Statistical process control

Statistical process control

1. Statistical Process Control
Statistical process control (SPC) is a statistical procedure using control charts to see if any part of
production process is not functioning properly and could cause poor quality.
SPC is a tools for identifying problems in order to make improvements. Process control is achieved by
taking periodic samples from the process and plotting these sample points on a chart, to see if the
process is within statistical control limits.
T he application of statistical techniques to determine whether a quantity of material should be accepted or rejected based on the inspection or test
of a sample.

statistical process control (SPC)

T he application of statistical techniques to determine whether a process is delivering what the customer wants.
Quality Measures: The quality of a product or service can be evaluated using either an attribute of the
product or service or variables measures.
An attribute is a product characteristic that can be evaluated with a discrete response such as good or
bad, yes or no, acceptable or not attributes could be color, cleanliness, surface, texture etc.
A variable measures is a product characteristics that is measured on continuous scale such as length,
weight, temperature etc.
Control Chart:
Control charts are graphs that visually show if a sample is within statistical control limits. They have two
basic purposes, to establish the control limits for a process and then to monitor the process to indicate
when it is out of control. Control charts exist for attributes and variables within each category there are
several different types of control charts. Among them p-chart, and c-chart are for attributes and mean )(x
and range (R) control charts are for variables.
UCL

Nominal


LCL
Assignable causes likely

2. Control Charts for Attributes

Two charts commonly used for performance measures based on attributes measures are the p- and c-chart. The p-
chart is used for controlling the proportion of defects generated by the process. The c-chart is used for controlling the
number of defects when more than one defect can be present in a service or product.

p-chart

A chart used for controlling the proportion of defective services or products generated by the process.
p-Charts The p-chart is a commonly used control chart for attributes. The performance char-acteristic is counted
rather than measured, and the entire service or item can be declared good or defective. For example, in the banking
industry, the attributes counted might be the number of nonendorsed deposits or the number of incorrect financial
statements sent to customers. The method involves selecting a random sample, inspecting each item in it, and
calculating the sam-ple proportion defective, p, which is the number of defective units divided by the sample size.
Sampling for a p-chart involves a “yes/no” decision: The process output either is or is not defective. The
underlying statistical distribution is based on the binomial distribution. However, for large sample sizes, the normal
distributionprovides a good approximation to it. The standard deviation of the distribution of proportion defectives, sp,
is

sp = npp /)1(
where
n = sample size
= central line on the chart, which can be either the historical average population
proportion defective or a target value

We can use sp to arrive at the upper and lower control limits for a p-chart:
UCLp = p + z sp and LCLp = p - z sp

where

z = normal deviate (number of standard deviations from the average)
Z is occasionally equal to 2.00 but most frequently is 3.00. A Z values of 2.00 corresponds to an overall normal
probability of 95 percent and z=3.00 corresponds to a normal probability of 99.74 percent. Management usually
selects z=3.00 because if the process is in control if wants a high probability that the sample values will fall within the
control limits.

3. Fig: Normal Distribution Curve
The Western Jeans company produces denim jeans. The company wants to establish a p-chart to
monitor the production process and maintain high quality. Western believes that approximately 99.74
percent of the variability in the production process (corresponding to 3-sigma limits, or z=3.00) is known
random and thus should be within control limits, where as .26 percent of the process variability is not
random and suggests that the process is out of control. The company has taken 20samples (one per day
for 20 days) each containing 100 pairs of jeans (n=100) and inspected them for defects, the results of
which are as follows;
Sample
Number of
Defects
Proportions of
Defects
1 6 0.06
2 0 0
3 4 0.04
4 10 0.1
5 6 0.06
6 4 0.04
7 12 0.12
8 10 0.1
9 8 0.08
10 10 0.1
11 12 0.12
12 10 0.1
13 14 0.14
14 8 0.08

4. 15 6 0.06
16 16 0.16
17 12 0.12
18 14 0.14
19 20 0.2
20 18 0.18
Total 200

The proportion defective for the population is not known. The company wants to construct p-chart to
determine when the production process might be out of control.
Solution:
Since p is not known, it can be estimated from the total sample; :
01.0)03.0(00.31.0
19.0)03.0(00.31.0
03.0
100
)1.01(1.0)1(
,
1.0
)100(20
200
Figure
zpLCL
zpUCL
n
pp
Now
nsobservatiosampleTotal
defectiveTotal
p
p
p
p














0
0.05
0.1
0.15
0.2
0.25
0 5 10 15 20 25
Series1

5. The process is below the lower control limits for sample 2 (i,e during day 2).
The above process was the upper limits during day 19.


