Download ITED 434 Quality Organization & Management Ch 10 & 11

ITED 434
Quality Organization &
Management Ch 10 & 11
Ch 10: Basic Concepts of
Statistics and Probability
Ch 11: Statistical Tools for
Analyzing Data
Chapter Overview





Statistical Fundamentals
Process Control Charts
Some Control Chart Concepts
Process Capability
Other Statistical Techniques in Quality
Management
Statistical Fundamentals

Statistical Thinking
– Is a decision-making skill demonstrated by
the ability to draw to conclusions based on
data.

Why Do Statistics Sometimes Fail in the
Workplace?
– Regrettably, many times statistical tools do
not create the desired result. Why is this
so? Many firms fail to implement quality
control in a substantive way.
Statistical Fundamentals

Reasons for Failure of Statistical Tools
– Lack of knowledge about the tools; therefore,
tools are misapplied.
– General disdain for all things mathematical
creates a natural barrier to the use of
statistics.
– Cultural barriers in a company make the use
of statistics for continual improvement difficult.
– Statistical specialists have trouble
communicating with managerial generalists.
Statistical Fundamentals

Reasons for Failure of Statistical Tools
(continued)
– Statistics generally are poorly taught,
emphasizing mathematical development
rather than application.
– People have a poor understanding of the
scientific method.
– Organization lack patience in collecting data.
All decisions have to be made “yesterday.”
Statistical Fundamentals

Reasons for Failure of Statistical Tools
(continued)
– Statistics are view as something to buttress
an already-held opinion rather than a
method for informing and improving
decision making.
– Most people don’t understand random
variation resulting in too much process
tampering.
Statistical Fundamentals

Understanding Process Variation
– Random variation is centered around a
mean and occurs with a consistent amount
of dispersion.
– This type of variation cannot be controlled.
Hence, we refer to it as “uncontrolled
variation.”
– The statistical tools discussed in this
chapter are not designed to detect random
variation.
Statistical Fundamentals

Understanding Process Variation (cont.)
– Nonrandom or “special cause” variation
results from some event. The event may
be a shift in a process mean or some
unexpected occurrence.

Process Stability
– Means that the variation we observe in the
process is random variation. To determine
process stability we use process charts.
Statistical Fundamentals

Sampling Methods
– To ensure that processes are stable, data
are gathered in samples.
• Random samples. Randomization is useful
because it ensures independence among
observations. To randomize means to sample
is such a way that every piece of product has
an equal chance of being selected for
inspection.
• Systematic samples. Systematic samples have
some of the benefits of random samples
without the difficulty of randomizing.
Statistical Fundamentals

Sampling Methods
– To ensure that processes are stable, data
are gathered in samples (continued)
• Sampling by Rational Subgroup. A rational
subgroup is a group of data that is logically
homogenous; variation within the data can
provide a yardstick for setting limits on the
standard variation between subgroups.
Standard normal distribution


The standard normal distribution is a normal
distribution with a mean of 0 and a standard deviation
of 1. Normal distributions can be transformed to
standard normal distributions by the formula:
X is a score from the original normal distribution,  is
the mean of the original normal distribution, and  is
the standard deviation of original normal distribution.
Standard normal distribution

A z score always reflects the number of standard deviations
above or below the mean a particular score is.

For instance, if a person scored a 70 on a test with a mean of 50
and a standard deviation of 10, then they scored 2 standard
deviations above the mean. Converting the test scores to z
scores, an X of 70 would be:

So, a z score of 2 means the original score was 2 standard
deviations above the mean. Note that the z distribution will only
be a normal distribution if the original distribution (X) is normal.
Applying the formula
Applying the formula will always produce a transformed variable
with a mean of zero and a standard deviation of one. However, the
shape of the distribution will not be affected by the transformation. If
X is not normal then the transformed distribution will not be normal
either. One important use of the standard normal distribution is for
converting between scores from a normal distribution and percentile
ranks.
Areas under portions of the standard
normal distribution are shown to the
right. About .68 (.34 + .34) of the
distribution is between -1 and 1 while
about .96 of the distribution is
between -2 and 2.
Area under a portion of the
normal curve - Example 1
If a test is normally distributed with a mean of
60 and a standard deviation of 10, what
proportion of the scores are above 85?
From the Z table, it is calculated that .9938 of
the scores are less than or equal to a score 2.5
standard deviations above the mean. It follows
that only 1-.9938 = .0062 of the scores are
above a score 2.5 standard deviations above the
mean. Therefore, only .0062 of the scores are
above 85.
Example 2
 Suppose you wanted to know the
proportion of students receiving scores
between 70 and 80. The approach is to
figure out the proportion of students
scoring below 80 and the proportion
below 70.
 The difference between the two
proportions is the proportion scoring
between 70 and 80.
 First, the calculation of the proportion
below 80. Since 80 is 20 points above the
mean and the standard deviation is 10, 80
is 2 standard deviations above the mean.
The z table is used to determine
that .9772 of the scores are
below a score 2 standard
deviations above the mean.
Example 2
To calculate the proportion below 70:
Assume a test is normally
distributed with a mean of 100 and
a standard deviation of 15. What
proportion of the scores would be
between 85 and 105?
The solution to this problem is
similar to the solution to the last
one. The first step is to calculate the
proportion of scores below 85.
Next, calculate the proportion of
scores below 105. Finally, subtract
the first result from the second to
find the proportion scoring between
85 and 105.
The z-table is used to determine
that the proportion of scores
less than 1 standard deviation
above the mean is .8413. So, if
.1587 of the scores are above 70
and .0228 are above 80, then
.1587 -.0228 = .1359 are between
70 and 80.
Example 2
Begin by calculating the proportion
below 85. 85 is one standard deviation
below the mean:
Using the z-table with the value of -1 for z, the area below -1 (or
85 in terms of the raw scores) is .1587.
Do the same for 105
Example 2
The z-table shows that the proportion scoring below .333 (105 in raw
scores) is .6304. The difference is .6304 - .1587 = .4714. So .4714 of the
scores are between 85 and 105.
Sampling Distributions
Sampling Distributions
If you compute the mean of a sample of 10 numbers, the
value you obtain will not equal the population mean
exactly; by chance it will be a little bit higher or a little
bit lower.
If you sampled sets of 10 numbers over and over again
(computing the mean for each set), you would find that
some sample means come much closer to the population
mean than others. Some would be higher than the
population mean and some would be lower.
Imagine sampling 10 numbers and computing the mean
over and over again, say about 1,000 times, and then
constructing a relative frequency distribution of those
1,000 means.
Sampling Distributions
The distribution of means is a very good approximation
to the sampling distribution of the mean.
The sampling distribution of the mean is a theoretical
distribution that is approached as the number of samples
in the relative frequency distribution increases.
With 1,000 samples, the relative frequency distribution
is quite close; with 10,000 it is even closer.
As the number of samples approaches infinity, the
relative frequency distribution approaches the sampling
distribution
Sampling Distributions
 The sampling distribution of the mean for a sample size of
10 was just an example; there is a different sampling
distribution for other sample sizes.
 Also, keep in mind that the relative frequency distribution
approaches a sampling distribution as the number of
samples increases, not as the sample size increases since
there is a different sampling distribution for each sample
size.
Sampling Distributions
 A sampling distribution can also be defined as the
relative frequency distribution that would be
obtained if all possible samples of a particular
sample size were taken.
 For example, the sampling distribution of the mean
for a sample size of 10 would be constructed by
computing the mean for each of the possible ways
in which 10 scores could be sampled from the
population and creating a relative frequency
distribution of these means.
 Although these two definitions may seem different,
they are actually the same: Both procedures
produce exactly the same sampling distribution.
Sampling Distributions
Statistics other than the mean have sampling
distributions too. The sampling distribution of the
median is the distribution that would result if the
median instead of the mean were computed in each
sample.
Students often define "sampling distribution" as the
sampling distribution of the mean. That is a serious
mistake.
Sampling distributions are very important since
almost all inferential statistics are based on sampling
distributions.
Sampling Distribution of the mean
The sampling distribution of the mean is a very important
distribution. In later chapters you will see that it is used to
construct confidence intervals for the mean and for significance
testing.
Given a population with a mean of  and a standard deviation of
, the sampling distribution of the mean has a mean of  and a
standard deviation of / N , where N is the sample size.
The standard deviation of the sampling distribution of the mean is
called the standard error of the mean. It is designated by the
symbol .
Sampling Distribution of the mean
Note that the spread of the sampling distribution of the mean
decreases as the sample size increases.
An example of the effect of sample size is shown above.
Notice that the mean of the distribution is not affected by
sample size.
Spread
A variable's spread is the degree scores on the variable differ
from each other.
If every score on the variable were
about equal, the variable would have
very little spread.
There are many measures of spread.
The distributions on the right side of
this page have the same mean but
differ in spread: The distribution on
the bottom is more spread out.
Variability and dispersion are
synonyms for spread.
5 Samples
10 Samples
15 Samples
20 Samples
100 Samples
1,000 Samples
10,000 Samples
Hypothesis Testing
Classical Approach

