ITED 434 Quality Organization

1
ITED 434Quality Organization Management Ch 10
11

• Ch 10 Basic Concepts of Statistics and
  Probability
• Ch 11 Statistical Tools for Analyzing Data

2
Chapter Overview

• Statistical Fundamentals
• Process Control Charts
• Some Control Chart Concepts
• Process Capability
• Other Statistical Techniques in Quality
  Management

3
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.

4
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.

5
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.

6
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 dont understand random variation
  resulting in too much process tampering.

7
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.

8
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.

9
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.

10
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.

11
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.

12
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.

13
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.
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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.

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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.
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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.
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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
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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.
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Sampling Distributions
20
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.

21
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

22
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.

23
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.

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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.

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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 s/? 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
  ?.

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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.
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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.
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5 Samples
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10 Samples
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15 Samples
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20 Samples
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100 Samples
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1,000 Samples
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10,000 Samples
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Hypothesis Testing
37
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.

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Left Tailed Test H1 parameter lt valueNotice the
inequality points to the left Decision Rule
Reject H0 if t.s. lt c.v.
Right Tailed Test H1 parameter gt valueNotice
the inequality points to the right Decision
Rule Reject H0 if t.s. gt c.v.
Two Tailed Test H1 parameter not equal
valueAnother way to write not equal is lt or
gtNotice the inequality points to both sides
Decision Rule Reject H0 if t.s. lt c.v. (left)
or t.s. gt 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)
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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.

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P-Value Approach (Contd)

• 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 p-value
  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.

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P-Value Approach (Contd)

• 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.

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Process Control ChartsSlide 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.

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Process Control ChartsSlide 2 of 37
Control Chart (Figure 10.3 in the Textbook)
The UCL, CL, and LCL are computed statistically
Each point represents data that are
plotted sequentially
Upper Control Limit (UCL)
Center Line (CL)
Lower Control Limit (LCL)
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Process Control ChartsSlide 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.

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Process Control ChartsSlide 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)
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Process Control ChartsSlide 5 of 37
Central Requirements for Properly Using Process
Charts
1.
You must understand the generic process for
implementing process charts. You must know how to
interpret process charts. You need to know when
different process charts are used. You need to
know how to compute limits for the different
types of process charts.
2.
3.
4.
47
Process Control ChartsSlide 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.

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Process Control ChartsSlide 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.

49
Process Control ChartsSlide 8 of 37

• A Generalized Procedure for Developing Process
  Charts (continued)
• Update the limits when changes have been made to
  the process.

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Process Control ChartsSlide 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.

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Process Control ChartsSlide 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.

52
Process Control ChartsSlide 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.

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Process Control ChartsSlide 12 of 37
X-bar and R Charts
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Process Control ChartsSlide 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.

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Process Control ChartsSlide 14 of 37
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Process Control ChartsSlide 15 of 37
Control Chart Evidence for Investigation (Figure
10.10 in the textbook)
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Process Control ChartsSlide 16 of 37
Control Chart Evidence for Investigation (Figure
10.10 in the textbook)
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Process Control ChartsSlide 17 of 37
Control Chart Evidence for Investigation (Figure
10.10 in the textbook)
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Process Control ChartsSlide 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.

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Process Control ChartsSlide 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.

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Process Control ChartsSlide 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.

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Process Control ChartsSlide 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.

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Process Control ChartsSlide 22 of 37
h and g Distributions (Figure 10.12 in the
textbook)
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Process Control ChartsSlide 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.

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Process Control ChartsSlide 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.

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Process Control ChartsSlide 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.

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Process Control ChartsSlide 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.

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Process Control ChartsSlide 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.

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Process Control ChartsSlide 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

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Process Control ChartsSlide 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.

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Process Control ChartsSlide 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.

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Process Control ChartsSlide 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.

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Process Control ChartsSlide 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.

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Process Control ChartsSlide 33 of 37
Summary of Chart Formulas (Table 10.2 in the
textbook)
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Process Control ChartsSlide 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.

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Process Control ChartsSlide 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.

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Process Control ChartsSlide 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.

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Process Control ChartsSlide 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.

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Process CapabilitySlide 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.

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Process CapabilitySlide 2 of 4
Six-Sigma Quality (Figure 10.21 in the textbook)
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Process CapabilitySlide 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.

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Process CapabilitySlide 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.

83
Determine characteristic to be charted.
How to choose the correct control chart
Non-conforming units? ( bad parts)
Nonconformities? (I.e., discrepancies per part.)
Is the data variable?
NO
NO
YES
YES
YES
NO
Constant sample size?
Is sample space constant?
NO
Use m chart.
Use p chart.
YES
YES
Is it homogeneous, or not conducive to subgroup
sampling? (e.g., chemical bath, paint batch,
etc.)
Use c or m chart.
Use np or p chart.
Can subgroup averages be conveniently computed?
Use median chart.
NO
NO
YES
YES
Next slide.
Use X - MR chart.
84
How to choose the correct control chart
Can subgroup averages be conveniently computed?
(from previous page)
NO
Use median chart.
YES
Is the subgroup size lt 9?
NO
YES
Can s be calculated for each group?
NO
YES
Use .
X - s chart