spc 0002

8/13/2019 Spc 0002

1/14

CHAPTER

Data may be collected by direct observation or indirectly through written or verba)questions. The latter technique is used extensively by market research personnel andpublic opinion pollsters. Data that are collected for quality control purposes are obtainedby direct observation and are classified as either variables or attributes. Variables arethose quality charact-:ristics that are measurable, such as a weight measured in grams.Attributes, on the other hand, are those quality characteristics that are c,Iassified as eitherconforming or not conforming to specifications, such as a go-no go gauge.

A histogram is sufficient for many quality control problems. However, with a broadrange of problems a graphical technique is either undesirable or needs the additional in.formation provided by analytical techniques. Analytical methods of describing a collec_tion of data have the advantage of occupying less space than a graph. They also have theadvantage of allO\ving for comparisons between collections of dala. They also allow foradditional calculations and inferences. There are two principal analytical methods of describing a collection of data: measures of central tendency and measures of dispersion.

A measure of central tendency of a distribution is a numerical value that describes thecentral position of th~ data or how the data tend to build up in the center. There are threemeasures in common use in quality: (I) th_ average, (2) the median, and (3) the mode.

The average is the sum of the observations divided by the number of observations. is the most common measure of central tendency and is represented by the equation2

IIX=~

where X = average and is read as X bar= number of observed valuesXi = observed valueL= sum of

Unless otherwise noted, X stands for the average of observed values, x The sameequation is used to find

X or Xx average of averagesR average of rangesp average of proportions, etc.

Another measure of central tendency is the median, Md, which is defined as th.evalue that divides a series of ordered observations so that the number of items above

C For data grouped into cells. the equation uses fjX where = cell frequency and X = cell midpoint.

8/13/2019 Spc 0002

2/14

ST TISTIC L PROCESS CONTROL

is equal to the number below it. Two situations are possible-when the number in theseries is odd and when the number in the series is even. When the number in the seriesis odd, the median is the midpoint of the values, provided the data are ordered. Thus.the ordered set of numbers 3, 4, 5, 6, 8, 8, and 10 has a median of 6. When the numberin the series is even, the median is the average of the two middle numbers. Thus, theordered set of numbers 3, 4, 5, 6, 8, and 8 has a median that is the average of 5 and 6,which is (5 6)/2 = 5.5.

The mode, Mo, of a set of numbers is the value that occurs with the greatest frequency. It is possible for the mode to be nonexistent in a series of numbers or to havemore than one value. To illustrate, the series of numbers 3, 3, 4, 5, 5, 5, and 7 has a modeof 5; the series of numbers 22, 23, 25, 30, 32, and 36 does not have a mode; and the series of numbers 105, 105, 105, 107, 108, 109, 109, 109, 110, and 112 has two modes,105 and 109. A series of numbers is referred to as unimodal if it has one mode, bimodalif it has two modes, and multimodal if there are more than two modes. When data aregrouped into a frequency distribution, the midpoint of the cell with the highest frequencyis the mode, because this point represents the highest point (greatest frequency) of thehistogram.

The average is the most commonly-used measure of central tendency. It is used whenthe distribution is symmetrical or not appreciably skewed to the right or left; when additional statistics, such as measures of dispersion, control charts, and so on, are to becomputed based on thf' average; and when a stable value is needed for inductive statistics. The median becomes an effective measure of the central tendency when the distribution is positively (to the right) or negatively (to the left) skewed. The median is usedwhen an exact midpoint of a distribution is desired. When a distribution has extreme values, the average will be adversely affected, whereas the median will remain unchanged.Thus, in a series of numbers such as 12, 13,14,15,16, the median and average are identical and are equal to 14. However, if the first value is changed to a 2, the median remains at 14, but the average becomes 12. A control chart based on the median isuser-friendly and excellent for monitoring quality. The mode is used when a quick andapproximate measure of the central tendency is desired. Thus, the mode of a histogramis easily found by a visual examination. In addition, the mode is used to describe themost typical value of a distribution, such as the modal age of a particu ar group ..easures of ispersionA second tool of statistics is composed of the measures of dispersion, which describehow the data are spread out or scattered on each side of the central value. Measures ofdispersion and measures of central tendency are both needed to describe a collection ofdata. To illustrate. the employees of the plating and the assembly departments of a factory have identical average weekly wages of 325.36; however, the plating departmenthas a high of 330.72 and a low of 319.43,whereas the assembly department has a highof 380.79 and a low of 273.54. The data for the assembly department are spread out.or dispersed, farther from the average than are those of the plating department.

