Quality Control Chart "QCees"
Greg Graue
Ellen Kidwell
Nancy Stout
John Ruebusch
Rose Mack
Class: Statistics or any statistical section in Algebra and above.
Material: Graph paper, graphing calculator.
Goals: Enhance student's understanding of statistical sampling, mean and range; demonstrate the practical applications of statistics in quality control; reinforce calculator and graphing capabilities; and develop skills in generating and interpreting quality control charts.
Time Required: At least two and a half days in class with one homework assignment.
Background: W. Edwards Deming developed statistical quality control methods in the 1950's. These methods were first embraced by the Japanese and continue to help firms around the world with quality control programs.
For any production process, some method of evaluating the quality of the process and/or product is needed. A control chart displays successive means of measurements from a process with upper and lower control limits drawn on either side of the process average. These limits are calculated by testing samples and substituting sample averages into an appropriate formula. You plot the averages onto the control chart to determine whether any points fall between or outside of the limits or form unlikely patterns. If either of these happen, then the process is out of control and a decision needs to be made about what the problem is and how to fix it.
Control Chart

UCL 

Zone A 

Zone B 
X=Average 
Zone C 
Zone C 


Zone B 

Zone A 

LCL 
The variables control chart is divided into six equal zones. The middle measure is the mean of the means of the samples selected during the production process. A good operation may have points in various position on either side of the average and will have not points outside of the limits. The process is said to be out of control if:
I. One or more points fall outside of the control limits
II. Two points out of three successive points, on the same side of the centerline in Zone A or beyond.
III. Four out of five successive points on the same side of the center line in Zone B or beyond.
IV. Eight successive points are on the same side of the center line.
V. The points begin to show a definite pattern.
The following figure illustrates the five characteristics from above:
Following is a control chart which exhibits characteristics of a natural pattern.
Quality Control Chart Investigations
Name __________________________________ Date ____________ Period __________
The following charts are practice for determining when and why production has instability and is out of control. In the space provided, write the reason for the process being out of control.
(1)
(1) ___________________________________________________________________________
___________________________________________________________________________
(2)
(2) ___________________________________________________________________________
___________________________________________________________________________
(3)
(3) ___________________________________________________________________________
___________________________________________________________________________
(4)
(4) ___________________________________________________________________________
___________________________________________________________________________
Now that you have had practice identifying when and why a production process is out of control, let's develop a quality control chart and analyze the results. It is important to remember that in industry a) you are selecting samples from an extremely large population, b) the means of successive samples are what are being used as points on the chart, c) deviation from the mean is a very small value, and d) the plotted points from the production process are usually within the control limits. Often the central measurement value (average of the means) is given to you.
To develop a control chart with no predetermined mean, take three to five samples, n, of up to 25 subgroups, k, and determine the mean,
, for each subgroup. Identify the largest and smallest value in each subgroup. Use these values to determine the range for each subgroup. Find the average of the range values. Substitute these values into the formula for upper control limit (UCL) and lower control limit (LCL) (formulas shown below). Use the chart below from the American Society for Testing and Materials to determine the appropriate value for A_{2}. The value A_{2} is dependent upon n, the number of samples per subgroup. Place the values for the mean of the means (
) at the centerline of your graph and put the upper control limit and lower control limit in the appropriate places on your graph. Divide the region between the limits and the center line into 3 equal units. Plot the mean value from each subgroup in successive order and connect accordingly. Evaluate the chart for places of instability. Of course a well designed process or product should not have many points that are out of control.
Multipliers for X Charts
Sample Size 
Control Limit Factors 

A 
A_{2} 

2 
2.121 
1.880 
3 
1.732 
1.023 
4 
1.500 
0.729 
5 
1.342 
0.577 
6 
1.225 
0.483 
7 
1.134 
0.419 
8 
1.061 
0.373 
9 
1.000 
0.337 
10 
0.949 
0.308 
11 
0.905 
0.285 
12 
0.866 
0.266 
13 
0.832 
0.249 
14 
0.802 
0.235 
15 
0.775 
0.223 
16 
0.750 
0.212 
17 
0.728 
0.203 
18 
0.707 
0.194 
19 
0.688 
0.187 
20 
0.671 
0.180 
21 
0.655 
0.173 
22 
0.640 
0.167 
23 
0.626 
0.162 
24 
0.612 
0.157 
25 
0.600 
0.153 
Variables Control Chart
Use these formulas when sample are of quantitative units of measurement, e.g. length, weight, time, etc.
Calculate the average and range of each subgroup:
n = # of samples
Calculate the Process Average and the Average Range:
k = # of subgroups
k = # of subgroups
Calculate the Control Limits:
Example:
Sample ID # 
1 
2 
3 
4 
5 
Average X 
Largest Sample 
Smallest Sample 
Range 
1 
55 
75 
65 
80 
80 
71 
80 
55 
25 
2 
90 
95 
60 
60 
55 
72 
95 
55 
40 
3 
100 
75 
75 
65 
65 
76 
100 
65 
35 
4 
70 
110 
65 
60 
60 
73 
110 
60 
50 
5 
55 
65 
95 
70 
70 
71 
95 
55 
40 
6 
75 
85 
65 
65 
65 
71 
85 
65 
20 
7 
120 
110 
65 
85 
70 
90 
120 
65 
55 
8 
65 
65 
90 
90 
60 
74 
90 
60 
30 
9 
70 
85 
60 
65 
75 
71 
85 
60 
25 
10 
100 
80 
65 
60 
80 
77 
100 
60 
40 
= 74.6
= 36
n = 5 k = 10
UCL =
+ A_{2}R LCL =
 A_{2}R
= 74.6 + (.58)(36) = 74.6  (.58)(36)
= 95.48 = 53.72
Quality Control Chart Project 1
Is Your Random Generator Generating Randomly?
Student Lab Sheet
Name __________________________________ Date _____________ Period ________
To determine the accuracy of the random number generator on your calculator, complete a control chart and analyze the graph to determine if the average of the means produces points that are out of control. Each member of your group should generate 25 random numbers between 12 and 14, to four significant digits. You may use the program included here for the TI82 or create your own. The number of people in the group will determine the sample size (25 is the number of subgroups k). Share your data with the other members of your group. Though space is provided on the chart for a maximum of five members, you may have as few as three on your team. If there are only three members in your group, only columns one through three should be completed as samples. Each person in the group should complete the chart to verify the accuracy of answers. After completing the chart, each person is to determine the values for upper control limit (UCL), lower control limit (LCL), and complete a graph of the averages.
1. Randomly generate numbers between 12 and 14 to be placed on your chart (see lab sheet).
2. a) n = _________ b) k = _________
c) Find
= _________ d) Find
=__________
e)
f)
= _________ = _________
3. Complete a graph of the (X) points on a graph paper
4. Comment on what you see in the graph. In particular, is this process in control? How do you know?
Program to Generate Random Numbers Between 12 and 14 on a TI82: This program requires two inputs: the first is the number of the list where you want the set of numbers stored (for example L1), and the second is the length or dimension (for example 5) of the list.
: Input "LIST NO", A
: Input "DIM", D
: Seq(int (201 rand + 1200)/100,X,1,D,1)
: If A = 1: Ans ï€³ L1
: If A = 2: Ans ï€³ L2
: If A = 3: Ans ï€³ L3
: If A = 4: Ans ï€³ L4
: If A = 5: Ans ï€³ L5
: If A = 6: Ans ï€³ L6
Project 1 Data Sheet
The following is an assignment for cooperative groups that simulates a production. Each person in the group will randomly generate a set of 25 numbers between 12 and 14. Each person will represent a column (1  5). Complete the chart and do a quality control chart of the means. After completing the chart, you will analyze the results.
Sample ID # 
1 
2 
3 
4 
5 
Average X 
Largest Sample 
Smallest Sample 
Range 
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 









Quality Control Chart Project 1
Is Your Random Generator Generating Randomly?
Sample Solution for a Group of Three
Sample ID # 
1 
2 
3 
4 
5 
Average X 
Largest Sample 
Smallest Sample 
Range 
1 
13.14 
12.44 
13.16 


