u-Charts: Attribute Control Chart
By Nathan Westover
Brigham Young University
November 2012
Agenda
1. U-Charts Defined
2. Brainstorming Exercise: How can this tool be used in your organization
3. Nuts and Bolts: What are Control Charts?
4. Nuts and Bolts: Attribute Control Charts vs. Variable control Charts
5. Nuts and Bolts: What information do u-Charts convey?
6. Nuts and Bolts: How u-Charts are developed?
7. How it works
8. Real World Example
9. Sample Exercise
10.Summary
11.Readings List
U-Charts Defined
u-Chart: A control chart that tacks the variation in the average number of
defects per unit.
Example: XXX company produces cold weather coats. For
the X123 model of coat, XXX uses a u-chart to track the
average number of defects each coat has from a sample.
Brainstorming Exercise: How can this tool be used
in your organization?
• Think of a few products or product categories in your company that
appear to constantly have defects
• Write these down at the top of your notepad
• Throughout the presentation think of how you can implement u-Charts
with these products
Nuts and Bolts: What are Control Charts?
Tools for monitoring process Variation
Types of Control Charts
Variables
x
x-bar
R
MR
s
Attributes
Process population average
p
Mean or Average
np
Proportion Defective
Number of Defective or Number
non-conforming
Range
c
Number nonconforming in a
consistent sample space
Moving Range
u
Number of defects per unit
Standard Deviation
Nuts and Bolts:
Attribute Control Charts vs. Variable Control Charts
Attribute Control Charts: Process control chart that tracks variation in either-or
situations
Example: XXX company produces flash memory used in digital
MP3 players. XXX uses an attribute control chart to track the
proportion of units that are defective
Variable Control Charts: Process control chart that tracks variation in continuous
measurements such as weight, height, or volume
Example: XXX company produces flash memory used in digital
MP3 players. XXX uses a variable control chart to track the
average number of flash memory that is produced per hour.
Defect Charts: u-Charts vs. c-Charts
u-Charts:
• Average number of defects
per unit
• Units do not have to be, but
can be from the same
sample space
• Ex: Average number of
defects in a sample of the
unibody casing for all sizes
of Apple Macbook Pro’s
c-Charts:
• Actual number of Defects
per unit
• Units must be from the
same sample space Ex. Size,
Height, Length
• Ex: Actual number of
defects in a sample of the
unibody casing for an Apple
17” Macbook Pro
What information does a u-chart convey and how
can it be used?
• Non-random Variation in the average number of defects from a given sample space.
That sample space can be the same or varied.
• This information can then be used in a quality rating system for rating vendors or
suppliers, depending on the purpose behind using the chart.
• If the chart is for internal use, it can help a company to see the whether the variation
is random or not, and can give insight as to what needs to be improved in the process
How are u-chart’s developed/work?
Step 1:
• Determine the sample space that is going to be used e.g. Sample amount, Type of
product, Varied number of units or standard number
Step 2:
• Collect sample data
Step 3:
• Create a control chart with upper and lower limits
Creating the Control Chart
• Using the sample data, Determine the sum of
the defects by adding up all the defects record
Sample Data
Item Number
1
2
3
4
5
Number of Defects
4
z2
3
1
5
Creating the Control Chart
• Use the Sum to Determine ū
Sample Data
Item Number Number of Defects
1
4
2
2
3
3
4
1
5
5
Sum of Data
15
Creating the Control Chart
• Use the u-Bar to determine UCL and LCL.
Because LCL ends up being negative and the LCL
cannot go below 0, LCL becomes 0.
Sample Data
Item Number Number of Defects
1
4
2
2
3
3
4
1
5
5
Sum of Data
15
u-Bar
3
Creating the Control Chart
• Using the sample data, the Upper and Lower
Control Limits (UCL and LCL), and u-Bar (Also
Known as the Center Line, CL), Create a u-Chart
Sample Data
Item Number of
Number Defects
1
4
2
2
3
4
5
Sum of
Data
u-Bar
3
1
5
15
3
U-Chart Example
UCL
8.1961
52423
8.1961
52423
8.1961
52423
8.1961
52423
8.1961
52423
LCL
CL
9
0
3
0
3
8
7
6
Number of Defects
5
0
3
UCL
4
LCL
3
0
3
0
3
CL
2
1
0
1
2
3
4
5
Real World Example
Libby’s Cups is a company that makes glass cups for household use. Recently
Libby’s managers have been concerned with their cups having too many bubbles
in the glass. However, they are unsure if this is just random variation in the
process, or if it is a non-random problem that can be addressed. In order to
determine whether or not this is non-random variation or not, Libby’s managers
decided to randomly select 25 samples from all of their styles of cups and count
the total number of defects per sample. The average sample size used is 2.
Real World Example (Sample Data)
The following data was taken from 25 randomly selected glass cups
Item Number
1
2
3
4
5
6
7
8
9
10
11
12
13
Number of Defects
2
3
1
10
4
5
7
4
3
8
2
3
1
Item Number
14
15
16
17
18
19
20
21
22
23
24
25
Number of Defects
5
4
9
5
2
5
6
10
11
9
7
3
Real World Example
Libby’s managers then used the data from the sample to calculate the sample
mean/Center Line (CL), the Upper Control Limit (UCL) and the Lower Control Limit
(LCL)
Equations
Results
Real World Example (excel Data)
Item Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Number of Defects
2
3
1
10
4
5
7
4
3
8
2
3
1
5
4
9
5
2
5
6
10
11
9
7
3
Sum of u
ū
129
5.16
UCL
9.978713521
9.978713521
9.978713521
9.978713521
9.978713521
9.978713521
9.978713521
9.978713521
9.978713521
9.978713521
9.978713521
9.978713521
9.978713521
9.978713521
9.978713521
9.978713521
9.978713521
9.978713521
9.978713521
9.978713521
9.978713521
9.978713521
9.978713521
9.978713521
9.978713521
LCL
0.341286479
0.341286479
0.341286479
0.341286479
0.341286479
0.341286479
0.341286479
0.341286479
0.341286479
0.341286479
0.341286479
0.341286479
0.341286479
0.341286479
0.341286479
0.341286479
0.341286479
0.341286479
0.341286479
0.341286479
0.341286479
0.341286479
0.341286479
0.341286479
0.341286479
CL
5.16
5.16
5.16
5.16
5.16
5.16
5.16
5.16
5.16
5.16
5.16
5.16
5.16
5.16
5.16
5.16
5.16
5.16
5.16
5.16
5.16
5.16
5.16
5.16
5.16
Real World Example (excel Data Visual)
Libby’s Managers then plotted the Data and the control limits into a control
chart
Libby's Cups Example
12
10
8
Number of Defects
UCL
6
LCL
CL
4
2
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Real World Example (Conclusion)
Interpretation:
After reviewing the chart, Libby’s managers were a bit concerned. It appeared
that at certain times there was non-random variation in the number of defects
in the cups. They concluded that they were probably overproducing to make
sure that they compensated for those that had too many defects. They decided
to evaluate the production process more thoroughly to try to reduce the
amount of defects or waste in the process.
Sample Exercise
You have recently taken a job as the Senior Quality Manager at Xtreme Toys. Xtreme
Toys specializes in making an off-road tricycle for kids. The tricycles it makes comes in
several different sizes and colors. In addition they have different size wheels depending
on what sort of terrain they are going to be used on.
Recently, one of Xtreme Toy’s retailors has been rejecting several lots of tricycles
claiming that they have too many defects. After inspecting the returned lots, it appears
that the defects seem to appear in the paint finish. Many of the tricycles have scratches
in the finish and it appears to be completely random.
You are tasked my senior management to determine the cause of these defects. To
assist in determining this, you decide to set up a u-chart to monitor the process. Each
sample you take will be on average 3 units.
Sample Exercise Data
Item Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Number of Defects
3
2
1
0
5
2
4
3
3
6
7
8
11
14
10
6
3
3
2
5
1
2
2
4
7
8
9
10
11
15
Calculate:
Sum of Defects
u-bar/ Center Line
Upper Control Limit
Lower Control Limit
Create:
u-Chart
Analyze:
Is the process in Control?
If not, Where is it out of
control?
Solutions:
Sample Exercise
Equations
Sum of Defects:
U-Bar/Center Line
Upper Control
Limit
Lower Control
Limit
Calculations
Sample Exercise
Item Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Sum of u
ū
Number of Defects
3
2
1
2
5
2
4
3
2
3
4
5
6
4
5
3
3
4
2
3
5
3
2
4
5
4
3
3
2
1
98
3.266666667
UCL
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
6.397161835
LCL
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
0.136171498
CL
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
3.266666667
Sample Exercise
Sample Exercise
7
6
5
4
Number of Defects
UCL
LCL
3
CL
2
1
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Conclusion:
Sample Exercise
The process appeared to be within the control limits that had been set, however it
still appeared to be trending out of control.
• Five sample means in a row were above the center line. This indicates that their
may be periods of sustained poor performance which could be the root cause of
the scratched or damaged lots
• Six sample means on a decreasing trend. Because the fewer defect the better,
this could indicate that a problem has been fixed and that the process is
improving. It would need to be monitored more closely to see if the mean has
shifted.
Recommendation:
• Shut down the production line and evaluate the cause of the sustained poor
performance.
Summary
• u-Charts are designed to track the variation in the average number of defects
• u-Charts fall into the Attribute category of Control Charts
• u-Charts do not have to be from the same sample space and can vary in the
number of units per sample.
• Three steps to make a u-chart
• Step 1: Determine the sample space
• Step 2: Collect sample data
• Step 3: Create control chart
Reading List
Foster, S. Thomas. Managing Quality: Integrating the Supply Chain. 4th ed.
Boston: Prentice Hall, 2010. Print.
Bhat, K. Shridhara. Total Quality Management.
Himalyaya: Himalaya Publishing House. Print.