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CHAPTER 6
Statistical Methods in Quality Management
Teaching Notes
This chapter describes concepts of statistics, statistical thinking, statistical methodology,
sampling, experimental design, and process capability. Students should be encouraged to take a
“big picture” perspective on this framework, rather than the approach of: “How do I get the
right answer?”
Although this chapter reviews many of the basic concepts and techniques of statistics that are
relevant to the technical areas of statistical process control (SPC), it is by no means
comprehensive. Students should be encouraged to consult a statistics textbook for further
insights on the topics in this chapter. These topics are typically covered in business or
engineering statistics course that students should have had prior to taking a course using this
text. Key objectives for this chapter include:
• To establish the importance of statistics as the “bridge” between quality of design and
quality of conformance. The proper use of statistics is highlighted as a quality
improvement tool.
Statistical Methods in Quality Management 2
• To introduce the concept of statistical measurement of service quality
The Instructor’s Resource folder on the website for this chapter has a number of Baldrige video
ANSWERS TO QUALITY IN PRACTICE QUESTIONS
Improving Quality of a Wave Soldering Process Through the Design of Experiments
1. The first experimental design at the HP plant did not achieve the true optimum
combination of factors, because not all combinations were tested. It is theoretically
2. Experimental design allows the experimenter to systematically evaluate two or more
methods to determine which is better, or to determine the levels of controllable factors
to optimize process yields or minimize variation of a response variable. Therefore, it is
generally faster and more efficient than using one at a time, trial-and-error methods.
Applying Statistical Analysis in a Six Sigma Project at GE-Fanuc
1. This case showed how wave soldering technology was applied to electronic circuit boards
in a process that is very similar to the “Improving Quality of a Wave Soldering Process
Through the Design of Experiments” case, above. There are many variables that must
be taken into account in order to improve the process, some of which are the “true” root
2. The Ni-Au boards could have originally been selected for any number of reasons.
Reports on the advantages of using Ni-Au boards could have been read in technical
Statistical Methods in Quality Management 3
3. The F-test shows that the vendors, finishes, and interaction effects are all highly
significant for this particular independent variable, “Wave Solder Skips.” Thus, the
team must further analyze the defects to find out why the vendors’ products vary from
their competitors’, how the finishes differ, and what the interaction effects are.
ANSWERS TO REVIEW QUESTIONS
1. Statistics is a science concerned with “the collection, organization, analysis,
interpretation, and presentation of data.” Statistics is essential for quality and for
implementing a continuous improvement philosophy. Statistical methods help managers
2. In statistical terminology, an experiment is a process that results in some outcome. The
outcome of an experiment is a result that we observe. The collection of all possible
outcomes of an experiment is called the sample space. A sample space may consist of a
small number of discrete outcomes or an infinite number of outcomes. Probability is the
likelihood that an outcome occurs.
3. The following rules apply to calculating probabilities of events:
Rule 1: The probability of any event is the sum of the probabilities of the outcomes that
compose that event.
4. The multiplication rule of probability is: P(A and B) = P(A | B) P(B) = P(B | A) P(A),
where P(A | B) reads as the conditional probability of A, given B. Conditional
probability is the probability of occurrence of one event A, given that another event B is
known to be true or have already occurred.
5. The two most important types of probability distributions are discrete and continuous
probability distributions. Under the discrete category, the binomial and Poisson
Statistical Methods in Quality Management 4
distributions are the most important. The binomial distribution calculates the
probability of exactly x successes in a sequence of n identical experiments, called trials.
The Poisson distribution is closely related to the binomial. It is derived by allowing the
sample size n to become very large, while the probability of success or failure (p) to
become very small (approaching 0).
6. A probability distribution can be either discrete or continuous, depending on the nature
of the random variable it models. For discrete probability distributions, a complete,
finite number of outcomes and their associated probabilities of occurrence can be listed.
These outcomes are called a list of mutually exclusive and collectively exhaustive
outcomes.
