Control charts for attributes

1. RME-085
Total Quality Management
Topic: Control charts for attributes
By:
Dr. Vinod Kumar Yadav
Department of Mechanical Engineering
G. L. Bajaj Institute of Technology and Management
Greater Noida
Email: vinod.yadav@glbitm.org

2. Theory of Control charts
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU
Lucknow in VIII semester 2019-20, and is not intended for wider circulation.
Control Charts for attributes
• Attribute – Quality characteristics that conform to
specifications or do not conform to specifications [1].
• Discrete data (Binary).
• Used where (i) measurements can’t be made. (ii)
Derived through judgements due to shortage of time
(Example: Accept / Reject - (Go/No Go gauges).
• Nonconformity - Departure of a quality
characteristic from its intended level – Hence,
product do not meet intended specification. Relevant
for specifications.
• Defect: Non-conformity concerned with satisfying
intended usage requirements. Relevant for usage
evaluation.
• Note: Variables can be converted into attributes – meeting the UCL/LCL
(Accept) or not meeting (Reject).
Limitations of variable control
chart:
1. Cannot be used for quality
characteristics like attributes.
2. Variables may be hundred or thousand
or even more- so, not feasible to draw
100 or 1000 X and R chart – time
consuming.
Attributes: Missing parts, incorrect color, scratches, roughness
Control
Charts
Variables
X , R etc.
Attributes
P, C etc.

3. Types of control charts for attributes
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU
Lucknow in VIII semester 2019-20, and is not intended for wider circulation.
1. For non-conforming units: (P charts)
• P chart – Expressed as a fraction or a
%. Used when each unit can be
considered pass or fail – no matter the
number of defects.
• P-chart shows the proportion
nonconforming in a sample or
subgroup.
• Based on the binomial distribution.
• Similar charts can be drawn for
conforming proportion.
• One more type: Number nonconforming
(np chart)
2. For non-conformities (c charts)
• c chart shows the count of
nonconformities in an inspected unit.
• u chart: Analogous to c chart: to count
nonconformities per unit [1].
• Based on the Poisson distribution.
• The c-chart allows the practitioner to
assign each sample more than one
defect.

4. Types of control charts for attributes
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU
Lucknow in VIII semester 2019-20, and is not intended for wider circulation.
1. For non-conforming units: (P charts)
• P chart – the proportion of the number of occurrences
of an event to the total number of occurrences.
• The fraction nonconforming is the proportion of the
number nonconforming in a sample or subgroup to the
total number in the sample or subgroup.
p =
𝒏𝒑
𝒏
Where,
p = proportion or fraction nonconforming in the sample
n = number in the sample
𝒏𝒑 = number nonconforming in the sample
P chart Versatile chart –used for one characteristics at
one time
The subgroup size of the p chart can be either
variable or constant.
Objectives of p charts:
1. Calculate average quality level for
benchmarking.
2. Help management in decision making
for quality improvement.
3. Performance evaluation of operator.
4. Identify the places where X and R
charts can be used.

5. P chart construction for constant subgroup size
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech
Mechanical Engineering at AKTU Lucknow in VIII semester 2019-20, and is not intended for wider circulation.
Step-1: Choose quality characteristics:
• P charts can be used for: non-conformation of (a) a single
quality characteristic (b) A group of quality characteristics (c)
Part (d) Product (e) Group of products.
• Performance evaluation of (a) Worker (b) work place (c)
Division (d) shift (e) Plant (f) Corporation
Step-2: Selection of group size and method:
• Subgroup size - function of the nonconforming proportion.
• Example: If proportion nonconforming (p) = 0.01 and n
(subgroup size) = 100, then the average number
nonconforming, np , would be one per subgroup. This would
not make a good chart, because a large number of values posted
to the chart would be 0. If a part has a proportion
nonconforming of 0.15 and a subgroup size of 50, the average
number of nonconforming would be 7.5, which would make a
good chart. A minimum average number of nonconforming
unit size of 50 is suggested as a starting point.
Method of determining sample size [1]
n = p (1-p) 𝒁α/𝟐
𝑬
2
n = sample size
p = non-conforming proportion
𝒁α/𝟐 = Normal distribution coefficient (Z
value) for area between two tails. The area
represents the decimal equivalent of the
confidence limit.
E = maximum allowable error in the estimate of
p , which is also referred to as the desired
precision.
𝒁α/𝟐 Confidence limit (%)
1.036 70.0
1.282 80.0
1.645 90.0
1.96 95.0
2.575 99.0
3.00 99.73
Table-1