8/13/2019 Spc 0002

3/14

CHAPTER

One of the measures of dispersion is the range which for a series of numbers is thedifference between the largest and smallest values of observations Symbolically it isrepresented by the equation

where R = range = highest observation in a seriesl = lowest observation in a series

The other measure of the dispersion used in quality is the standard deviation It is anumerical value in the units of the observed values that measures the spreading tendencyof the data A large standard deviation shows greater variability of the data than does asmall standard deviation In symbolic terms it is represented by the equation

n lwhere S = sample standard deviation

Xi = observed valueX = averagen = number of observed values

Unless otherwise noted s stands for sx the sample standard deviation of observedvalues The same equation is used to findSx = sample standard deviation of averagessp = sample standard deviation of proportionsSs = sample standard deviation of standard deviations etc

The standard deviation is a reference value that measures the dispersion in the dataIt is best viewed as an index that is defined by the formula The smaller the value of thestandard deviation the better the quality because the distribution is more closely com-pacted around the central value The standard deviation also helps to define populationsas discussed in the next section

In quality control the range is a very common measure of the dispersion It is used inone of the principal control charts The primary advantage of the range is in providing aknowledge of the total spread of the data It is also valuable when the amouht of data IStoo small or too scattered to justify the calculation of a more precise measure of disper-sion As the number of observations increases the accuracy of the range decreases be-cause it becomes easier for extremely high or low readings to occur It is suggested t~althe use of the range be limited to a maximum of ten observations The standard deVIa-tion is used when amore precise measure is desired

The average and standard deviation are easily calculated with a hand calculator

8/13/2019 Spc 0002

4/14

~TATISTICAL PROCESSCONTROL 8

AMPLE PROBLEMDetermine the average, median, mode, range, and standard deviation for the height ofseven people. Data are 1.83, 1.91. 1.78, 1.80, 1.>\3, 1.85, 1.87 meters.

X = L X n = (1.83 1.91 ... 1.87)/7 = 1.84Md = {1.9l, 1.87, 1.85, 1.83, 1.83, 1.80, 1.78} = 1.83Mo = 1.83 = h XI = 1.91 - 1.78 = 0.13

~ X; Xf ~(1.91 -1.84)2 ... (1.78 ~ 1.84)2S = ,/---- = ------------n l 7= 0.04

opulation and ampleAt this point, it is desirable to examine the concept of a population and a sample. In orderto construct a frequency distribution of the weight of steel shafts; a small portion, or ~ampIe, is selected to represent all the steel shafts. The population is the whole collection of steelshafts. When averages, standard deviations, and other measures are computed from samples, they are referred to as statistics. Because the composition of samples will fluctuate, thecomputed statistics will be larger or smaller than their true population values, or parameters. Parameters are considered to be fixed reference (standard) values or the best estimateof these values available at a particular time. The population may have a finite number ofitems, such as a day s production of steel shafts. It may be infinite or almost infinite, suchas the number of rivets in a year s production of jet airplanes. The population may be defined differently, depending on the particular situation. Thus, a study of a product could involve the population of an hour s production, a week s production, 5,000 pieces, and so on.Because it is rarely possible to measure all of the population, a sample is selected. Sam

pling is necessary when it may be impossible to measure the entire population; when theexpense to observe all the data is prohibitive; when the required inspection destroys theproduct; or when a test of the entire population may be too dangerous, as would be the casewith a new medical drug: Actually, a,nanalysis of the entire population may not be as accurate as sampling. It has been shown that 100 manual inspection of low percent nonconforming product is not as accurate as sampling. This is probably due to the fact thatboredom and fatigue cause inspectors to prejudge each inspected item as being acceptable.When designating a population, the corresponding Greek letter is used. Thus, the

sample average has the symbol X and the population mean the symbol 11 (mu). Note thatthe word average changes to mean when used for the population. The symbol X o is the~tandard_or reference value. Mathematical concepts are based on 11, which is the truevalue-Xo represents a practical equivalent in order to use the concepts. The sample

8/13/2019 Spc 0002

5/14

CHAPTER 18-------------------------------------------------------TABLE 18 3omparison of Sample and Population

Sample PopulationStatisticX-averages-sample standard deviation

ParameterIl(Xo)-meana(so)-standard deviation

standard deviation has the symbol s, and the population standard deviation the symbolo (sigma). The symbol So is the standard or reference value and has the same relationship to 0 th2t 0 has to J.l.The true population value may never be known; therefore, thesymbol 1and are sometimes used to indicate estimate of. A comparison of sampleand population is given in Table 18-3. A sample frequency distribution is represented bya histogram, whereas a population frequency distribution is represented by a smoothcurve. To some extent, the sample represents the real world and the population represents the mathematical world. The equations and concepts are based on the population.