A continuous random variable is defined over one or more intervals of real numbers,
and therefore, has an infinite number of possible outcomes. A curve that characterizes
outcomes of a continuous random variable is called a probability density function, and
Statistical Methods in Quality Management 5
7. The three basic elements of statistical methodology are descriptive statistics, statistical
inference and predictive statistics. The methods for the efficient collection,
organization, and description of data are called descriptive statistics. Statistical
inference is the process of drawing conclusions about unknown characteristics of a
population from which the data were taken. Predictive statistics is used to develop
8. Methods of sample selection, or sampling schemes, include: simple random sampling,
stratified sampling, systematic sampling and cluster sampling. Simple random sampling
is useful where one needs to gather information from a moderately large, homogeneous
9. Any sampling procedure can result in two types of errors: sampling error and
systematic error. Sampling error occurs naturally and results from the fact that a sample
may not always be representative of the population, no matter how carefully it is
Statistical Methods in Quality Management 6
10. A population is a complete set or collection of objects of interest. A sample is a subset
of objects taken from a population.
11. Measures of location, are essentially those that focus on “central tendency,” such as the
12. Measures of dispersion are used to indicate the degree of “scatter” of data. They include
the range, variance, and standard deviation. The latter two statistics measure scatter
around the mean of the sample or population.
13. The proportion, usually denoted as p, is used to measure the fraction of data that have a
14. The standard error of the mean is the (estimated) standard deviation of the population
15. The central limit theorem is extremely useful in that it states (approximately) that a
sampling distribution can be defined as the distribution obtained by taking a large
16. Microsoft Excel provides data analysis tools, called the Analysis ToolPak, that are
useful in complex statistical analyses. You provide the data and parameters for each
Statistical Methods in Quality Management 7
17. A confidence interval (CI) is an interval estimate of a population parameter that also
specifies the likelihood that the interval contains the true population parameter. This
probability is called the level of confidence, denoted by 1 – , and is usually expressed
as a percentage. Used together, these statistical tools can help clarify what we know and
18. Some applications of hypothesis testing that might be applied to the topics in Chapters
3, 4, and 5 are varied. For example, a hypothesis might be tested concerning the
19. ANOVA is a methodology for drawing conclusions about equality of means of multiple
populations. In its simplest form – one-way ANOVA – it compares means of observed
20. Correlation is a measure of a linear relationship between two variables, X and Y, and is
measured by the (population) correlation coefficient. Correlation coefficients will range
from -1 to +1. A correlation of 0 indicates that the two variables have no linear
relationship to each other. Thus, if one changes, we cannot reasonably predict what the
other variable might do using a linear equation (we might, however, have a well-
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21. The purpose of design of experiments is to set up a test or series of tests to enable the
22. A factorial experiment is a specific type of experimental design that considers all
combinations of levels of each factor. For example, a factorial experiment might be set
SOLUTIONS TO PROBLEMS – CHAPTER 6
Note: Data sets for several problems in this chapter are available in the Excel workbook
C06Data.xlsx on the Student Companion Site accompanying this chapter. Click on the
appropriate worksheet tab as noted in the problem (e.g., Prob. 6-1) to access the data. In
addition, the templates in the Template folder are provided to assist in solving example
problems in the body of the chapter are also available to aid in solving many of these problems.
1. A new production process at Fabulast, Inc., has two in-line stages. The probability of
defective components being produced in stage 1 is 15 percent and 10 percent in stage 2.
Assembled units that have defective components only from stage 1 OR only from stage
2 are considered repairable. However items that have defective components from both
stage 1 and stage 2 (completely defective) must be scrapped.
a. Use a probability tree diagram and calculate the probabilities that the Fabulast
assembled units are: (i) defective in stage 1 and defective in stage 2 (are completely
Statistical Methods in Quality Management 9
Answer
Test indicates completely defective
(0.15) (0.10) = 0.015
a) As shown on the tree diagram, the probabilities that the Fabulast units are:
1) defective in stage one AND defective in stage two (are completely defective) = 0.015
The probability for repairable assembled units = 0.220
b) Using the multiplication rule, the probability that any Fabulast product coming off
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2. Auditors at the Numeros Verdes Partners, P.S.C. took a sample of 200 accounts payable
bills, as shown in the table found in the Excel worksheet Prob. 6-2 in the Excel
workbook C06Data.xlsx.
