Quality is defined as customers' perception of how well a product or service meets their expectations. There are three types of quality: quality of design, quality of performance, and quality of conformance. Statistical quality control uses statistical techniques to control, improve, and maintain quality. Control charts are used to determine if a process is in or out of control by monitoring for random or assignable variation. Process capability indices like Cp and Cpk compare process variability to specification limits to determine if a process is capable of meeting specifications.

1. Quality Control and Analysis
Quality is define as customers perception about the degree to which a product or a service meets his expectations.
1.Types of Quality
• Quality of Design
It is concerned with the tightness of specification for manufacturing any product.
• Quality of Performance
It is concerned with how well a product gives its performance. It depends upon quality of design
and quality of conformance.
2. Parameters Governing Quality
• Performance
• Range and type of features
• Reliability and durability
• Maintainability and serviceability

2. 3. Statistical Quality Control (SQC)
It is defined as the quality control system where
statistical techniques are used to control, improve
and maintain quality.
Quantitative aspects of quality management
Statistical quality control Statistical process control
(Acceptance sampling) (Process Control Charts)

3. Descriptive Statistics
• Descriptive Statistics include:
– The Mean- measure of central
tendency
– The Range- difference between
largest/smallest observations in a
set of data
– Standard Deviation measures the
amount of data dispersion around
mean
– Distribution of Data shape
– Normal or bell shaped or
– Skewed
n
x
x
n
1i
i∑=
=
( )
1n
Xx
σ
n
1i
2
i
−
−
=
∑=
meansampleofmeanstandardσ
deviationstandardProcessσsizesamplen;where
x =
==

4. Control Charts and Their Types
• The basis of control charts is to checking whether the variation
in the magnitude of a given characteristic of a manufactured
product is arising due to random variation or assignable
variation.
• Random variation: Natural variation or allowable variation,
small magnitude. e.g. length, weight, diameter, time
• Assignable variation: Non-random variation or preventable
variation, relatively high magnitude.
If the variation is arising due to random variation, the process is
said to be under control. But, if the variation is arising due to
assignable variation then the process is said to be out of control.

5. Types of Control Charts
x
Chart
R
Chart
s
Chart
c
Chart
np
Chart
p
Chart
Variables Attributes
Control Chart

6. Control Chart for Variable
• The Mean Chart (x-Chart): It shows the centering of the
process and shows the variation in the averages of individual
samples. e.g. length, weight, diameter, time.
• R Chat: It show the variation in the range of the sample.
• Control Unit: For plotting control charts generally ±3σ
selected. Therefore such control charts are known as 3σ
control charts.
• Percentage of values under normal curve

7. Major Parts of Control Chart
CL
UCL
LC
L
3σ
3σ
Out of control
Out of control
1091 2 3 4 5 6 7 8
Sample Number
QualityScale
Central Line (CL): This indicates the
desired standard or the level of the
process.
Upper Control Limit (UCL): This
indicates the upper limit of tolerance.
Lower Control Limit (LCL): This
indicates the lower limit of tolerance.
If m is the underlying statistic so that
&
CL =
UCL =
LCL =
( ) mE m = µ ( ) 2
Var mm = σ
mµ
3m mµ + σ
3m mµ − σ

8. Calculation Procedure for x-Bar & R Chart
• Calculate the x-bar and Range for each samples.
• Calculate the grand average ( ) and average range ( ).
Let sample size(n)=5
x R
S. No. 1 2 3 4 5
1
2
. . .
. . .
. . .
N
x R
1X
2X
NX
1R
2R
NR

9. •For X-Chart
sizesampleson thedependsdanddofvaluethewhere;
RDLCL
RDUCL
N
R......RR
R
43
3
4
N21
R
R
=
=
+++
=
• For R- Chart
N
x
X
N
1i
i∑=
=
x
x
σ
σ
3XLCL
3XUCL
−=
+=
sizesample
deviationstandardProcess
meansampleofmeanstandardwhere;
Here,
=
=
=
=
n
n
x
x
σ
σ
σ
σ

