Statistical Quality Control-QC
Charts
Quality Control Charts using Excel

Statistical Quality Control
(SCQ)
Process
 Any activity or set of activities that takes inputs and
create a product. For example: an industrial plant takes
raw materials and creates a finished product.

Division of SQC
SQC can be divided into three categories
1- Descriptive statistics
That is used to describe quality
characteristics and their relationship
Includes:
1)- Mean, 2)- SD, 3)- Range, 4)- Measure of
distribution of data

Division of SQC
2- Statistical process control (SPC)
 Involves inspection of a random sample of an
output of a process and deciding whether
process is producing products within the
predetermined range of characteristics
3- Acceptance sampling
 It is a process of randomly inspecting a sample
of a goods and deciding whether to accept or
reject the entire lot based on the results

Controlled Variation
 The variation that you can never eliminate totally. There
are many small, unobservable chance effects that
influence the outcome.
 This kind of variation is said to be "in control" not
because the operator is able to control the factors
absolutely, but rather because the variation is the result
of normal disturbances, called common causes, within
a process.
 This type of variation can be predicted. In other words,
given the limitations of the process, these common
causes are controlled to the greatest extent possible.

Uncontrolled Variation
 Variation that arise at irregular intervals and for which
reasons are outside the normal functioning process,
induced by a special cause.
 Special causes include differences between machines,
different skill or concentration levels of workers,
changes in atmospheric conditions, and variation in the
quality of inputs.
 Unlike controlled variation, uncontrolled variation can
be reduced by eliminating its special cause.

Variations: in short
 Controlled variations are
native to the process, resulting from normal
factors called "common causes“
 Uncontrolled variations are
the result of "special causes" and need not
be inherent in the process

Control Charts
 As long as the points remain between the lower and upper
control limits, we assume that the observed variation is
controlled variation and that the process is in control

Control Chart
 The process is out of control. Both the fourth and the twelfth
observations lie outside of the control limits, leading us to believe that
their values are the result of uncontrolled variation.

Control Chart
 Even control charts in
which all points lie
between the control limits
might suggest that a
process is out of control.
In particular, the existence
of a pattern in eight or
more consecutive points
indicates a process out of
control, because an
obvious pattern violates
the assumption of random
variability.

Control Chart
 The first eight
observations are
below the center
line, whereas the
second seven
observations all lie
above the center
line. Because of
prolonged periods
where values are
either small or
large, this process
is out of control.

Control Chart
 Other types of suspicious patterns may appear in
control charts
 Control chart makes it very easy to identify visually
points and processes that are out of control without
using complicated statistical tests
 This makes the control chart an ideal tool for the
quick and easy quality control

Chart and Hypothesis testing
 The idea underlying control charts is closely related to
confidence intervals and hypothesis testing. The
associated null hypothesis is that the process is in
control; you reject this null hypothesis if any point lies
outside the control limits or if any clear pattern appears
in the distribution of the process values.
 Another insight from this analogy is that the possibility
of making errors exists, just as errors can occur in
standard hypothesis testing. Occasionally a point that
lies outside the control limits does not have any special
cause but occurs because of normal process variation.

Variable and Attribute Charts
 Categories of control charts:
 that monitor variables and
 that monitor attributes.
 Variable charts display continuous measurements such as weight,
diameter, thickness, purity and temperature. Its statistical analysis
focuses on the mean values of such measures.
 Attribute charts differ from variable charts in that they describe a
feature of the process rather than a continuous variable. Attributes
can be either discrete quantities, such as the number of defects in a
sample, or proportions, such as the percentage of defects per lot.

Attribute charts
Often they can be evaluated with a simple
yes or no decision. Examples include colour,
taste, or smell. The monitoring of attributes
usually takes less time than that of variables
because a variable needs to be measured
(e.g., the bottle of soft drink contains 15.9
ounces of liquid).

An attribute requires only a single decision, such
as yes or no, good or bad, acceptable or
unacceptable (e.g., the tablet colour is good or
bad) or counting the number of defects (e.g., the
number of broken bottles in the box).

Using Subgroups
 In order to compare process levels at various points in time, we usually
group individual observations together into subgroups
 The purpose of the subgrouping is to create a set of observations in
which the process is relatively stable with controlled variation
 For example, if we are measuring the results of a manufacturing
process, we might create a subgroup consisting of values from the
same machine closely spaced in time
 A control chart might then answer the question "Do the averages
between the subgroups vary more than the expected?”