The primary objective in selecting a sample is to learn something about the population that will aid in making some type of decision. The sample selected must be of sucha nature that it tends to resemble or represent the population. How successfully the sample represents the population is a function of the size of the sample, chance, the samplingmethod, and whether or not the conditions change.

ormal urveAlthough there are as many different populations as there are conditions, they can be descrihed by a few general types. One type of population that is quite common is called thenormal curve, or Gaussian distribution. The normal curve is a symmetrical, unimodal,bel1-shaped distribution with the mean. median, and mode having the same value. curve of the normal population for the resistance in ohms of an electrical device withpopulation mean, J.l, of 90 Q and population standard deviation, 0, of 2 Q is shown inFigure 18-10. The interval between dotted lines is equal to one standard deviation, 0.

Much of the variation in nature and in industry fol1ows the frequency distribution ofthe normal curves. Thus. the variations in the weights of elephants, the speeds of antelopes, and the heights of human beings wil1 follow a normal curve. Also, the variationsfound in industry, such as the weights of gray 'iron castings, the lives of 60-watt lightbulbs, and the dimensions of steel piston rings, will be expected to follow the normalcurve. When considering the heights of human beings, we can expect a small percent

8/13/2019 Spc 0002

6/14

STATISTICAL PROCESS CONTROL 9

~ =90cr = 2 0

.- - 00 84 86 88 90 92 94 96 00 ----.Figure 18 1 Normal Distribution for Resistance of an Electrical Device

~ = 14 ~ = 20 ~ = 29Figure 18 11 Normal Curves with Different Means but Identical Standard Deviations

age of them to be extremely tall and a small percentage to be extremely short, with themajority of human heights clustering about the average value. The normal curve is sucha good description of the variations that occur to most quality characteristics in industrythat it is the basis for many quality control techniques.There is a definite relationship among the mean, the standard deviation, and the nor

mal curve. Figure 18-11 shows three normal curves with different mean values; note thatthe only change is in the location. Figure 18-12 shows three nonnal curves with the samemean but different standard deviations. The figure illustrates the principle that the largerthe standard deviation, the flatter the curve data are widely dispersed), and the smallerthe standard deviation, the more peaked the curve data are nan-owly dispersed). If thestandard deviation is zero, all values are identical to the mean and there is no curve.The normal distribution is fully defined by the population mean and population stan

dard deviation. Also, as seen by Figures 18-11 and 18-12, these two parameters are independent. In other words, a change in one parameter has no effect on the other.A relationship exists between the standard deviation and the area under the normal curve,

8/13/2019 Spc 0002

7/14

CHAPTER 8 -x

8 11 14 17 20 23 26 29 32 352Normal Curves with Different Standard Deviations but Identical Means

-30- -20- -10- ~ 10- 20- 3 1Figure 8 3Percent of Values Included Between Certain Values of the StandardDeviation

duction to ontrol h rtsVariation _One of the axioms, or truisms, of production is that no two objects are ever made exactlyalike. In fact, the variation concept is a law of nature because no two natural items in anycategory are the same. The variation may be quite large and easily noticeable, such asthe beight of human beings, or the variation may be very small, such as the weights offibeHipped pens or the shapes of snowflakes. When variations are very small, it mayappear that items are identical; however, precision instruments will show differences. Iftwo items appear to have the same measurement, it is due to the limits of our measuring

8/13/2019 Spc 0002

8/14

ST TISTI L PRO ESS ONTROL 9

instruments. As measuring instruments have become more refined, variation has continued to exist; only the increment of variation has changed. The ability to measure variation is necessary before it can be controlled.

There are three categories of variations in piece part production:1. Within-piece variation is illustrated by the surface roughness of a piece, wherein

one portion of the surface is rougher than another portion or the width of one end of akeyway varies from the other end.