12.91 
13.16 
12.44 
.72 
2 
12.15 
12.06 
12.70 


12.30 
12.06 
12.70 
.64 
3 
12.77 
12.35 
13.07 


12.73 
12.37 
13.07 
.70 
4 
12.66 
12.42 
12.38 


12.49 
12.38 
12.66 
.28 
5 
13.14 
12.85 
12.76 


12.92 
12.76 
13.14 
.22 
6 
12.91 
12.77 
12.44 


12.71 
12.44 
12.91 
.47 
7 
12.59 
13.36 
12.48 


12.81 
12.48 
13.36 
.88 
8 
12.29 
12.79 
12.27 


12.45 
12.27 
12.79 
.52 
9 
12.46 
12.48 
12.06 


12.33 
12.06 
12.48 
.42 
10 
12.43 
12.04 
12.35 


12.27 
12.04 
12.43 
.39 
11 
12.89 
12.36 
12.72 


12.66 
12.36 
12.89 
.53 
12 
12.74 
12.95 
12.55 


12.75 
12.55 
12.95 
.40 
13 
12.88 
12.71 
12.77 


12.79 
12.71 
12.88 
.17 
14 
12.80 
12.54 
13.32 


12.89 
12.54 
13.32 
.78 
15 
12.37 
12.72 
12.88 


12.99 
12.72 
13.37 
.65 
16 
12.13 
13.39 
12.52 


12.68 
12.13 
13.39 
.26 
17 
12.08 
12.80 
12.53 


12.47 
12.08 
12.80 
.72 
18 
12.89 
12.03 
12.19 


12.37 
12.03 
12.89 
.86 
19 
12.81 
12.38 
12.80 


12.66 
12.38 
12.81 
.43 
20 
12.39 
12.59 
12.94 


12.64 
12.39 
12.94 
.55 
21 
12.71 
13.28 
12.26 


12.75 
12.26 
13.28 
1.02 
22 
12.66 
12.94 
13.17 


12.92 
12.66 
13.17 
.51 
23 
12.94 
12.71 
12.01 


12.55 
12.01 
12.94 
.93 
24 
12.91 
12.36 
12.40 


12.57 
12.36 
12.91 
.55 
25 
12.39 
12.35 
12.86 


12.52 
12.35 
12.86 
.51 
k = 25 n = 3 A_{2} = 1.023
= 12.64
= .60
UCL = 12.64 + (1.023)(.60) = 13.25 LCL = 12.64  (1.023)(.60) = 12.03
Quality Control Chart Project 2
"The Paper Cutter Production Company"
Teacher Notes: This project was designed as a model to demonstrate a production process. The intent is for the students to actively participate in an activity upon which they would collect data and analyze the results. The students will cut paper and check the quality of their work by using the techniques of quality control charts.
With the class divided into groups of 3, 4, or 5 students, each group will complete its own quality control chart. Each student in the group will cut strips of paper into 10 cm lengths without using a ruler except for the first cut. They should rotate tasks after the cutter has completed 7 groups of 5. The job is complete when every member of the group has completed their 7 groups of 5. Students will record the data, complete a graph, and then analyze the data.
Materials: Precut strips of paper that are 82 inches long and any width. (We found that cutting 2 inch widths from 82 by 11 inch paper worked well.) You need 35 strips per student. One pair of scissors and a metric ruler per group are also needed.
Job Descriptions:
1. Supervisor: The supervisor will collect the ruler and give it to the measurer after the first cut. It is his or her responsibility to maintain a clean area and verify that the number of strips cut, held by the selector and measured by the measurer, totals 35.
2. Cutter: Will use the ruler for the first cut. (That's all of the onthejob training they get.) All other cuts will be done by visualization. They are to hand the ruler to the supervisor. After the cutter has completed 7 groups of 5 students rotate jobs.
3. Selector: The selector receives the cut strips from the cutter. He or she would select 3 out of 5 cut strips to hand to the measurer. It is the responsibility of the selector to also keep an account of the 7 groups of 5 strips they receive.
4. Measurer: The measurer will measure the lengths of the strips they receive from the selector. Each student has a chart to record data. Because each student will be a measurer, they are to record their data in the respective section. In other words, the first measurer will record data in rows 17, the second measurer will record data in rows 814 on their own sheet, and so on.
Closing Time: After all students have rotated as cutter, they are to collect information from their teammates to complete their respective charts. Each student is expected to find the means, UCL, LCL, and do a graph. They are to analyze the data to determine if, when, and why the production process was out of control. This analysis should include comments about how they would respond to the results. Have each company branch prepare an oral presentation which should include the graph.
Setup: Arrange the students in groups of 4. Have at each station the above mentioned materials. You may wish to provide some reward to the production site with the best record. Use the chart included in this packet as the chart on which to place this data.
Evaluation:
1. Completion of introductory charts 10 pts
2. Completion of modeling project 1 50 pts
a. generate #s 10 pts
b. Find values 15 pts
c. Chart 10 pts
d. Interpretation 15 pt
3. Completion of modeling project 2 50 pts
4. Mini individual quizCgive students five charts to evaluate points for "instability" and have them identify why it is out of control.
Extensions:
1. Have students write a paper on where and how quality control charts could be used in their school. For example, carnation sales, candy sales, fund raisers, etc.
2. Have students actually do their suggestions as a project.
3. Investigate Attributes Control Charts and formulas for when samples reflect qualitative characteristics such as is/is not defective, go/no go (see charts enclosed)
4. Investigate Variable Control Charts and formulas using range (see charts enclosed)
5. Obtain some quantitative data from a local business or the library. Use the data set to construct quality control charts and have students write a one page summary of the situation for management.
6. Take a field trip to some industrial plant where they are using control charts.
7. Have students do a report on Deming and the impact he has had on Japan.
Teacher Options:
1. Included in this packet are some actual charts from a company. You may wish to show these to your students. There is also a blank form included in case you wish to have them do their graphs on these instead of on regular graph paper.
2. The mean, UCL, and LCL obtained from the paper cutting project could be given to the students to use for them to just graph the mean of their sample points.
Resources:
Deming, W. Edwards, 1986. Out of the Crisis. Cambridge, Mass: MIT Center for Advanced Engineering Studies.
Deming, W. Edwards, 1990. Sample Design in Business Research. A. Wiley Interservice Publications.
Gitlow, H., Gitlow, S., Oppenheim, A., & Oppenheim, R., 1989. Tools and Methods for the Improvement of Quality. Homewood, Ill: Richard D. Irving.
GOAL/QPC 1985, 1988. The Memory Jogger. 13 Branch Street, Metheun, MA 01844.
Quality Control Chart Project 2
"The Paper Cutter Production Company"
Sample Solution for a Group of Three
Sample ID # 
1 
2 
3 
4 
5 
Average X 
Largest Sample 
Smallest Sample 
Range 
1 
10.0 
9.8 
10.9 


10.2 


1.1 
2 
10.4 
9.7 
10.8 


10.3 


1.1 
3 
10.8 
8.4 
10.2 


9.8 


2.4 
4 
10.1 
9.8 
8.9 


9.6 


1.2 
5 
8.2 
10.4 
9.6 


9.4 


2.2 
6 
10.4 
9.7 
10.3 


10.1 


.7 
7 
10.8 
10.6 
10.3 


10.6 


.5 
8 
9.3 
9.5 
10.0 


9.6 


.7 
9 
9.8 
9.6 
9.4 


9.6 


.4 
10 
9.7 
9.5 
10.0 


9.7 


.3 
11 
9.6 
9.6 
9.4 


9.5 


.2 
12 
9.8 
9.6 
10.0 


9.8 


.4 
13 
10.1 
9.6 
9.5 


9.7 


.6 
14 
9.6 
10.1 
10.8 


10.2 


1.2 
15 
11.2 
10.2 
9.8 


10.4 


1.4 
16 
10.4 
9.8 
9.5 


9.9 


.9 
17 
10.3 
10.4 
10.5 


10.4 


.2 
18 
10.2 
11.0 
11.8 


11.2 


1.1 
19 
10.8 
11.4 
11.0 


11.1 


.6 
20 
11.2 
10.6 
12.0 


11.3 


1.4 
21 
10.0 
11.0 
10.8 


10.6 


1.0 
22 









23 









24 









25 









k = 21 n = 3 A_{2} = 1.023
= 10.1
= .9
UCL = 10.1 + (1.023)(.9) = 11.0 LCL = 10.1  (1.023)(.9) = 9.2
Quality Control Chart Project 2
"The Paper Cutter Production Company"
Student Lab Sheet
This is the Paper Cutter Production Company upon which each of your groups represent a production site or branch. As company CEO, we would like to know the quality of our product, the effectiveness of the process and which branch should be recognized as "Plant of the Year." Quality control charts are required at each site. These will be used to evaluate the company. Today is the day for evaluation. Good luck!
Our Product: We cut strips of paper that are 10 cm in length. We believe in sharing tasks. So the work schedule is as follows.
Job Descriptions:
1. Supervisor: The supervisor will collect the ruler and give it to the measurer after the first cut. It is his or her responsibility to maintain a clean area and verify that the number of strips cut, held by the selector and measured by the measurer, totals 35.
2. Cutter: Will use the ruler for the first cut (that's all of the job training you get). All other cuts will be done by visualization. They are to hand the ruler to the supervisor. After the cutter has completed 7 groups of 5, they rotate jobs.
3. Selector: The selector receives the cut strips from the cutter. They would select 3 out of 5 cut strips to hand to the measurer. It is the responsibility of the selector to also keep an account of the 7 groups of 5 strips they receive.
4. Measurer: The measurer will measure the lengths of the strips they receive from the selector. Each student has a chart to record data. Because each student will be a measurer, they are to record their data in the respective section. In other words, the first measurer will record data in rows 17, the second measurer will record data in rows 814 on their own sheet, and so on.
Closing Time: After all students have rotated as cutter, they are to collect information from their teammates to complete their respective charts. Each student is expected to find the means, UCL, LCL and do a graph. They are to analyze the data to determine if, when and why the production process was out of control. This analysis should include comments about how they would respond to the results. Have each company branch present an oral presentation which should include the graph and turn in a one page analysis from the group.
"The Paper Cutter Production Company"
Student Data Sheet
Name _____________________________ Date ______________ Period ____________
Closing Time Report:
1. Complete chart
2. n = _________
= ___________
3. k = _________
= __________
4.
=________ = ________
5. Complete a graph of the
points on graph paper.
6. Analysis:___________________________________________________________________
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Funded in part by the National Science Foundation and Indiana University 1995