a. Find the proportion of the accounts payable in the sample that are classified as
overdue by using the Excel COUNTIF function.
b. If an auditor takes a random sample of only 10 accounts from this population,
assuming that they follow a binomial distribution, what is the probability that: (i)
exactly 5 bills will be overdue? (ii) 4 or fewer bills will be overdue? (iii) 6 or more
bills will be overdue? Use the binomial probability distribution formula and verify
your result using Excel Binomial spreadsheet template.
Answer
a) The 25 percent overdue bills count is verified in the P06-02-03-10BiPoisExp.xlsx
spreadsheet.
The output shows:
Statistical Methods in Quality Management 11
3. The Turkuman Rug Company buys medium grade carpet in 100-foot rolls. The average
number of defects per roll is 1.8. Assuming that these data follow a Poisson
distribution, use the Poisson spreadsheet template to answer the following questions.
a) What is the probability of finding exactly 7 defects in a carpet roll chosen at random?
b) What is the probability of finding 4 or fewer defects in a carpet roll?
Answer
The Poisson distribution’s function has this form: 𝑓(𝑥)= 𝒆−𝝀 𝝀𝒙
𝒙!
4. Southwestern Punch was made by Frutayuda, Inc. and sold in 12-ounce cans to benefit
victims of Hurricane Zero. The mean number of ounces placed in a can by an automatic
fill pump is 11.7 with a standard deviation of 0.18 ounce. Assuming a normal
distribution, determine the probability that the filling pump will cause an overflow in a
can, that is, the probability that more than 12 ounces will be released by the pump and
overflow the can.
Statistical Methods in Quality Management 12
Answer
4. For cans of Southwestern Punch the mean, µ = 11.7; the standard deviation, = 0.18
z = 𝑥− 𝜇
𝜎
z = 12− 11.7
0.18 = 1.67
5. Los Alamos Green Tea is sold in 500 milliliter bottles. The standard deviation for the
filling process is 7 milliliters. What must the target mean for the process be to ensure
that the probability of overfilling more than 500 ml in a bottle is at most 1 percent?
Answer
5. For Los Alamos Green Tea’s bottling process, the values for the 1 percent cutoff and
the standard deviation are:
x = 480 ml; = 7 ml
Statistical Methods in Quality Management 13
6. Kiwi Oil is sold in 950 milliliter (ml) cans. The mean volume of oil placed in a can is
920 ml with a standard deviation of 12 ml. Assuming a normal distribution of the data,
what is the probability that the filling machine will cause an overflow in a can, that is,
the probability that more than 950 ml will be placed in the can?
Answer
6. The mean, for the Kiwi oil product is µ = 920; the standard deviation, = 12, x = 950.
z = 𝑥− 𝜇
𝜎
7. Wayback Cleaning Co. has found that standard size offices have a standard deviation of
5 minutes for their cleaning time. The operations manager knows that 95 percent of the
offices require more than 120 person-minutes to clean. However, she wishes to find out
the average cleaning time for the offices. Can you calculate that for her?
Answer
7. Given that the standard deviation for Wayback Cleaning Co. offices, is = 5 min., x =
120, and P (z > x) = 0.95. Using the NORM.INV function, we have to solve for z when:
Statistical Methods in Quality Management 14
8. The mean time to pour and process 5 cubic yards of concrete by the Piedra
Cretebuilders Co. is 15.5 minutes. If 2 percent of the projects with 5 yards of concrete
require more than 15.75 minutes, what is the standard deviation of the time for such
projects?