2. Piece-to-piece variation occurs among pieces produced at the same time. Thus, thelight intensity of four consecutive light bulbs produced from a machine will be different.

3. Time-to-time variation is illustrated by the difference in product produced at different times of the day. Thus, product produced in the early morning is different from thatproduced later in the day, or as a cutting tool wears, the cutting characteristics change.

Categories of variation for other types of processes such as a continuous and batchare not exactly the same; however, the concept is similar.

Variation is present in every process due to a combination of the equipment, materials, environment, and operator. The first source of variation is the equipment. Thissource includes tool wear, machine vibration, workholding-device positioning, and hydraulk and electrical fluctuations. When all these variations are put together, there is acertain capability or precision within which the equipment operates. Even supposedlyidentical machines will have different capabilities. This fact becomes a very importantconsideration when scheduling the manufacture of critical parts.The second source of variation is the material. Because variation occurs in the finished product, it must also occur in the raw material which was someone else s finishedproduct). Such quality characteristics as tensile strength, ductility, thickness, porosity,and moisture content can be expected to contribute to the overall variation in the finalproduct.

A third source of variation is the environment. Temperature, light, radiation, particlesize, pressure, and humidity all can contribute to variation in the product. In order to control environmental variations, products are sometimes manufactured in white rooms.Experiments are conducted in outer space to learn more about the effect of the environment on product variation.

A fourth source is the operator. This source of variation includes the method by whichthe operator performs the operation. The operator s physical and emotional well-beingalso contribute to the variation. A cut finger, a twisted ankle, a personal problem, or aheadache can make an operator s quality performance vary. An operator s lack of understanding of equipment and material variations due to lack of training may lead to frequent machine adjustments, thereby compounding the variability. As our equipment hasbecome more automated, the operator s effect on variation has lessened.

The preceding four sources account for the true variation. There is also a reported

8/13/2019 Spc 0002

9/14

HAPTER 18 --the cause of the incorrect reporting of variation. In general, variation due to inspectionshould be one-tenth of the four other sources of variations. Note that three of thesesources are present in the inspection activity-an inspector or appraiser, inspectionequipment, and the environment.Run ChartA run chart, which is shown in Figure 18-14, is a very simple technique for analyzingthe process in the development stage or. for that matter, when other charting techniquesare not applicable. The important point is to draw a picture of the process and let it talkto you. A picture is worth a thousand words. provided someone is listening. Plotting thedata points is a very effective w;:.yof finding out about the process. This activity shouldbe done as the first step in data analysis. Without a run chart, other data analysis toolssuch as the average, sample standard deviation, and histogram--can lead to erroneousconclusions.

The particular run chart shown in Figure 18-14 is referred to as an X chart and is usedto record the variation in the average value of samples. Other charts, such as the chartrange) or chart proportion) would have also served for explanation purposes. Thehorizontal axis is labeled Subgroup Number, which identifies a particular sample consisting of a fixed number of observations. These subgroups are plotted by order of production, with the first one inspected being 1 and the last one on this chart being 25. Thevertical axis of the graph is the variable, which in this particular case is weight measuredin kilograms.

Each small solid Jiamond represents the average value within a subgroup. Thus, subgroup number 5 consists of. say, four observations, 3.46, 3.49, 3.45, and 3.44, and their

3.523.5.48

3.46eIJ

3.44 t. 3,42::l E 3.4IJ .c:l 3.38Il 3.36 -.343.32 t.3

II III 10505

Subgroup number

8/13/2019 Spc 0002

10/14


average is 3.46 kg. This value is the one posted on the chart for subgroup number 5. Averages are used on control charts rather than individual observations because averagevalues will indicate a change in variation much faster. Also, with two or more observations in a sample, a measure of the dispersion can be obfained for a particular subgroup.

The solid line in the center of the chart can have three different interpretations, depending on the available data. First, it can be the average of the plotted points, which inthe case of an X chart is the average of the averages or X-double bar. Second, it canbe a standard or reference value, Xo based on representative prior data, an economicvalue based on production costs or service needs, or an aimed-at value based on specifications. Third, it can be the population mean, f1, if that value is known.Control Chart ExampleOne danger of using a run chart is its tendency to show every variation in data as beingimportant. In order to indicate when observed variations in quality are greater than couldbe left to chance, the control chart method of analysis and presentation of data is used.The control chart method for variables is a means of visualizing the variations that occur in the central tendency and dispersion of a set of observations. It is a graphical recordof the quality of a particular characteristic. It shows whether or not the process is in astable state by adding statistically determined control limits to the run chart.

