good / bad, works / doesn’t

count % bad (P chart)

count # defects / item (C chart)

Variables:

measure length, weight, temperature (x-bar

chart)

measure variability in length (R chart)

p Chart Control Limits

# Defective

Items in

Sample i

Sample i

Size

n

ppzpUCLp

1

p

X i

i1

k

ni

i1

k

p Chart Control Limits

# Defective

Items in

Sample i

Sample i

Size

z = 2 for

95.5% limits;

z = 3 for

99.7% limits

# Samples

n

ppzpUCLp

1

p

X i

i1

k

ni

i1

k

n

ni

i1

k

k

p Chart Control Limits

# Defective

Items in

Sample i

# Samples

Sample i

Size

z = 2 for

95.5% limits;

z = 3 for

99.7% limits

n

ppzpUCLp

1

n

ppzpLCLp

1

n

ni

i1

k

k

p

X i

i1

k

ni

i1

k

p Chart Example

You’re manager of a 1,700

room hotel. For 7 days,

you collect data on the

readiness of all of the

rooms that someone

checked out of. Is the

process in control (use z =

3)?

© 1995 Corel Corp.

p Chart Hotel Data # Rooms No. Not Proportion

Day n Ready p

1 1,300 130 130/1,300 =.100

2 800 90 .113

3 400 21 .053

4 350 25 .071

5 300 18 .06

6 400 12 .03

7 600 30 .05

p Chart Control Limits

079.0150,4

326

150,4

30...90130

1

1

k

i

i

k

i

i

n

X

p

8.5927

150,4

7

600...80013001

k

n

n

k

i

i

068.7/)05.0...113.010.0( p

p Chart Solution

8.592,069.0 np

8.592

079.01079.03079.0

1CL

n

ppzp

0457.0LCL,1123.0UCL

0333.0079.00111.0*3079.0

Hotel Room Readiness P-Bar

0

0.02

0.04

0.06

0.08

0.1

0.12

1 2 3 4 5 6 7

UCL

Actual

LCL

R Chart

Type of variables control chart

Interval or ratio scaled numerical data

Shows sample ranges over time

Difference between smallest & largest values

in inspection sample

Monitors variability in process

Example: Weigh samples of coffee &

compute ranges of samples; Plot

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?

Hotel Example

Hotel Data

Day Delivery Time

1 7.30 4.20 6.10 3.45 5.55

2 4.60 8.70 7.60 4.43 7.62

3 5.98 2.92 6.20 4.20 5.10

4 7.20 5.10 5.19 6.80 4.21

5 4.00 4.50 5.50 1.89 4.46

6 10.10 8.10 6.50 5.06 6.94

7 6.77 5.08 5.90 6.90 9.30

R &X Chart Hotel Data

Sample

Day Delivery Time Mean Range

1 7.30 4.20 6.10 3.45 5.55 5.32

7.30 + 4.20 + 6.10 + 3.45 + 5.55

5 Sample Mean =

R &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

7.30 - 3.45 Sample Range =

Largest Smallest

R &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 8.70 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

UCL D R

LCL D R

R

R

k

R

R

i

i

k

4

3

1

Sample Range

at Time i

# Samples

Table 10.3, p.433

Control Chart Limits

n A2 D3 D4

2 1.88 0 3.278

3 1.02 0 2.57

4 0.73 0 2.28

5 0.58 0 2.11

6 0.48 0 2.00

7 0.42 0.08 1.92

R Chart Control Limits

894.37

22.4...27.485.31

k

R

R

k

i

i

0894.3*0*

232.8894.3*11.2*

3

4

RDLCL

RDUCL

R

R

10.3 Table from , 43 DD

02

46

8

1 2 3 4 5 6 7

R, Minutes

Day

R Chart Solution

UCL

X Chart Control Limits

k

R

Rk

X

X

RAXUCL

k

i

i

k

i

i

X

11

2

Sample

Range

at Time i

# Samples

Sample

Mean at

Time i

X Chart Control Limits A2 from

Table 10-3

k

R

Rk

X

X

RAXLCL

RAXUCL

k

i

i

k

i

i

X

X

11

2

2

Table 10.3 Limits

n A2 D3 D4

2 1.88 0 3.278

3 1.02 0 2.57

4 0.73 0 2.28

5 0.58 0 2.11

6 0.48 0 2.00

7 0.42 0.08 1.92

R &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 8.70 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

894.37

22.4...27.485.3

813.57

79.6...59.632.5

1

1

k

R

R

k

X

X

k

i

i

k

i

i

566.3894.3*58.0813.5*

060.8894.3*58.0813.5*

2

2

RAXLCL

RAXUCL

X

X

X Chart Solution*

0

2

4

6

8

1 2 3 4 5 6 7

X, Minutes

Day

UCL

LCL

Thinking Challenge

You’re manager of a 500-

room hotel. The hotel owner

tells you that it takes too

long to deliver luggage to the

room (even if the process

may be in control). What do

you do?

Redesign the luggage delivery process

Use TQM tools

Cause & effect diagrams

Process flow charts

Pareto charts

Solution

Method People

Material Equipment

Too

Long