Answer
8. Given that the mean process time is µ = 15.5 minutes for the Piedra’s standard pour of
5 cubic yards of concrete, we find z by taking P (z < x) = P(1.0000) – P (0.0200) =
0.9800. Using the NormINVTemplate.xlsx,
z = 2.054
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9. The dimension of a machined part has a nominal specification of 11.9 cm. The process
that produces the part can be controlled to have a mean value equal to this specification,
but has a standard deviation of 0.05 cm. What is the probability that a part will have a
dimension:
a) exceeding 12 cm?
b) between 11.9 and 11.95 cm?
c) less than 11.83 cm?
Answer
9. The mean value for the machined part, in cm is: µ = 11.9; the standard deviation, =
0.05
a) P(x > 12.0 cm) = 1.0000 – P ( x < 12.0)
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10. Genjeteye, Inc. makes aircraft engines. The mean time to failure has been found to be
100,000 hours and is exponentially distributed.
a) What is the failure rate, , per hour?
b) What is the cumulative probability of failure after 10,000 hours or fewer? Between
10,000 and 15,000 hours?
c) If Genjeteye wishes to provide a warranty that no more than 5 percent of the units
will fail, how many hours of operation without failure should the company guarantee?
Answer
10. a) 𝜆= 1
𝑀𝑇𝑇𝐹= 1
100,000=0.00001
Statistical Methods in Quality Management 17
11. Use the data for Twentyfirst Century Laundry for the weights of loads of clothes
processed through their washing department in a week. (See Prob. 6-11 in C06Data
workbook).
a. Apply the Excel Descriptive Statistics tool to compute the mean, standard deviation,
and other relevant statistics, and interpret the results in a meaningful fashion.
b. Use the Frequency Distribution and Histogram Excel template to construct a
frequency distribution and histogram for the data. From what type of distribution might
you suspect the data are drawn? Experiment with the number of cells to create a
visually-appealing histogram and use the Excel Histogram tool to verify the results.
Answer
11. The following results were obtained from the Twentyfirst Century Laundry data
Descriptive Statistics
Mean
32.920
Standard Error
2.590
Median
25.500
Mode
14.000
Standard Deviation
25.899
Kurtosis
0.233
Skewness
0.994
Range
106.000
Minimum
1.000
Maximum
107.000
Sum
3292.000
Count
100.000
Largest(1)
107.000
Smallest(1)
1.000
Confidence Level (95.0 percent)
5.139
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12. The times for carrying out a blood test at Rivervalley Labs for 100 tests, found in the
Prob. 6-12, in the C06Data Excel workbook, were studied in order to better understand
the process. Apply the Descriptive Statistics tool to compute summary statistics and
explain the results. Also, construct a frequency distribution and histogram, for the data
taken from set. From what type of distribution might you suspect the data are drawn?
Answer
Summary statistics using Excel follow. Also shown is the histogram constructed by
the Data Analysis tool. For best results in constructing the histogram, it is suggested
Descriptive Statistics
Mean
3.578
25
30
35
Histogram
Statistical Methods in Quality Management 19
Standard Error
0.081
Median
3.600
13. The data for Prob. 6-13 found in C06Data Excel workbook shows the weight of a set of
castings (in kilograms) being made in the Fillmore Metalwork foundry. Construct an
Excel spreadsheet to compute the mean and the sample standard deviation using
formulas (6.14) and (6.18). Verify your results using Excel functions.
Answer
10
15
20
Histogram
Mode
3.600
Standard Deviation
0.812
Sample Variance
0.660
Kurtosis
Skewness
Range
3.600
Minimum
1.700
Maximum
5.300
Sum
Count
Largest(1)
5.300
Smallest(1)
1.700
Confidence Level (95.0 percent)
0.161
Statistical Methods in Quality Management 20
14. A warehouse manager at Wherehousing, Inc. maintains a large inventory of video
games. The company’s database states that the mean value of the games in inventory is
$50, with a standard deviation of $5. The manager is concerned about pilfering the
more expensive games by the warehouse employees. She picked a random sample of
100 games and found the mean value to be $48.50. Assuming a normal distribution,
what is the probability that the sample mean would be $48.50 or less, if all the
inventory can actually be accounted for? What conclusions would you reach?
Answer
a) Since this probability is on the lower tail of the normal distribution, we must
calculate: