glossary - remoclicmeteo.edu.vn/giaotrinhxs/e-book/pq220-6234f.glo.pdf · 2009-04-02 · 689...

689

Glossary

Acceptance region. In hypothesis testing, a region inthe sample space of the test statistic such that if thetest statistic falls within it, the null hypothesis isaccepted (better terminology is that the null hypothe-sis cannot be rejected, since rejection is always astrong conclusion and acceptance is generally a weakconclusion).

Addition rule. A formula used to determine the proba-bility of the union of two (or more) events from theprobabilities of the events and their intersection(s).

Additivity property of �2. If two independent randomvariables X1 and X2 are distributed as chi-square with v1

and v2 degrees of freedom respectively, Y � X1 � X2 isa chi-square random variable with u � v1 � v2 degreesof freedom. This generalizes to any number of inde-pendent chi-square random variables.

Adjusted R2. A variation of the R2 statistic that com-pensates for the number of parameters in a regressionmodel. Essentially, the adjustment is a penalty for in-creasing the number of parameters in the model.

Alias. In a fractional factorial experiment when certainfactor effects cannot be estimated uniquely, they are saidto be aliased.

All possible (subsets) regressions. A method of vari-able selection in regression that examines all possiblesubsets of the candidate regressor variables. Efficientcomputer algorithms have been developed for imple-menting all possible regressions.

Alternative hypothesis. In statistical hypothesis test-ing, this is a hypothesis other than the one that is beingtested. The alternative hypothesis contains feasible con-ditions, whereas the null hypothesis specifies conditionsthat are under test.

Analysis of variance. A method of decomposing thetotal variability in a set of observations, as measured by

the sum of the squares of these observations from theiraverage, into component sums of squares that are asso-ciated with specific defined sources of variation.

Analytic study. A study in which a sample from apopulation is used to make inference to a future popu-lation. Stability needs to be assumed. See enumerativestudy.

Arithmetic mean. The arithmetic mean of a set ofnumbers x1, x2, p , xn is their sum divided by the num-ber of observations, or . The arithmeticmean is usually denoted by , and is often called theaverage.

Assignable cause. The portion of the variability in aset of observations that can be traced to specific causes,such as operators, materials, or equipment. Also called aspecial cause.

Attribute. A qualitative characteristic of an item orunit, usually arising in quality control. For example,classifying production units as defective or nondefec-tive results in attributes data.

Attribute control chart. Any control chart for a dis-crete random variable. See variables control charts.

Average. See Arithmetic Mean.

Average run length, or ARL. The average number of samples taken in a process monitoring or inspec-tion scheme until the scheme signals that the processis operating at a level different from the level in whichit began.

Axioms of probability. A set of rules that probabilitiesdefined on a sample space must follow. See probability.

Backward elimination. A method of variable selec-tion in regression that begins with all of the candidateregressor variables in the model and eliminates theinsignificant regressors one at a time until only signifi-cant regressors remain.

x11�n2 g n

i�1xi

PQ220 6234F.Glo 5/16/02 5:58 PM Page 689 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH 1 14 FIN L: PPEND

690 GLOSSARY

Bayes’ theorem. An equation for a conditional proba-bility such as in terms of the reverse conditionalprobability .

Bernoulli trials. Sequences of independent trials withonly two outcomes, generally called “success” and “fail-ure,” in which the probability of success remains constant.

Bias. An effect that systematically distorts a statisticalresult or estimate, preventing it from representing thetrue quantity of interest.

Biased estimator. See Unbiased estimator.

Bimodal distribution. A distribution with two modes.

Binomial random variable. A discrete random vari-able that equals the number of successes in a fixednumber of Bernoulli trials.

Bivariate normal distribution. The joint distributionof two normal random variables.

Block. In experimental design, a group of experimentalunits or material that is relatively homogeneous. Thepurpose of dividing experimental units into blocks is toproduce an experimental design wherein variabilitywithin blocks is smaller than variability between blocks. This allows the factors of interest to be com-pared in a environment that has less variability than inan unblocked experiment.

Box plot (or box and whisker plot). A graphical dis-play of data in which the box contains the middle 50%of the data (the interquartile range) with the mediandividing it, and the whiskers extend to the smallest andlargest values (or some defined lower and upper limits).

C chart. An attribute control chart that plots the totalnumber of defects per unit in a subgroup. Similar to adefects-per-unit or U chart.

Categorical data. Data consisting of counts or obser-vations that can be classified into categories. Thecategories may be descriptive.

Causal variable. When y � f(x) and y is considered tobe caused by x, x is sometimes called a causal variable.