The Classical Approach to hypothesis testing is to
compare a test statistic and a critical value. It is best
used for distributions which give areas and require
you to look up the critical value (like the Student's t
distribution) rather than distributions which have you
look up a test statistic to find an area (like the normal
distribution).

The Classical Approach also has three different
decision rules, depending on whether it is a left tail,
right tail, or two tail test.

One problem with the Classical Approach is that if a
different level of significance is desired, a different
critical value must be read from the table.
Left Tailed Test
H1: parameter < value
Notice the inequality points to the left
Decision Rule: Reject H0 if t.s. < c.v.
Right Tailed Test
H1: parameter > value
Notice the inequality points to the right
Decision Rule: Reject H0 if t.s. > c.v.
Two Tailed Test
H1: parameter not equal value
Another way to write not equal is < or >
Notice the inequality points to both sides
Decision Rule: Reject H0 if t.s. < c.v. (left) or
t.s. > c.v. (right)
The decision rule can be summarized as follows:
Reject H0 if the test statistic falls in the critical region
(Reject H0 if the test statistic is more extreme than the critical value)
P-Value Approach

The P-Value Approach, short for Probability Value, approaches
hypothesis testing from a different manner. Instead of comparing
z-scores or t-scores as in the classical approach, you're
comparing probabilities, or areas.

The level of significance (alpha) is the area in the critical region.
That is, the area in the tails to the right or left of the critical
values.

The p-value is the area to the right or left of the test statistic. If it
is a two tail test, then look up the probability in one tail and
double it.

If the test statistic is in the critical region, then the p-value will be
less than the level of significance. It does not matter whether it is
a left tail, right tail, or two tail test. This rule always holds.

Reject the null hypothesis if the p-value is less
than the level of significance.
P-Value Approach (Cont’d)

You will fail to reject the null hypothesis if the p-value is greater
than or equal to the level of significance.

The p-value approach is best suited for the normal distribution
when doing calculations by hand. However, many statistical
packages will give the p-value but not the critical value. This is
because it is easier for a computer or calculator to find the
probability than it is to find the critical value.

Another benefit of the p-value is that the statistician immediately
knows at what level the testing becomes significant. That is, a pvalue of 0.06 would be rejected at an 0.10 level of significance,
but it would fail to reject at an 0.05 level of significance.
Warning: Do not decide on the level of significance after
calculating the test statistic and finding the p-value.
P-Value Approach (Cont’d)
 Any proportion equivalent to the following
statement is correct:
The test statistic is to the p-value as the
critical value is to the level of significance.
Process Control Charts
Slide 1 of 37

Process Charts
– Tools for monitoring process variation.
– The figure on the following slide shows a
process control chart. It has an upper limit,
a center line, and a lower limit.
Process Control Charts
Slide 2 of 37
Control Chart (Figure 10.3 in the Textbook)
The UCL, CL, and
LCL are computed
statistically
Upper Control
Limit (UCL)
Center
Line (CL)
Lower Control
Limit (LCL)
Each point represents
data that are plotted
sequentially
Process Control Charts
Slide 3 of 37

Variables and Attributes
– To select the proper process chart, we must
differentiate between variables and attributes.
• A variable is a continuous measurement
such as weight, height, or volume.
• An attribute is the result of a binomial
process that results in an either-or-situation.
– The most common types of variable and
attribute charts are shown in the following
slide.
Process Control Charts
Slide 4 of 37
Variables and Attributes
Variables
Attributes
X (process population average)
P (proportion defective)
X-bar (mean for average)
np (number defective)
R (range)
C (number conforming)
MR (moving range)
U (number nonconforming)
S (standard deviation)
Process Control Charts
Slide 5 of 37
Central Requirements for Properly Using
Process Charts
1. You must understand the generic process for implementing
process charts.
2. You must know how to interpret process charts.
3. You need to know when different process charts are used.
4. You need to know how to compute limits for the different types
of process charts.
Process Control Charts
Slide 6 of 37
 A Generalized Procedure for
Developing Process Charts
– Identify critical operations in the process where
inspection might be needed. These are
operations in which, if the operation is performed
improperly, the product will be negatively affected.
– Identify critical product characteristics. These are
the attributes of the product that will result in either
good or poor function of the product.
Process Control Charts
Slide 7 of 37

A Generalized Procedure for Developing
Process Charts (continued)
– Determine whether the critical product
characteristic is a variable or an attribute.
– Select the appropriate process control chart
from among the many types of control charts.
This decision process and types of charts
available are discussed later.
– Establish the control limits and use the chart to
continually improve.
Process Control Charts
Slide 8 of 37

A Generalized Procedure for
Developing Process Charts (continued)
– Update the limits when changes have been
made to the process.
Process Control Charts
Slide 9 of 37
 Understanding Control Charts
– A process chart is nothing more than an
application of hypothesis testing where the
null hypothesis is that the product meets
requirements.
• An X-bar chart is a variables chart that monitors
average measurement.
• An example of how to best understand control
charts is provided under the heading
“Understanding Control Charts” in the textbook.
Process Control Charts
Slide 10 of 37

X-bar and R Charts
– The X-bar chart is a process chart used to monitor
the average of the characteristics being
measured. To set up an X-bar chart select
samples from the process for the characteristic
being measured. Then form the samples into
rational subgroups. Next, find the average value of
each sample by dividing the sums of the
measurements by the sample size and plot the
value on the process control X-bar chart.
Process Control Charts
Slide 11 of 37

X-bar and R Charts (continued)
– The R chart is used to monitor the variability or
dispersion of the process. It is used in
conjunction with the X-bar chart when the
process characteristic is variable. To develop
an R chart, collect samples from the process
and organize them into subgroups, usually of
three to six items. Next, compute the range,
R, by taking the difference of the high value in
the subgroup minus the low value. Then plot
the R values on the R chart.
Process Control Charts
Slide 12 of 37
X-bar and R Charts
Process Control Charts
Slide 13 of 37

Interpreting Control Charts
– Before introducing other types of process charts,
we discuss the interpretation of the charts.
– The figures in the next several slides show
different signals for concern that are sent by a
control chart, as in the second and third boxes.
When a point is found to be outside of the control
limits, we call this an “out of control” situation.
When a process is out of control, the variation is
probably not longer random.
Process Control Charts
Slide 14 of 37
Process Control Charts
Slide 15 of 37
Control Chart Evidence for Investigation
(Figure 10.10 in the textbook)
Process Control Charts
Slide 16 of 37
Control Chart Evidence for Investigation
(Figure 10.10 in the textbook)
Process Control Charts
Slide 17 of 37
Control Chart Evidence for Investigation
(Figure 10.10 in the textbook)
Process Control Charts
Slide 18 of 37

Implications of a Process Out of Control
– If a process loses control and becomes
nonrandom, the process should be
stopped immediately.
– In many modern process industries where
just-in-time is used, this will result in the
stoppage of several work stations.
– The team of workers who are to address
the problem should use a structured
problem solving process.
Process Control Charts
Slide 19 of 37

X and Moving Range (MR) Charts for
Population Data
– At times, it may not be possible to draw
samples. This may occur because a
process is so slow that only one or two
units per day are produced.
– If you have a variable measurement that
you want to monitor, the X and MR charts
might be the thing for you.
Process Control Charts
Slide 20 of 37

X and Moving Range (MR) Charts for
Population Data (continued)
– X chart. A chart used to monitor the mean
of a process for population values.
– MR chart. A chart for plotting variables
when samples are not possible.
– If data are not normally distributed, other
charts are available.
Process Control Charts
Slide 21 of 37

g and h Charts
– A g chart is used when data are geometrically
distributed, and h charts are useful when data are
hypergeometrically distributed.
– The next slide presents pictures of geometric and
hypergeometric distributions. If you develop a
histogram of your data, and it appears like either
of these distributions, you may want to use either
an h or a g chart instead of an X chart.
Process Control Charts
Slide 22 of 37
h and g Distributions (Figure 10.12 in the textbook)
Process Control Charts
Slide 23 of 37

Control Charts for Attributes
– We now shift to charts for attributes. These charts
deal with binomial and Poisson processes that are
not measurements.
– We will now be thinking in terms of defects and
defectives rather than diameters or widths.
• A defect is an irregularity or problem with a
larger unit.
• A defective is a unit that, as a whole, is not
acceptable or does not meet specifications.
Process Control Charts
Slide 24 of 37

p Charts for Proportion Defective
– The p chart is a process chart that is used to
graph the proportion of items in a sample that are
defective (nonconforming to specifications)
– p charts are effectively used to determine when
there has been a shift in the proportion defective
for a particular product or service.
– Typical applications of the p chart include things
like late deliveries, incomplete orders, and clerical
errors on written forms.
Process Control Charts
Slide 25 of 37

np Charts
– The np chart is a graph of the number of
defectives (or nonconforming units) in a
subgroup. The np chart requires that the
sample size of each subgroup be the same
each time a sample is drawn.
– When subgroup sizes are equal, either the
p or np chart can be used. They are
essentially the same chart.
Process Control Charts
Slide 26 of 37

np Charts (continued)
– Some people find the np chart easier to
use because it reflects integer numbers
rather than proportions. The uses for the
np chart are essentially the same as the
uses for the p chart.
Process Control Charts
Slide 27 of 37

c and u Charts
– The c chart is a graph of the number of
defects (nonconformities) per unit. The
units must be of the same sample space;
this includes size, height, length, volume
and so on. This means that the “area of
opportunity” for finding defects must be the
same for each unit. Several individual
unites can comprise the sample but they
will be grouped as if they are one unit of a
larger size.
Process Control Charts
Slide 28 of 37

c and u Charts (continued)
– Like other process charts, the c chart is
used to detect nonrandom events in the life
of a production process. Typical
applications of the c chart include number
of flaws in an auto finish, number of flaws
in a standard typed letter, and number of
incorrect responses on a standardized test
Process Control Charts
Slide 29 of 37

c and u Charts (continued)
– The u chart is a graph of the average number of
defects per unit. This is contrasted with the c
chart, which shows the actual number of defects
per standardized unit.
– The u chart allows for the units sampled to be
different sizes, areas, heights and so on, and
allows for different numbers of units in each
sample space. The uses for the u chart are the
same as the c chart.
Process Control Charts
Slide 30 of 37

Other Control Charts
– s Chart. The s (standard deviation) chart is
used in place of the R chart when a more
sensitive chart is desired. These charts
are commonly used in semiconductor
production where process dispersion is
watched very closely.
Process Control Charts
Slide 31 of 37

Other Control Charts (continued)
– Moving Average Chart. The moving
average chart is an interesting chart that is
used for monitoring variables and
measurement on a continuous scale.
– The chart uses past information to predict
what the next process outcome will be.
Using this chart, we can adjust a process
in anticipation of its going out of control.
Process Control Charts
Slide 32 of 37

Other Control Charts (continued)
– Cusum Chart. The cumulative sum, or
cusum, chart is used to identify slight but
sustained shifts in a universe where there
is no independence between observations.
Process Control Charts
Slide 33 of 37
Summary of Chart Formulas (Table 10.2 in the textbook)
Process Control Charts
Slide 34 of 37

Some Control Chart Concepts
– Choosing the Correct Control Chart
• Obviously, it is key to choose the correct control
chart. Figure 10.19 in the textbook shows a
decision tree for the basic control charts. This
flow chart helps to show when certain charts
should be selected for use.
Process Control Charts
Slide 35 of 37

Some Control Chart Concepts (continued)
– Corrective Action. When a process is out of
control, corrective action is needed. Correction
action steps are similar to continuous
improvement processes. They are
• Carefully identify the problem.
• Form the correct team to evaluate and solve
the problem.
• Use structured brainstorming along with
fishbone diagrams or affinity diagrams to
identify causes of the problem.
Process Control Charts
Slide 36 of 37

Some Control Chart Concepts (continued)
– Corrective Action (continued)
• Brainstorm to identify potential solutions to
problems.
• Eliminate the cause.
• Restart the process.
• Document the problem, root causes, and
solutions.
• Communicate the results of the process to all
personnel so that this process becomes
reinforced and ingrained in the operations.
Process Control Charts
Slide 37 of 37

Some Control Chart Concepts (continued)
– How Do We Use Control Charts to Continuously
Improve?
• One of the goals of the control chart user is to
reduce variation. Over time, as processes are
improved, control limits are recomputed to
show improvements in stability. As upper and
lower control limits get closer and closer
together, the process improving.
• The focus of control charts should be on
continuous improvement and they should be
updated only when there is a change in the
process.
Process Capability
Slide 1 of 4
 Process Stability and Capability
– Once a process is stable, the next emphasis is
to ensure that the process is capable.
– Process capability refers to the ability of a
process to produce a product that meets
specifications.
– Six-sigma program such as those pioneered
by Motorola Corporation result in highly
capable processes.
Process Capability
Slide 2 of 4
Six-Sigma Quality (Figure 10.21 in the textbook)
Process Capability
Slide 3 of 4

Process Versus Sampling Distribution
– To understand process capability we must first
understand the differences between population and
sampling distributions.
• Population distributions are distributions with all the
items or observations of interest to a decision
maker.
• A population is defined as a collection of all the
items or observations of interest to a decision
maker.
• A sample is subset of the population. Sampling
distributions are distributions that reflect the
distributions of sample means.
Process Capability
Slide 4 of 4

The Difference Between Capability and
Stability?
– Once again, a process is capable if
individual products consistently meet
specifications.
– A process is stable if only common
variation is present in the process.
Determine
characteristic
to be charted.
Is the data
variable?
YES
How to choose the correct control chart
NO
NO
Non-conforming
units? (% bad
parts)
Nonconformities?
(I.e., discrepancies
per part.)
YES
YES
NO
Constant
sample size?
NO
YES
Use X - MR chart.
Use np or
p chart.
NO
Use

chart.
YES
YES
Is it homogeneous,
or not conducive to
subgroup sampling?
(e.g., chemical
bath, paint
batch, etc.)
Is sample
space
constant?
Use
p chart.
Can subgroup
averages be
conveniently
computed?
YES
Next slide.
Use
c or 
chart.
NO
Use
median
chart.
How to choose the correct control chart
(from previous page)
Can subgroup
averages be
conveniently
computed?
NO
Use
median
chart.
YES
Is the subgroup
size < 9?
NO
Use
X - R chart
.
YES
Can s be calculated
for each group?
YES
Use
X - s chart
.
NO
Use
X - R chart
.