6. P chart construction for constant subgroup size contd.
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU Lucknow in VIII semester 2019-
20, and is not intended for wider circulation.
Example: A gear company wants to calculate the proportion of gears that are not conforming the desired specifications.
As per previous record, the company assumes the percent nonconforming = 30%. The expected precision is 20% and a
confidence level of 99%. Determine the sample size.
Solution:
Given p = 0.3; Precision = 20%; Confidence level = 99%
E = 20% of p = 0.2*0.3 = 0.06
𝒁α/𝟐 = 2.575 (From Table 1)
n = 0.3 (1-0.3) (2.575/0.06)2 = 386.78
Hence, after 387 samples, the value of p = 0.6997. Then, the true value will be between 0.3 and 0.69 (p ± E) 99% of the
time.
n = p (1-p) 𝒁α/𝟐
𝑬
2
Step-3: Collect the data:
• P charts are effective during the starpt-up phase of a new item or process, when the process is very erratic [1].
Step-4: Calculate the trial central line and control limits.
𝑝 = Average proportion of nonconforming for many subgroups
n = number inspected in a subgroup 𝒑 =
𝒏𝒑
𝒏 UCL/LCL Ref [1]

7. P chart construction for constant subgroup size contd.
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU Lucknow in VIII semester 2019-
20, and is not intended for wider circulation.
Step-5: Establish the revised central line and
control limits if the process is out of control:
• Note: only those subgroups with assignable causes are discarded.
• Then new UCL and LCL are computed
Step-6: Achieve the objective
• The last step involves action and leads to the achievement of the
objective.
Figure-1: p-chart [1]
Out of control
𝒑 =
𝒏𝒑 −𝒏𝒑𝒅
𝒏 −𝒏𝒅
Where,
𝒏𝒑 𝒅 = number nonconforming in the discarded group
𝒏 𝒅 = number inspected in the discarded subgroups

8. P chart construction for variable subgroup size
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU Lucknow in VIII semester 2019-20, and is not intended for wider
circulation.
• p charts should be developed and used with a constant subgroup size.
• This situation is not possible when the p chart is used for 100% inspection of output that
varies from day to day.
• Data for p –chart use from sampling inspection might vary for a variety of reasons. Because
the control limits are a function of the subgroup size, n , the control limits will vary with the
subgroup size. Therefore, they need to be calculated for each.
• The procedures of data collection, trial central line and control limits, and revised central
line and control limits are the same as those for a p chart with constant subgroup size.

9. P chart construction for variable subgroup size (Example – Juice production plant)
Subgroup
No. of
inspections
(n)
Number
Nonconformi
ng (np)
Fraction
Nonconformin
g (p)
Limit
UCL LCL
01.01.2020 238 5 0.021 0.023 0.000
02.01.2020 140 2 0.014 0.017 0.000
03.01.2020 193 5 0.026 0.028 0.000
04.01.2020 245 4 0.016 0.018 0.000
05.01.2020 200 4 0.020 0.022 0.000
06.01.2020 216 5 0.023 0.025 0.000
07.01.2020 194 5 0.026 0.028 0.000
08.01.2020 196 4 0.020 0.023 0.000
09.01.2020 200 7 0.035 0.038 0.000
10.01.2020 201 4 0.020 0.022 0.000
11.01.2020 195 5 0.026 0.028 0.000
12.01.2020 175 4 0.023 0.025 0.000
13.01.2020 199 6 0.030 0.033 0.000
14.01.2020 200 3 0.015 0.017 0.000
15.01.2020 205 7 0.034 0.037 0.000
16.01.2020 189 4 0.021 0.023 0.000
17.01.2020 180 5 0.028 0.031 0.000
18.01.2020 185 4 0.022 0.024 0.000
19.01.2020 260 8 0.031 0.033 0.000
20.01.2020 199 6 0.030 0.033 0.000
21.01.2020 255 6 0.024 0.025 0.000
22.01.2020 245 3 0.012 0.014 0.000
23.01.2020 271 10 0.037 0.039 0.000
24.01.2020 150 5 0.033 0.037 0.000
25.01.2020 201 5 0.025 0.027 0.000
Summation 5132 126
- Sample Size – day wise inspection (25 samples from 25 days data).
- The variation in the number inspected per day can be due to -
Machines breakdown, worker unavailability or any other reason
- Lowest inspection on 02.01.2020 due to major breakdown in plant.
Highest on 23.01.2020 due to the deployment of extra labor
(overtime)
- UCL and LCL are calculated using the same procedures and formulas as
for a constant subgroup. However, because the subgroup size changes
each day, limits must be calculated for each day.
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
P chart for variable subgroup
Fraction Nonconforming (p) UCL LCL p bar
Date (Subgroup)
simplified

10. Number non-conforming chart (np chart)
- np chart is almost same as the p chart But, not used
for the same objective as p chart
- The np chart is easier for operating personnel to
understand than the p chart
- Inspected results are plotted directly on to the chart
- Applicable only to constant subgroups
- The sample size should be shown on the chart so
viewers have a reference point.
- Central line and control limits are changed by a
factor of n.
- UCL = n𝒑 𝒐 + 3 𝒏𝒑 𝒐(𝟏 − 𝒑 𝒐)
- LCL = n𝒑 𝒐 - 3 𝒏𝒑 𝒐(𝟏 − 𝒑 𝒐)
If 𝒑 𝒐 is unknown, then it is determined by collecting
data, calculating trial UCL/LCL, and obtaining the best
estimate of 𝒑 𝒐.
Example: Given : Sample size = 100 per day in a lot of
3000 𝒑 𝒐 = 0.07
Then n𝒑 𝒐 = 100 * 0.07 = 7
UCL = 7 + 3*√ (7 (1-0.07) = 14.65
LCL = 7 - 3*√ (7 (1-0.07) = - 0.6544
2
5
7
8
10
9
7
3
8 8
12
-2
0
2
4
6
8
10
12
14
16
1 2 3 4 5 6 7 8 9 10 11
np Chart
npo UCL LCL Number of nonconformity
Nonconforming
number
sample size po npo UCL LCL
2 100 0.07 7 14.65441 -0.65441
5 100 0.07 7 14.65441 -0.65441
7 100 0.07 7 14.65441 -0.65441
8 100 0.07 7 14.65441 -0.65441
10 100 0.07 7 14.65441 -0.65441
9 100 0.07 7 14.65441 -0.65441
7 100 0.07 7 14.65441 -0.65441
3 100 0.07 7 14.65441 -0.65441
8 100 0.07 7 14.65441 -0.65441
8 100 0.07 7 14.65441 -0.65441
12 100 0.07 7 14.65441 -0.65441
UCL
LCL

11. Non-conformities chart (c chart)
- C chart is used to count non-conformities
- U chart count of nonconformities per unit
- UCL = 𝑪 + 3 𝑪
- LCL = 𝑪 - 3 𝑪
Where,
𝑪 = 𝐂𝐞𝐧𝐭𝐫𝐚𝐥 𝐥𝐢𝐧𝐞 𝐚𝐯𝐞𝐫𝐚𝐠𝐞 𝐯𝐚𝐥𝐮𝐞
UCL
LCL
3
6
5
4
5
3
2
4 4 4
-4
-2
0
2
4
6
8
10
12
1 2 3 4 5 6 7 8 9 10
AxisTitle
Group No.
c chart
Central line (average) UCL LCL No. of defects
Group
No.
No. of
Defects
Central line
(average) UCL LCL
1 3 4 10 -2
2 6 4 10 -2
3 5 4 10 -2
4 4 4 10 -2
5 5 4 10 -2
6 3 4 10 -2
7 2 4 10 -2
8 4 4 10 -2
9 4 4 10 -2
10 4 4 10 -2
sum 40

12. References:
[1] Dale H. Besterfiled. A Text book on Quality Improvement. 9th Edition. Pearson (ISBN 10: 0-13-262441-9) pp: 123-148.
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU
Lucknow in VIII semester 2019-20, and is not intended for wider circulation.

13. Thank you