IMPROVEMENT

4-1 Introductionandchapterobjectives 4-2 Populationandsample

4-3 Parameterandstatistic 4-4 Probability

4-5 Descriptivestatistics:describingproductorprocesscharacteristics 4-6 Probabilitydistributions

4-7 Inferentialstatistics:drawingconclusionsonproductandprocessquality Summary

Appendix:Approximationstosomeprobabilitydistributions Symbols

P^�(A) ProbabilityofeventA f(x) μ Populationmeanofaqualitycharacteristic

X Sampleaverage F(x)

s Samplestandarddeviation E(X^�)

n Samplesize

Xi ith observation in a sample Z

N Populationsize p

T(α) α% trimmed mean

σ Populationstandarddeviation λ

s2^� Samplevariance λ

σ² Populationvariance

R Range γ

γ1 Skewnesscoefficient

γ2 Kurtosiscoefficient α

r Samplecorrelationcoefficient

M Samplemedian β

sm Standarddeviationofthesamplemedian p(x) Probabilitydistribution(ormassfunction) Γ(u)

foradiscreterandomvariable 1 α

Probabilitydensityfunctionfora continuousrandomvariable Cumulativedistributionfunction Expectedvalueormeanofarandom

variableX

Standardnormalrandomvariable Probabilityofsuccessonatrialina

binomialexperiment

MeanofaPoissonrandomvariable Failurerateforanexponential

distribution

LocationparameterforaWeibull distribution

ScaleparameterforaWeibull distribution

ShapeparameterforaWeibull distribution

Gammafunctionforthevariableu Level of confidence for confidence

intervals Fundamentals of Quality Control and Improvement,Fourth Edition. Amitava Mitra

2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

Companionwebsite:www.wiley.com\go\mitra\QualityControl4e

149

4-1 INTRODUCTION AND CHAPTER OBJECTIVES

In this chapter we build a foundation for the statistical concepts and techniques used in quality control and improvement. Statistics is a subtle science, and it plays an important role in quality programs. Only a clear understanding of statistics will enable you to apply it properly. They are often misused, but a sound knowledge of statistical principles will help you formulate correct procedures in different situations and will help you interpret the results properly. When we analyze a process, we oftenfind it necessary to study its characteristics individually. Breaking the process down allows us to determine whether some identifiable cause has forced a deviation from the expected norm and whether a remedial action needs to be taken. Thus, our objective in this chapter is to review different statistical concepts and techniques along two major themes.

Thefirst deals withdescriptive statistics, those that are used to describe products or processes and their characteristic features, based on collected data. The second theme is focused on inferential statistics, whereby conclusions on product or process parameters are made through statistical analysis of data. Such inferences, for example, may be used to determine if there has been a significant improvement in the quality level of a process, as measured by the proportion of nonconforming product.

4-2 POPULATION AND SAMPLE

Apopulationis the set of all items that possess a certain characteristic of interest.

Example 4-1 Suppose that our objective is to determine the average weight of cans of brand A soup processed by our company for the month of July. The population in this case is the set of all cans of brand A soup that are output in the month of July (say, 50,000). Other brands of soup made during this time are not of interest; here only the population of brand A soup cans is considered.

Asampleis a subset of a population. Realistically, in many manufacturing or service industries, it is not feasible to obtain data on every element in the population. Measurement, storage, and retrieval of large volumes of data are impractical, and the costs of obtaining such information are high. Thus, we usually obtain data from only a portion of the population—a sample.

Example 4-2 Consider our brand A soup. To save ourselves the cost and effort of weigh­

ing 50,000 cans, we randomly select a sample of 500 cans of brand A soup from the July output.

4-3 PARAMETER AND STATISTIC

Aparameteris a characteristic of a population, something that describes it.

Example 4-3 For our soup example, we will be looking at the parameter average weight of all 50,000 cans processed in the month of July.

Astatisticis a characteristic of a sample. It is used to make inferences on the population parameters that are typically unknown.

Example 4-4 Our statistic then is the average weight of a sample of 500 cans chosen from the July output. Suppose that this value is 300 g; this would then be anestimateof the average weight of all 50,000 cans. A statistic is sometimes called anestimator.

