6.6. Control charts for number of occurrences: c-chart
The p-chart just discussed is used when you select sample of items and you determine the number of the sample items that possesses a specific attribute of interest. Each item either has or does not have that attribute. In practice often we meet the situations that involve attribute data but differ from the p-chart. Each sampling unit could have one or more of the attributes of interest. Number of attributes of interest, called the number of occurrences , counts number of imperfections per item over time. This is called a c-chart.
As with the other control charts studied in this chapter, some general notations used for control charts for number of occurrences are necessary.
Assume
that, a sequence of K
items is inspected over time, and for each item, the number of
occurrences of some event, such as imperfections, is recorded. These
numbers of occurrences denoted
for
.
The sample mean of occurrences is
A 3-sigma (3 standard deviation) control chart for the number of occurrences (c-chart) can be constructed in the usual way:
Control chart for c- number of occurrences
The
chart
is a time plot of the number of occurrences over the time.
The central line is
The lower control limit is
if
if
The upper control limit is
Example:
Handheld calculators are manufactured and checked for defects. If a calculator is not defective, it is packaged and shipped to a retail store. Any defective calculators are repaired before they are shipped. Twelve of the defective calculators are checked for the number of defects per calculator. The numbers of defects per calculator are:
6, 3, 2, 5, 6, 7, 4, 3, 7, 8, 9 and 5
a) Find the central line and lower and upper limits for c-chart.
b) Draw the c-chart and discuss its features.
Solution:
a) First of all, let us find the mean number of defects per calculator, .
The lower and upper control limits are
since
b) Figure 6.4 represents c-chart
`
Since, all points fall within the upper and lower control limits, the process is in control. That is, in the defective calculators, the number of defects per calculator is not excessive.
Remark:
To use MINITAB menu follow the following instructions:
1. Select Stat>Control charts>Select C
2. Enter variable location (C1; C2; etc.)
3. Enter subgroup size (200, 300 etc.)
4. Click OK.
Exercises
1. Sixty minute cassette tapes are checked for defects. The number of defects in each of 8 tapes is shown below
-
Tape
1
2
3
4
5
6
7
8
Number of defects
1
2
1
1
1
3
6
2
a) Find the central line and lower and upper limits for c-chart.
b) Draw the c-chart and discuss its features.
2. Workers are painting a large apartment building. An inspector checks several walls for paint defects. The number of defects per wall is shown below
Sample |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
Number of defects |
6 |
12 |
9 |
8 |
8 |
6 |
12 |
10 |
11 |
8 |
a) Find the central line and lower and upper limits for c-chart.
b) Draw the c-chart and discuss its features.
3. A reader has very carefully read local paper for 15 weeks. For each Sunday’s edition he has counted the number of typographical and spelling errors. The results are shown below
-
Week
Errors
Week
Errors
1
2
3
4
5
6
7
8
13
15
14
17
12
20
7
14
9
10
11
12
13
14
15
12
13
21
9
17
19
22
a) Find the sample mean number of errors for these 15 weeks.
b) Find the central line and lower and upper limits for c-chart.
c) Draw the c-chart and discuss its features.
4. In a large market, the number of complaints in a week was recorded over 16 weeks. The results are shown in accompanying table
-
Week
Errors
Week
Errors
1
2
3
4
5
6
7
8
17
11
15
22
21
29
8
18
9
10
11
12
13
14
15
16
23
15
10
16
17
22
23
20
a) Find the sample mean number of complaints per week.
b) Find the central line and lower and upper limits for c-chart.
c) Draw the c-chart and discuss its features.
Answers
1.
a)
2.125;
;
;2.
a)
9.00;
;
;
3.
a)15; b)
15;
;
;
4.
a) 17.94; b)
17.94;
.