10. xx
xx
n21
zσxLCL
zσxUCL
sampleeachw/innsobservatioof#theis
(n)andmeanssampleof#theis)(where
n
σ
σ,
...xxx
x x
−=
+=
=
++
=
k
k
Constructing an X-bar Chart:
A quality control inspector at the Cocoa Fizz soft drink company has taken three
samples with four observations each of the volume of bottles filled. If the standard
deviation of the bottling operation is .2 ounces, use the below data to develop
control charts with limits of 3 standard deviations for the 16 oz. bottling operation.
Center line and control limit
formulas
Time 1 Time 2 Time 3
Observation 1 15.8 16.1 16.0
Observation 2 16.0 16.0 15.9
Observation 3 15.8 15.8 15.9
Observation 4 15.9 15.9 15.8
Sample
means (X-
bar)
15.875 15.975 15.9
Sample
ranges (R)
0.2 0.3 0.2

11. Solution and Control Chart (x-bar)
• Center line (x-double bar):
• Control limits for±3σ limits:
15.92
3
15.915.97515.875
x =
++
=
15.62
4
.2
315.92zσxLCL
16.22
4
.2
315.92zσxUCL
xx
xx
=





−=−=
=





+=+=

12. Control Chart for Range (R)
• Center Line and Control Limit
formulas:
• Factors for three sigma control
limits
0.00.0(.233)RDLCL
.532.28(.233)RDUCL
.233
3
0.20.30.2
R
3
4
R
R
===
===
=
++
=
Factor for x-Chart
A2 D3 D4
2 1.88 0.00 3.27
3 1.02 0.00 2.57
4 0.73 0.00 2.28
5 0.58 0.00 2.11
6 0.48 0.00 2.00
7 0.42 0.08 1.92
8 0.37 0.14 1.86
9 0.34 0.18 1.82
10 0.31 0.22 1.78
11 0.29 0.26 1.74
12 0.27 0.28 1.72
13 0.25 0.31 1.69
14 0.24 0.33 1.67
15 0.22 0.35 1.65
Factors for R-Chart
Sample Size
(n)
• R- Bar Control Chart

13. Control Charts for Attributes
P-Charts & C-Charts
• Attributes are discrete events: yes/no or pass/fail
– Use P-Charts for quality characteristics that are discrete and involve
yes/no or good/bad decisions
– Number of leaking caulking tubes in a box of 48
– Number of broken eggs in a carton
– Use C-Charts for discrete defects when there can be more than one
defect per unit
– Number of flaws or stains in a carpet sample cut from a production run
– Number of complaints per customer at a hotel

14. •P- Chart:
It is also known as fraction defective chart.
This is made for the situation where the sample
size is varying.
Sample No. Size No.(n) No. of
Defectives
Fraction
Defective
1 n1 d1 p1
2 n2 d2 p2
3 n3 d3 p3
: : : :
: : : :
N nn dn pn
InspectedTotal
Defectives#
pCL ==
N
n
)n(sizesampleAverage
N
1i
i∑=
=
( )
( )σzpLCL
σzpUCL
p
p
−=
+=

15. P-Chart Example:
A production manager for a tire company has inspected the number of defective tires in
five random samples with 20 tires in each sample. The table below shows the number of
defective tires in each sample of 20 tires. Calculate the control limits.
Sample Number
of
Defective
Tires
Number of
Tires in
each
Sample
Proportion
Defective
1 3 20 .15
2 2 20 .10
3 1 20 .05
4 2 20 .10
5 2 20 .05
Total 9 100 .09
Solution:
( )
( ) 0.1023(.064).09σzpLCL
.2823(.064).09σzpUCL
0.64
20
(.09)(.91)
n
)p(1p
σ
.09
100
9
InspectedTotal
Defectives#
pCL
p
p
p
=−=−=−=
=+=+=
==
−
=
====