Figure 18-15, is the run chart of Figure 18-14 with the control limits added. They arethe two dashed outer lines and are called the upper and lower control limits. These limits are established to assist in judging the significance of the variation in the quality ofthe product. Control limits are frequently confused with specification limits, which arethe permissible limits of a quality characteristic of each individual unit of a product.However, control limits are used to evaluate the variations in quality from subgroup to

3.523.5.48il

3.46C 0

3.44b/j ... 3.42 ;> 3.4- = 3.38... b/j.0 3.36=

8/13/2019 Spc 0002

11/14

4 r:HAPTER 18 --subgroup. Therefore, for the X chart, the control limits are a function of the subgrouaverages. A frequency distribution of the subgroup averages can be determined with it~corresponding ayerage and standard deviation.

The control limits are then established at 30 from the central line. Recall, from thediscussion of the nonnal curve, that the number of items between 30 and - 30 equals99.73 . Therefore, it is expected that more than 997 times out of 1,000. the subgroupvalues will fall between the upper and lower limits. When this situation OCcurs, theprocess is considered to be in control. When a subgroup value falls outside the limitsthe process is considered to be out of control, and an assignable cause for the variatio~is present. Subgroup number lOin Figure 18-15 is beyond the upper control limit; therefore, there has been a change in the stable nature of the process, causing the out-of-controlpoint. As long as the sources of variation fluctuate in a natural or expected manner, a stable pattern of many chance causes (random causes) of variation develops. Chance causesof variation are inevitable. Because they are numerous and individually of relativelysmall importance, they are difficult to detect or identify. Those causes of variation thatare large in magnitude, and therefore readily identified, are classified as assignablecauses.3 When only chance causes are present in a process, the process is considered tobe in a state of statistical control. It is stable and predictable. However, when an assignable cause of variation is also pr~sent, the variation will be excessive, and the process isclassified as out of control or beyond the expected natural variation of the process.

Unnatural variation is the result of assignable causes. Usually, but not always, it requires corrective action by people close to the process, such as operators, technicians,clerks, maintenance workers, and first-line supervisors. Natural variation is the result ofchance causes-it requires management intervention to achieve quality improvement.In this regard, between 80 and 85 of the quality problems are due to management orthe system and 15 to 20 are due to operations. Operating personnel are giving a quality performance as long as the plotted points are within the control limits. If this performance is not satisfactory, the solution is the responsibility of the system rather thanof the operating personnel.

iable ontrol harts In practice, control charts are posted at individual machines or work centers to control aparticular quality characteristic. Usually, an X chart for the central tendency and an Rchart for the dispersion are used together. An example of this dual charting is illustratedin Figure 18-16, which shows a method of charting and reporting inspection results fOfrubber durometer.

At work center number 365- 2 at 8:30 A.M .. the operator selects four items for test~gand records the observations of 55.52.51, and 53 in the rows marked Xj, X X3 and 4'

3 Dr. w. Edwards Deming uses the words ommon and spe i l for chance and assignable.

8/13/2019 Spc 0002

12/14

X nd h rtsWork Center Number 7 7~QualityCharacteristic v ,,~eIBF Date WdZ

ff 7i Jf7i Ji J, tl-i J; 7i J7i JMMMMPM 13590123 771--77 7 774- 1-731P71131P-7- 731P4- 1-731P7117IIP1 16167IPI3 7.7.74- 1-4-31P. 7-77

X Chart55

11 500E2:l 4540

DCLi

LCLi

----------------------------------- UCLR

5

81___...

0 4~ :l 20

5

10

10

h rt

15

15

20

20

25

25Figure 8 6 Example of a Method of Reporting Inspection Results

8/13/2019 Spc 0002

13/14

HAPTER 18

respectively. A subgroup average value of 52.8 is obtained by summing the observatioand dividing by 4 and the range value of 4 is obtained by subtracting the low value 51nfrom the high value 55. The operator places a small solid circle at 52.8 on the X ch~and a small solid circle at 4 on the chart and then proceeds with his other duties.

