### Document

```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
Values
UCL
avg
LCL
Sample Number
Definitions of Out of Control
1.
2.
3.
4.
No points outside control limits
Same number above & below center line
Points seem to fall randomly above and
below center line
Most are near the center line, only a few are
close to control limits
1.
2.
3.
8 Consecutive pts on one side of centerline
2 of 3 points in outer third
4 of 5 in outer two-thirds region
Control Charts
Normal
5 above, or below
Too Low
Too high
Run of 5
Extreme variability
Control Charts
UCL
2σ
1σ
avg
1σ
2σ
LCL
Control Charts
2 out of 3 in the outer third
Out of Control Point?

Is there an “assignable cause?”


If not usual variability, GET IT OUT


Or day-to-day variability?
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
UCL
p
 p  z sp
LCL
p
 p  z sp
z = 2 for 95.5% limits
z = 3 for 99.7% limits
p = avg defect rate
n = avg sample size
sp = sample std dev
k
k
sp 
p (1  p )
n
n
n 
i 1
k
# Samples
i
X
p
i
i 1
k
n
i 1
i
# Defective
Items in
Sample i
Sample i
Size
p Chart Example
You’re manager of a 1,700
room hotel. For 7 days,
you collect data on the
rooms that someone
checked out of. Is the
process in control (use z =
3)?
p Chart Hotel Data
Day
1
2
3
4
5
6
7
# Rooms
n
1,300
800
400
350
300
400
600
No. Not Proportion
p
130
130/1,300 =.100
90
.113
21
.053
25
.071
18
.06
12
.03
30
.05
p Chart Control Limits
k
n 
n
i
i 1

1300  800 ...  600
k

4 ,150
7
 592 . 8
7
k
p 

Xi
i 1
k
n

130  90 ...  30
4 ,150

326
 0 . 079
4 ,150
i
i 1
p  ( 0 . 10  0 . 113  ...  0 . 05 ) / 7  . 068
p Chart Solution
p  0 . 079 ,
sp 
n  592 . 8
p  1  p 

n
0 . 079  1  0 . 079
592 . 8
CL  p  z  s p  0 . 079  3 * 0 . 111
 0 . 079  0 . 0333
UCL  0 . 1123 ,
LCL  0 . 0457

 0 . 0111
0.12
0.1
0.08
UCL
0.06
Actual
LCL
0.04
0.02
0
1
2
3
4
5
6
7
R Chart

Type of variables control chart


Shows sample ranges over time



Interval or ratio scaled numerical data
Difference between smallest & largest values
in inspection sample
Monitors variability in process
Example: Weigh samples of coffee &
compute ranges of samples; Plot
Why do we need 2 charts?
Consistent, but the average is in the wrong place
UCL
UCL
LCL
LCL
X-Bar Chart
R Chart
The average works out ok, but way too much variability between points
UCL
UCL
LCL
LCL
X-Bar Chart
R Chart
Hotel Example
You’re manager of a 500room 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 Data
Day
Delivery Time
1
7.30 4.20 6.10 3.45
2
4.60 8.70 7.60 4.43
3
5.98 2.92 6.20 4.20
4
7.20 5.10 5.19 6.80
5
4.00 4.50 5.50 1.89
6 10.10 8.10 6.50 5.06
7
6.77 5.08 5.90 6.90
5.55
7.62
5.10
4.21
4.46
6.94
9.30
R &X Chart Hotel Data
Day
Delivery Time
1 7.30 4.20 6.10 3.45 5.55
Sample
Mean Range
5.32
7.30 + 4.20 + 6.10 + 3.45 + 5.55
Sample Mean =
5
R &X Chart Hotel Data
Day
Delivery Time
1 7.30 4.20 6.10 3.45 5.55
Largest
Smallest
Sample
Mean Range
5.32
3.85
Sample Range = 7.30 - 3.45
R &X Chart Hotel Data
Day
1 7.30
2 4.60
3 5.98
4 7.20
5 4.00
6 10.10
7 6.77
Delivery Time
4.20 6.10 3.45
8.70 7.60 4.43
2.92 6.20 4.20
5.10 5.19 6.80
4.50 5.50 1.89
8.10 6.50 5.06
5.08 5.90 6.90
5.55
7.62
5.10
4.21
4.46
6.94
9.30
Sample
Mean Range
5.32
3.85
6.59
4.27
4.88
3.28
5.70
2.99
4.07
3.61
7.34
5.04
6.79
4.22
R Chart Control Limits
U C LR  D 4  R
Table 10.3, p.433
L C LR  D 3  R
k
 Ri
R 
i 1
k
Sample Range
at Time i
# Samples
Control Chart Limits
n
A2
D3
D4
2
1 .8 8
0
3 .2 7 8
3
1 .0 2
0
2 .5 7
4
0 .7 3
0
2 .2 8
5
0 .5 8
0
2 .1 1
6
0 .4 8
0
2 .0 0
7
0 .4 2
0 .0 8
1 .9 2
R Chart Control Limits
k
R 

Ri
i 1

k
3 . 85  4 . 27  ...  4 . 22
 3 . 894
7
UCL
R
 D 4 * R  2 . 11 * 3 . 894  8 . 232
LCL
R
 D 3 * R  0 * 3 . 894  0
D 3 , D 4 from Table 10.3
R Chart Solution
9
8
7
6
5
UCL
Range
LCL
4
3
2
1
0
1
2
3
4
5
6
7
X Chart Control Limits
UCL
 X  A2  R
X
Sample
Mean at
Time i
k

X 
i 1
k
k
R
Xi
R 
i 1
k
i
Sample
Range
at Time i
# Samples
X Chart Control Limits
A2 from
Table 10-3
UCL
X
 X  A2  R
LCL
X
 X  A2  R
k
X
X 
i 1
k
k
R
i
R 
i 1
k
i
Control Chart Limits
n
A2
D3
D4
2
1 .8 8
0
3 .2 7 8
3
1 .0 2
0
2 .5 7
4
0 .7 3
0
2 .2 8
5
0 .5 8
0
2 .1 1
6
0 .4 8
0
2 .0 0
7
0 .4 2
0 .0 8
1 .9 2
R &X Chart Hotel Data
Day
1 7.30
2 4.60
3 5.98
4 7.20
5 4.00
6 10.10
7 6.77
Delivery Time
4.20 6.10 3.45
8.70 7.60 4.43
2.92 6.20 4.20
5.10 5.19 6.80
4.50 5.50 1.89
8.10 6.50 5.06
5.08 5.90 6.90
5.55
7.62
5.10
4.21
4.46
6.94
9.30
Sample
Mean Range
5.32
3.85
6.59
4.27
4.88
3.28
5.70
2.99
4.07
3.61
7.34
5.04
6.79
4.22
X Chart Control Limits
k

Xi
i 1
X 

k
5 . 32  6 . 59  ...  6 . 79
 5 . 813
7
k
R
i 1
R 
k
i

3 . 85  4 . 27  ...  4 . 22
 3 . 894
7
UCL
X
 X  A 2 * R  5 . 813  0 . 58 * 3 . 894  8 . 060
LCL
X
 X  A 2 * R  5 . 813  0 . 58 * 3 . 894  3 . 566
X Chart Solution*
9
8
7
6
5
UCL
Mean
LCL
4
3
2
1
0
1
2
3
4
5
6
7
Summary




Overview of “In Control”
Attribute vs Continuous Control Charts
P Charts
X-bar and R charts
```