16. Control Chart for Defects (C-Chart)
• C-Chart is made of number of defects which are present in a
sample and is made for the situation where the sample size (n) is
constant, n can be equal to 1 or more than one.
• Consider the occurrence of defects in an inspection of product(s).
Suppose that defects occur in this inspection according to Poisson
distribution; that is
Where x is the number of defects and c is known as mean and/or variance of the
Poisson distribution
When the mean number of defects c in the population from which samples are taken
is known
( ) , 0,1, 2, ,
!
c x
e c
P x x
x
−
= = K
ccLCL
ccUCL
samplesof#
complaints#
CL
c
c
z
z
−=
+=
=
Note: If this calculation yields a negative value of
LCL then set LCL=0.

17. C-Chart Example:
The number of weekly customer complaints are monitored in a
large hotel using a c-chart. Develop three sigma control limits
using the data table below.
Week Number of
Complaints
1 3
2 2
3 3
4 1
5 3
6 3
7 2
8 1
9 3
10 1
Solution:
02.252.232.2ccLCL
6.652.232.2ccUCL
2.2
10
22
samplesof#
complaints#
CL
c
c
=−=−=−=
=+=+=
===
z
z

18. •np-Chart (Number of Defective chart):
This is known as number of defective chart and is made for the cases
where the sample size (n) is constant.
Sample
number
Sample
size (n)
No. of
defective
(d)
P=d/n
1 n d1 P1=d1/n
2 n d2 P2=d2/n
3 n d3 P3=d3/n
: : : :
: : : :
, n dn Pn=dn/n
N
p
p
N
1i
i∑=
=
pnCL =
)p-(1pn3pnLCL
)p-(1pn3pnUCL
−=
+=

19. Process Capability
• Product Specifications
– Preset product or service dimensions, tolerances: bottle fill might be 16 oz. ±.2 oz. (15.8oz.-16.2oz.)
– Based on how product is to be used or what the customer expects
• Process Capability – Cp and Cpk
– Assessing capability involves evaluating process variability relative to preset product or service specifications
– Cp assumes that the process is centered in the specification range
– Cpk helps to address a possible lack of centering of the process 6σ
LSLUSL
widthprocess
widthionspecificat
Cp
−
==





 −−
=
3σ
LSLμ
,
3σ
μUSL
minCpk
 Process capability compares the output of in-
control process to the specification limits.

20. Relationship between Process Variability
and Specification Width
• Three possible ranges for Cp
– Cp = 1, as in Fig. (a), process
variability just meets specifications
– Cp ≤ 1, as in Fig. (b), process not capable of
producing within specifications
– Cp ≥ 1, as in Fig. (c), process
exceeds minimal specifications
• One shortcoming, Cp assumes that the process is
centered on the specification range
• Cp=Cpk when process is centered

21. Example of Process Capability
Computing the Cp Value at Cocoa Fizz: 3 bottling machines are being evaluated for possible use at the Fizz plant.
The machines must be capable of meeting the design specification of 15.8-16.2 oz. with at least a process
capability index of 1.0 (Cp≥1)
The table below shows the information
gathered from production runs on each
machine. Are they all acceptable?
Solution:
– Machine A
– Machine B
Cp=
– Machine C
Cp=
Machine σ USL-
LSL
6σ
A .05 .4 .3
B .1 .4 .6
C .2 .4 1.2
1.33
6(.05)
.4
6σ
LSLUSL
Cp ==
−

22. Computing the Cpk Value at Cocoa Fizz
• Design specifications call for a target value of 16.0
±0.2 OZ.
(USL = 16.2 & LSL = 15.8)
• Observed process output has now shifted and has a
µ of 15.9 and a
σ of 0.1 oz.
where: µ= the mean of the process
• Cpk is less than 1, revealing that the process is not
capable.
.33
.3
.1
Cpk
3(.1)
15.815.9
,
3(.1)
15.916.2
minCpk
==





 −−
=