The frequency with which the operator inspects a product at a particular machine 0work center is determined by the quality of the product. When the process is in contro~and no difficulties are being encountered fewer inspections may be required. Conversely when the process is out of control or during start-up more inspections may beneeded. The inspection frequency at a machine or work center can also be determinedby the amount of time that must be spent on noninspection activities. In the exampleproblem the inspection frequency appears to be every 60 or 65 minutes.

At 9:30 A M the operator performs the activities for subgroup 2 in the same manneras for subgroup 1 It is noted that the range value of 7 falls on the upper control limit.Whether to consider this in control or out of control would be a matter of organizationpolicy. It is suggested that it be classified as in control and a cursory examination for anassignable cause be conducted by the operator. A plotted point that falls exactly on thecontrol limit is a rare occurrence.

The inspection results for subgroup 2 show that the third observation X has a valueof 57 which exceeds the upper control limit. It is important to remember the earlier discussion on control limits and specifications. In other words the 57 value is an individual observation and does not relate to the control limits. Therefore the fact that anindividual observation is greater than or less than a control limit is meaningless.

Subgroup 4 has an average value of 44 which is less than the lower control limit of45. Therefore subgroup 4 is out of control and the operator will report this fact to thedepartmental supervisor. The operator and supervisor will then look for an assignablecause and if possible take corrective action. Whatever corrective action is taken will benoted by the operator on the X and charts or on a separate form. The control chart indicates when and where trouble has occurred. The identification and elimination of thedifficulty is a production problem. Ideally the control chart should be maintained].. heoperator provided time is available and proper training has been given. When the operator cannot maintain the chart then it is maintained by quality control.

The control chart is used to keep a continuing record of a particular quaHj characteristic. It is a picture of the process over time. When the chart is completed and storedin an office file it is replaced by a fresh chart. The chart is used to improve the processquality to determine the process capability to determine when to leave the process aloneand when to make adjustme~ts and to investigate causes of unacceptable or margin~quality. It is also used to make decisions on product or service specifications and deCIsions on the acceptability of a recently-produced product or service.

uality haracteristic The variable that is chosen for the X and charts must be a quality characteristic that ismeasurable and can be expressed in numbers. Quality characteristics that can be eX

8/13/2019 Spc 0002

14/14


pressed in terms of the seven basic units (length, mass, time, electrical current, temperature, substance, or luminous intensity), as well as any of the derived units, such aspower, velocity, force, energy, density, and pressure, are appropriate.

Those quality characteristics affecting the performance of the product would normallybe given first attention. These may be a function of the raw materials, component parts,subassemblies, or finished parts. In other words, high priority is given to the selection ofthose characteristics that are giving difficulty in terms of production problems and/orcost. An excellent opportunity for cost savings frequently in~olves situations wherespoilage and rework costs are high. A Pareto analysis is also useful for establishing priorities. Another possibility occurs where destructive testing is ust d t inspect a product.

In any organization, a large number of variables make up a product or service. It is,therefore, impossible to place X and charts on all variables. A judicious selection ofthose quality characteristics is required.

ubgroup ize and ethodAs previously mentioned, the data that are plotted on the control chart consist of groupsof items called rational subgroups. It is important to understand that data collected in arandom manner do not qualify as rational. A rational subgroup is one in which the variation within the group is due only to chance causes. This within-subgroup variation isused to determine the control limits. Variation between subgroups is used to evaluatelong-term stability. Subgroup samples are selected from product or a service producedat one instant of time or as close to that instant as possible, such as four consecutive partsfrom a machine or four documents from a tray. The next subgroup sample would be similar, but for product or a service produced at a later time-say, one hour later.

Decisions on the size of the sample or subgroup require a certain amount of empirical judgment; however, some helpful guidelines are:

1. As the subgroup size increases, the control limits become closer to the centralvalue, which makes the control chart more sensitive to small variations in theprocess average.

2. As the subgroup size increases, the inspection cost per subgroup increases. Doesthe increased cost of larger subgroups justify the greater sensitivity?

3. When costly and/or destructive testing is used and the item is expensive, a smallsubgroup size of two or three is necessary, because it will minimize the destruction of expensive product.

4. Because of the ease of computation, a sample size of five is quite common in industry; however, when inexpensive electronic hand calculators are used, this reason is no longer valid.

5. From a statistical basis a distribution of subgroup averages, X s, is nearly normal for subgroups of four or more even wherr~the samples are taken from a non

spc 0002

Documents