PROBABILITY 151 4-4 PROBABILITY

Our discussion of the concepts of probability is intentionally brief. For an in-depth look at probability, see the references at the end of the chapter. Theprobabilityof an event describes the chance of occurrence of that event. A probability function is bounded between 0 and 1, with 0 representing the definite nonoccurrence of the event and 1 representing the certain occurrence of the event.

The set of all outcomes of an experiment is known as thesample spaceS.

Relative Frequency Definition of Probability

If each event in the sample space is equally likely to happen, the probability of an eventAis given by

nA

P…A† ˆ N …4-1†

whereP(A)ˆprobability of eventA,n_Aˆnumber of occurrences of eventA, andNˆsize of the sample space.

This definition is associated with the relative frequency concept of probability. It is applicable to situations where historical data on the outcome of interest are available. The probability associated with the sample space is 1 [i.e.,P(S)ˆ1].

Example 4-5 A company makes plastic storage bags for the food industry. Out of the hourly production of two thousand 500-g bags, 40 were found to be nonconforming. If the inspector chooses a bag randomly from the hour’s production, what is the probability of it being nonconforming?

Solution We define eventAas getting a bag that is nonconforming. The sample spaceS consists of 2000 bags (i. e.,Nˆ2000). The number of occurrences of eventA(n_A) is 40. Thus, if the inspector is equally likely to choose any one of the 2000 bags,

P…A† ˆ 40 ˆ0:02 2000 Simple and Compound Events

Simple eventscannot be broken down into other events. They represent the most elementary form of the outcomes possible in an experiment.Compound eventsare made up of two or more simple events.

Example 4-6 Suppose that an inspector is sampling transistors from an assembly line and identifying them as acceptable or not. Suppose the inspector chooses two transistors. What are the simple events? Give an example of a compound event. Find the probability offinding at least one acceptable transistor.

Solution Consider the following outcomes:

A₁: event that thefirst transistor is acceptable D1: event that thefirst transistor is unacceptable

A2: event that the second transistor is acceptable D2: event that the second transistor is unacceptable Four simple events make up the sample spaceS:

Sˆ fA1A2;A1D2;D1A2;D1D2g

These events may be described as follows:

E₁ˆ{A1A₂}: event that thefirst and second transistors are acceptable

E₂ˆ{A1D₂}: event that thefirst transistor is acceptable and the second one is not E₃ˆ{D1A₂}: event that thefirst transistor is unacceptable and the second one is

acceptable

E₄ˆ{D₁D₂}: event that both transistors are unacceptable

Compound event B is the event that at least one of the transistors is acceptable. In this case, eventBconsists of the following three simple events:Bˆ{E1,E2,E3}. Assuming that each of the simple events is equally likely to happen, P…B† ˆP…E₁† ‡P…E₂†‡

P…E₃† ˆ¹₄‡¹₄‡¹₄ˆ³₄. Figure 4-1 shows a Venn diagram, which is a graphical representation of the sample space and its associated events.

Complementary Events

Thecomplementof an eventAimplies the occurrence of everything butA. If we defineA^cto be the complement ofA, then

P…A^c† ˆ1 P…A† …4-2† Figure 4-2 shows the probability of the complement of an event by means of a Venn diagram. Continuing with Example 4-6, suppose that we want tofind the probability of the event that both transistors are unacceptable. Note that this is the complement of eventB, which was defined as at least one of the transistors being acceptable. So

3 1

P…B^c† ˆ1 P…B† ˆ1 4 ˆ 4

FIGURE 4-1 Venn diagram.

PROBABILITY 153

FIGURE 4-2 An event and its complement.Note: The shaded area representsP(A^c).

Additive Law

Theadditive lawof probability defines the probability of the union of two or more events happening. If we have two eventsAandB, the union of these two implies thatAhappensor B happensorboth happen. Figure 4-3 shows the union of two events,AandB. The hatched area in the sample space represents the probability of the union of the two events. Theadditive law is as follows:

P…A∪B† ˆP…AorBor both†

ˆP…A† ‡P…B† P…A∩B† …4-3† Note thatP(A∩B) represents the probability of the intersection of eventsAandB: that is, the occurrence of bothAandB. The logic behind the additive law can easily be seen from the Venn diagram in Figure 4-3, whereP(A) represents the area within the boundary-defining eventA.

Similarly, P(B) represents the area within the boundary-defining event B. The overlap (crosshatched) between areas A and B represents the probability of the intersection, P(A∩B). WhenP(A) is added toP(B), this intersection is included twice, so eq. (4-3) adds P(A) toP(B) and subtractsP(A∩B) once.

FIGURE 4-3 Union of two events.

Multiplicative Law

Themultiplicative lawof probability defines the probability of the intersection of two or more events. Intersection of a group of events means that all the events in that group occur. In general, for two eventsAandB,

P…A∩B† ˆP…AandB† ˆP…A†P…BjA†

ˆP…B†P…AjB† …4-4†

The termP(B|A) represents the conditional probability ofBgiven that eventAhas happened (i.e., the probability thatBwill occur ifAhas). Similarly,P(A|B) represents the conditional probability ofAgiven that eventBhas happened. Of the two forms given by eq. (4-4), the problem will dictate which version to use.

Independence and Mutually Exclusive Events

Two eventsAandBare said to beindependentif the outcome of one has no influence on the outcome of the other. IfAandBare independent, thenP(B|A)ˆP(B); that is, the conditional probability of B given that A has happened equals the unconditional probability of B.

Similarly,P(A|B)ˆP(A) ifAandBare independent. From eq. (4-4), it can be seen that ifA andBare independent, the general multiplicative law reduces to

P…A∩B† ˆP…AandB† ˆP…A†P…B† ifAandBare independent …4-5† Two eventsAandBare said to bemutually exclusiveif they cannot happen simulta­

neously. The intersection of two mutually exclusive events is the null set, and the probability of their intersection is zero. Notationally,P(A∩B)ˆ0 ifAandBare mutually exclusive.

Figure 4-4 shows a Venn diagram for two mutually exclusive events. Note that whenAandB are mutually exclusive events, the probability of their union is simply the sum of their individual probabilities. In other words, the additive law takes on the following special form:

P…A∪B† ˆP…A† ‡P…B† ifAandBare mutually exclusive

If eventsA and B are mutually exclusive, what can we say about their dependence or independence? Obviously, ifAhappens,Bcannot happen, and vice versa. Therefore, ifAand Bare mutually exclusive, they are dependent. IfAandBare independent, the additive rule from eq. (4-3) becomes

P…AorBor both† ˆP…A† ‡P…B† P…A†P…B† …4-6†

FIGURE 4-4 Mutually exclusive events.

PROBABILITY 155 Example 4-7

(a) In the production of metal plates for an assembly, it is known from past experience that 5% of the plates do not meet the length requirement. Also, from historical records, 3% of the plates do not meet the width requirement. Assume that there are no dependencies between the processes that make the length and those that trim the width. What is the probability of producing a plate that meets both the length and width requirements?

Solution LetAbe the outcome that the plate meets the length requirement andBbe the outcome that the plate meets the width requirement. From the problem statement, P(A^c)ˆ0.05 andP(B^c)ˆ0.03. Then

P…A† ˆ1 P…A^c† ˆ1 0:05ˆ0:95 P…B† ˆ1 P…B^c† ˆ1 0:03ˆ0:97

Using the special case of the multiplicative law for independent events, we have P…meeting both lengthandwidth requirements† ˆP…A∩B†

ˆP…A†P…B† …sinceAandBare independent events†

ˆ …0:95†…0:97† ˆ0:9215

(b) What proportion of the parts will not meet at least one of the requirements?

Solution The required probabilityˆP(A^corB^cor both). Using the additive law, we get

A^c A^c∩B^c†

ˆ0:05‡0:03 …0:03†…0:05† ˆ0:0785 P… orB^cor both† ˆP…A^c† ‡P…B^c† P…

Therefore, 7.85% of the parts will have at least one characteristic (length, width, or both) not meeting the requirements.

(c) What proportion of parts will meet neither length nor width requirements?

Solution We want tofindP(A^c∩B^c).

A^c A^c†P…

P… ∩B^c† ˆP… B^c† ˆ …0:05†…0:03† ˆ0:0015

(d) Suppose the operations that produce the length and the width are not independent. If the length does not satisfy the requirement, it causes an improper positioning of the part during the width trimming and thereby increases the chances of nonconforming width. From experience, it is estimated that if the length does not conform to the requirement, the chance of producing nonconforming widths is 60%. Find the proportion of parts that will not conform to either the length or the width requirement.

Solution The probability of interest isP(A^c∩B^c). The problem states thatP(B^c|A^c)ˆ 0.60. Using the general form of the multiplicative law

P…A^c∩B^c† ˆP…A^c†P…B^cjA^c†

ˆ …0:05†…0:60† ˆ0:03

So 3% of the parts will meet neither the length nor the width requirements. Notice that this value is different from the answer to part (c), where the events were assumed to be independent.

(e) In part (a), are eventsAandBmutually exclusive?

Solution We have foundP(A)ˆ 0.95,P(B)ˆ 0.97, andP(A∩B)ˆ 0.9215. IfAandB were mutually exclusive,P(A∩B) would have to be zero. However, this is not the case, since P(A∩B)ˆ 0.9215. SoAandBarenotmutually exclusive.

(f) Describe two events in this example setting that are mutually exclusive.

Solution EventsAandA^caremutuallyexclusive,tonameoneinstance,andP(A∩A^c)ˆ 0, becauseA and A^ccannot happen simultaneously. This means that it is not possible to produce a part that both meets and does not meet the length requirement.

4-5 DESCRIPTIVE STATISTICS: DESCRIBING PRODUCT OR PROCESS CHARACTERISTICS

Statisticsis the science that deals with the collection, classification, analysis, and making of inferences from data or information. Statistics is subdivided into two categories:descriptive statisticsandinferential statistics.

Descriptive statisticsdescribes the characteristics of a product or process using informa­

tion collected on it. Suppose that we have recorded service times for 500 customers in a fast- food restaurant. We can plot this as a frequency histogram where the horizontal axis represents a range of service time values and the vertical axis denotes the number of service times observed in each time range, which would give us some idea of the process condition. The average service time for 500 customers could also tell us something about the process.

Inferential statisticsdraws conclusions on unknown product or process parameters based on information contained in a sample. Let’s say that we want to test the validity of a claim that the average service time in the fast-food restaurant is no more than 3 minutes (min). Suppose wefind that the sample average service time (based on a sample of 500 people) is 3.5 minutes.

We then need to determine whether this observed average of 3.5 minutes is significantly greater than the claimed mean of 3 minutes.

Such procedures fall under the heading of inferential statistics. They help us draw conclusions about the conditions of a process. They also help us determine whether a process has improved by comparing conditions before and after changes. For example, suppose that the management of the fast-food restaurant is interested in reducing the average time to serve a customer. They decide to add two people to their service staff. Once this change is implemented, they sample 500 customers andfind that the average service time is 2.8 minutes. The question then is whether this decrease is a statistically significant decrease or whether it is due to random variation inherent to sampling. Procedures that address such problems are discussed later.

Data Collection

To control or improve a process, we need information, or data. Data can be collected in several ways. One of the most common methods is throughtdirect observation. Here, a measurement of the quality characteristic is taken by an observer (or automatically by an instrument); for instance, measurements on the depth of tread in automobile tires taken by an inspector are direct observations. On the other hand, data collected on the performance of a particular brand of hair dryer through questionnaires mailed to consumers areindirect observations. In this case, the data reported by the consumers have not been observed by the experimenter, who has no control over the data collection process. Thus, the data may beflawed because errors can

DESCRIPTIVE STATISTICS: DESCRIBING PRODUCT OR PROCESS CHARACTERISTICS 157 arise from a respondent’s incorrect interpretation of a question, an error in estimating the satisfactory performance period, or an inconsistent degree of precision among respondents’ answers.

Data on quality characteristics are described by arandom variableand are categorized as continuousordiscrete.

Continuous Variable A variable that can assume any value on a continuous scale within a range is said to becontinuous. Examples of continuous variables are the hub diameter of lawn mower tires, the viscosity of a certain resin, the specific gravity of a toner used in photocopying machines, the thickness of a metal plate, and the time to admit a patient to a hospital. Such variables are measurable and have associated numerical values.

Discrete Variable Variables that can assume afinite or countably infinite number of values are said to bediscrete. These variables are counts of an event. The number of defective rivets in an assembly is a discrete random variable. Other examples include the number of paint blemishes in an automobile, the number of operating capacitors in an electrical instrument, and the number of satisfied customers in an automobile repair shop.

Counting events usually costs less than measuring the corresponding continuous variables. The discrete variable is merely classified as being, say, unacceptable or not;

this can be done through a go/no-go gage, which is faster and cheaper thanfinding exact measurements. However, the reduced collection cost may be offset by the lack of detailed information in the data.

Sometimes, continuous characteristics are viewed as discrete to allow easier data collec­

tion and reduced inspection costs. For example, the hub diameter in a tire is actually a continuous random variable, but rather than precisely measuring the hub diameter numeri­

cally, a go/no-go gage is used to quickly identify the characteristic as either acceptable or not.

Hence, theacceptabilityof the hub diameter is a discrete random variable. In this case, the goal is not to know the exact hub diameter but rather to know whether it is within certain acceptable limits.

Accuracy and Precision Theaccuracyof a data set or a measuring instrument refers to the degree of uniformity of the observations around a desired value such that, on average, the target value is realized. Let’s assume that the target thickness of a metal plate is 5.25 mm.

Figure 4-5a shows observations spread on either side of the target value in almost equal

FIGURE 4-5 Accuracy and precision of observations.

proportions; these observations are said to be accurate. Even though individual observations may be quite different from the target value, a data set is considered accurate if the average of a large number of observations is close to the target.

For measuring instruments, accuracy is dependent oncalibration. If a measuring device is properly calibrated, the average output value given by the device for a particular quality characteristic should, after repeated use, equal the true input value.

Theprecisionof a data set or a measuring instrument refers to the degree of variability of the observations. Observations may be off the target value but still be considered precise, as shown in Figure 4-5b. A sophisticated measuring instrument should show very little variation in output values if a constant value is used multiple times as input. Similarly, sophisticated equipment in a process should be able to produce an output characteristic with as little variability as possible. The precision of the data is influenced by the precision of the measuring instrument. For example, the thickness of a metal plate may be 12.5 mm when measured by calipers; however, a micrometer may yield a value of 12.52 mm, while an optical sensor may give a measurement of 12.523 mm.

Having both accuracy and precision is desirable. In equipment or measuring instruments, accuracy can usually be altered by changing the setting of a certain adjustment. However, precision is an inherent function of the equipment itself and cannot be improved by changing a setting.

Measurement Scales

Four scales of measurement are used to classify data: the nominal, ordinal, interval, and ratio scales. Notice that each scale builds on the previous scale.

1. Nominal scale. The scale of measurement isnominalwhen the data variables are simply labels used to identify an attribute of the sample element. Labels can be

“conforming and nonconforming”or“critical, major, and minor.”Numerical values, even though assigned, are not involved.

2. Ordinal scale. The scale of measurement isordinalwhen the data have the properties of nominal data (i.e., labels) and the data rank or order the observations. Suppose that customers at a clothing store are asked to rate the quality of the store’s service. The customers rate the quality according to these criteria: 1 (outstanding), 2 (good), 3 (average), 4 (fair), 5 (poor). These are ordinal data. Note that a rating of 1 does not necessarily imply that the service is twice as good as a rating of 2. However, we can say that a rating of 1 is preferable to a rating of 2, and so on.

3. Interval scale. The scale of measurement isintervalwhen the data have the properties of ordinal dataandafixed unit of measure describes the interval between observations.

Suppose that we are interested in the temperature of a furnace used in steel smelting.

Four readings taken during a 2-hour interval are 2050, 2100, 2150, and 2200°F.

Obviously, these data values ranked (like ordinal data) in ascending order of temperature, indicating the coolest temperature, the next coolest, and so on. Further­

more, the differences between the ranked values can then be compared. Here the interval between the data values 2050 and 2100 represents a 50°F increase in temperature, as do the intervals between the remaining ranked values.

4. Ratio scale. The scale of measurement isratiowhen the data have the properties of interval dataanda natural zero exists for the measurement scale. Both the order of and difference between observations can be compared and there exists a natural zero for the