Example-2: The operation manager of the booking services department of hometown bank is concerned
about the number of wrong customer account numbers recorded by hometown personnel. Each week a
random sample of 2500 deposits is taken and the number incorrect account numbers recorded. The
results for the past 12 weeks are shown in the following table. Is the booking process out of statistical
control? Use three sigma control limits.
Sample Number Wrong account Number Proportions of Defects
1 15 0.006
2 12 0.0048
3 19 0.0076
4 2 0.0008
5 19 0.0076
6 4 0.0016
7 24 0.0096
8 7 0.0028
9 10 0.004
10 17 0.0068
11 15 0.006
12 3 0.0012

Solution:

6. defectivesproportionSampleFigure
zpLCL
zpUCL
n
pp
nsobservatioofNumberTotal
defectivesTotal
p
pcalculatetopastgU
p
p
p
:
0007.0)0014.0(30049.0
0091.0)0014.0(30049.0
0014.0
2500
)0049.01(0049.0)1(
0049.0
)2500(12
147
sin












Management explores the circumstance when sample 7 was taken. The encoding machine used to print
the account numbers on the checks was defective that week. The following week the machine was
repaired; however, the recommended preventive maintenance was not performed for months prior to
the failure. Management reviewed the performance of the maintenance department and instituted
changes to the maintenance procedures for the encoding machine. After the problem was corrected, an
analyst recalculated the control limits using the data without sample 7. Subsequent weeks were
sampled and the booking process was determined to be in statistical control.
C-Chart:
A c-chart is used when it is not possible to compute a proportion defective and the actual number of
defects must be used, for example, when automobiles inspected the number of defects in the paint job
can be counted for each car, but a proportion cannot be computed, since the total number of possible
defects is not known.
The underlying sampling distribution for a c-chart is the Poisson distribution. It is based on the
assumption that defects occur over a continuous region on the surface of the product provision of
0
0.002
0.004
0.006
0.008
0.01
0.012
0 2 4 6 8 10 12 14
Series1

7. service and that the probability on the surface or at any instant of time is negligible. The mean of the
distribution is c and standard deviation is c .
The control limits are; c
c
zcczcLCL
zcczcUCL





Exercise-1: The Ritz hotel has 240 rooms. The hotel’s housekeeping department is responsible for
maintaining the quality of the rooms’ appearance and cleanliness. Each individual housekeeper is
responsible for an area encompassing 20 rooms. Every room in use is thoroughly cleaned and its
supplies, toiletries and so on are restocked each day. Any defects that the housekeeping staffs notice
that are not part of the normal housekeeping service are supposed to be reported to hotel
maintenance. Every room is briefly inspected each day by a housekeeping supervisor. However, hotel
management also conducts inspection tours at random for detailed through inspection for quality
control purposes. The housekeeping service defects like an inoperative or missing TV remote, poor TV
picture quality or reception defective lamps, a malfunctioning clock, tears or stains in the bedcover or
curtains or a malfunctioning curtain pull. An inspection sample includes 12 rooms i,e one room selected
at random from each of the twelve 20 room blocks serviced by a house keeper. Following are the results
from 15 inspection samples conducted at random during a one month period.
Sample
No. of
Defects Sample
No. of
Defects
1 12 8 14
2 8 9 13
3 16 10 15
4 14 11 12
5 10 12 10
6 11 13 14
7 9 14 17
15 15
The hotel believes that approximately 99 percent of the defects (corresponding to 3 sigma limits) are
caused by non random variability. They want to construct a c-chart to monitor the housekeeping service
Solution:
The population process average is not known, the sample estimate, c , can be used instead

8. 99.167.12367.12
35.2367.12367.12
00.3sinlim
67.12
15
190




czczcLCL
czczcUCL
followaszgucomputedareitscontrolThe
c
c
c


Figure:


All the sample observations are within the control limits. Suggesting that the room quality is in control.

Exercise-2:
The woodland paper company produces paper for the newspaper industry. As a final step in the process,
the paper passes through a machine that measures various product quality characteristics when the
paper production process is in control it averages 20 defects per roll.
a) Set up control chart for the number of defects per roll use 2 sigma control limits.
b) Five rolls had the following number of defects: 16, 21, 17, 22, 21 and 24 respectively. The sixth
roll using pulp from a different supplier had 5 defects. Is the paper production process is
control?
Solution:
a) The average number of defects per roll is 20 , therefore
0
2
4
6
8
10
12
14
16
18
0 5 10 15 20
Series1

9. 06.1120220
94.2820220


czcLCL
czcUCL
Figure:

The supplier for the first 5 samples has been used by wood land paper for many years. The
supplier for the sixth sample is new to the company. Management decided to continue using the
new supplier for a while, monitoring the number of defects to see if it stays low. If the number
remains below the lcl for 20 consecutive samples, management will make the switch permanent
and recalculate the control chart parameter.

0
5
10
15
20
25
30
0 1 2 3 4 5 6 7
Series1