Mean (X-bar) X −Chart
 Each point in the x-chart displays the subgroup average
against the subgroup number: subgroup 2 occurring
after subgroup 1 and before subgroup 3.
 As an example, consider a medical store in which the
owner monitors the length of time customers wait to be
served. He decides to calculate the average wait-time in
half-hour increments. The first half-hour (for instance,
customers who were served between 9 a.m. and 9:30
a.m.) forms the first subgroup, and the owner records
the average wait-time during this interval. The second
subgroup covers the time from 9:30 a.m. to 10:00 a.m.,
and so forth.

The X −Chart
 It is based on the standard normal distribution.
 The standard normal distribution underlies the
mean chart, because the Central Limit Theorem
states that the subgroup averages
approximately follow the normal distribution
even when the underlying observations are not
normally distributed.

The X −Chart
 The applicability of the normal distribution allows the
control limits to be calculated very easily when the
standard deviation of the process is known. 99.74% of
the observations in a normal distribution fall within 3
standard deviations of the mean. In SPC, this means that
points that fall more than 3 standard deviations from the
mean occur only 0.26% of the time. Because this
probability is so small, points outside the control limits
are assumed to be the result of uncontrolled special
causes.

Control Limits when σ is known

The X −Chart Example
Semester Score 1 Score 2 Score 3 Score 4 Score 5 Mean
1 97 89 80 81 82
2 74 100 94 65 86
3 85 100 88 62 65
4 100 91 77 67 71
5 83 92 88 79 75
6 72 79 85 100 78
7 80 83 93 88 96
8 80 100 100 79 84
9 87 70 84 96 83
10 75 77 84 75 85
11 55 95 89 100 100
12 75 73 100 72 78
13 75 100 89 66 100
14 69 88 100 84 84
15 100 84 95 80 92
16 91 100 99 77 79
17 92 90 93 87 90
18 82 80 80 79 76
19 54 89 97 84 71
20 83 66 69 100 82
Mean/n

Scores Control Chart
94.716
90.878
89.716
Values
84.716
84.17 are in
control
79.716
77.462
74.716
69.716
0 5 10 15 20 25

Analysis
 No mean score falls outside the control
limits. The lower control limit is 77.462,
the mean subgroups average is 84.17,
and the upper control limit is 90.878.
There is no evident trend to the data or
non-random pattern.
 Then, there is no reason to believe the
teaching process is out of control.

Control Limits when σ is unknown

In many instances, the value of σ is not known.
The normal distribution does not strictly apply for
analysis when σ is unknown.
 When σ is unknown, the control limits are
estimated using the average range of
observations within a subgroup as the measure
of the variability of the process.

Control Limits when σ is unknown
 The control limits are
 R represents the average of the subgroup ranges, and
X is the average of the subgroup averages. A2 is a
correction factor that is used in quality-control charts.
There are many correction factors for different types
of control charts.

Range (R) Charts
 A control chart that monitors changes in the
dispersion or variability of the process.
 The method for developing and using R-
charts is the same as that for x-bar charts.
 The center line of the control chart is the
average range, and the upper and lower
control limits are computed as follows:

Control charts for attributes
Control charts for attributes are used to measure
quality characteristics that are counted rather than
measured. Attributes are discrete in nature and
require simple yes-or-no decisions. For example, the
proportion of broken bottles or the number leakage
bottles in a carton.
Two of the most common types of control charts for
attributes are
p-charts (proportion charts)
c-charts (count charts)

p-charts
P-charts are used to measure the proportion
of items in a sample that are defective. For
example: the proportion of broken ampules in
a batch.
P-charts are appropriate when both the
number of defectives and the size of the total
sample can be counted. Then a proportion
can be computed and used as the statistic of
measurement

C-charts
C-charts count the actual number of defects.
For example: we can count the number of
complaints from customers in a month, the
number of bacteria on a petri, number of
spots on a tablet. However, we cannot
compute the proportion of complaints from
customers, the proportion of bacteria on a
petri dish.

P-Charts
 The computation of the center line as well as the
upper and lower control limits is similar to the
computation for the other kinds of control charts.
 The center line is computed as the average
proportion defective in the population. This is
obtained by taking a number of samples of
observations at random and computing the average
value of across all samples.

 To construct the upper and lower
control limits for a p-chart, we use the
following formulas:

C-charts
C-charts are used to monitor the number of
defects per unit. Example: the number of
defective injections per box.

 The average number of defects is the
center line of the control chart. The
upper and lower control limits are
computed as follows:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Total
3 2 3 1 3 3 2 1 3 1 3 4 2 1 1 1 3 2 2 3 44
 The average number of defective per
observation is 44/20=2.2 Therefore, C-
bar is 2.2 which is CL

Reference
 Data Analysis with Excel. Berk & Carey,
Duxbury, 2000, chapter 12, p. 475-488