Cause-and-effect diagram. A chart used to organizethe various potential causes of a problem. Also called afishbone diagram.

Center line. A horizontal line on a control chart at thevalue that estimates the mean of the statistic plotted onthe chart.

Center line. See Control chart.

Central composite design (CCD). A second-orderresponse surface design in k variables consisting of a

P1B 0 A2P1A 0 B2

two-level factorial, 2k axial runs, and one or more cen-ter points. The two-level factorial portion of a CCD canbe a fractional factorial design when k is large. TheCCD is the most widely used design for fitting a second-order model.

Central limit theorem. The simplest form of the cen-tral limit theorem states that the sum of n independentlydistributed random variables will tend to be normallydistributed as n becomes large. It is a necessary andsufficient condition that none of the variances of theindividual random variables are large in comparison totheir sum. There are more general forms of the centraltheorem that allow infinite variances and correlatedrandom variables, and there is a multivariate version ofthe theorem

Central tendency. The tendency of data to clusteraround some value. Central tendency is usually ex-pressed by a measure of location such as the mean, me-dian, or mode.

Chance cause of variation. The portion of the vari-ability in a set of observations that is due to only randomforces and which cannot be traced to specific sources,such as operators, materials, or equipment. Also called acommon cause.

Chebyshev’s inequality. A result that provides boundsfor certain probabilities for arbitrary random variables.

Chi-square (or chi-squared) random variable. Acontinuous random variable that results from the sum ofsquares of independent standard normal random vari-ables. It is a special case of a gamma random variable.

Chi-squared test. Any test of significance based on the chi-square distribution. The most common chi-square tests are (1) testing hypotheses about thevariance or standard deviation of a normal distributionand (2) testing goodness of fit of a theoretical distribu-tion to sample data.

Coefficient of determination. See R2.

Completely randomized design. A type of experi-mental design in which the treatments or design factorsare assigned to the experimental units in a randommanner. In designed experiments, a completely random-ized design results from running all of the treatmentcombinations in random order.

Components of variance. The individual componentsof the total variance that are attributable to specificsources. This usually refers to the individual variancecomponents arising from a random or mixed modelanalysis of variance.


GLOSSARY 691

Conditional mean. The mean of the conditional prob-ability distribution of a random variable.

Conditional probability. The probability of an eventgiven that the random experiment produces an outcomein another event.

Conditional probability density function. The proba-bility density function of the conditional probabilitydistribution of a continuous random variable.

Conditional probability distribution. The distributionof a random variable given that the random experimentproduces an outcome in an event. The given eventmight specify values for one or more other randomvariables.

Conditional probability mass function. The proba-bility mass function of the conditional probabilitydistribution of a discrete random variable.

Conditional variance. The variance of the conditionalprobability distribution of a random variable.

Confidence coefficient. The probability 1 � � associ-ated with a confidence interval expressing the probabilitythat the stated interval will contain the true parametervalue.

Confidence interval. If it is possible to write a proba-bility statement of the form

where L and U are functions of only the sample data and� is a parameter, then the interval between L and U iscalled a confidence interval (or a 100(1 � �)% confi-dence interval). The interpretation is that a statementthat the parameter � lies in this interval will be true100(1 � �)% of the times that such a statement is made.

Confidence level. Another term for the confidencecoefficient.

Confounding. When a factorial experiment is run inblocks and the blocks are too small to contain a com-plete replicate of the experiment, one can run a fractionof the replicate in each block, but this results in losinginformation on some effects. These effects are linkedwith or confounded with the blocks. In general, whentwo factors are varied such that their individual effectscannot be determined separately, their effects are said tobe confounded.

Consistent estimator. An estimator that converges inprobability to the true value of the estimated parameteras the sample size increases.

P1L � U 2 � 1 � �

Contingency table. A tabular arrangement expressingthe assignment of members of a data set according totwo or more categories or classification criteria.

Continuity correction. A correction factor used to im-prove the approximation to binomial probabilities froma normal distribution.

Continuous distribution. A probability distributionfor a continuous random variable.

Continuous random variable. A random variablewith an interval (either finite or infinite) of real numbersfor its range.

Continuous uniform random variable. A continuousrandom variable with range of a finite interval and aconstant probability density function.

Contour plot. A two-dimensional graphic used for abivariate probability density function that displayscurves for which the probability density function isconstant.

Control chart. A graphical display used to monitor aprocess. It usually consists of a horizontal center linecorresponding to the in-control value of the parameterthat is being monitored and lower and upper control limits. The control limits are determined by statisticalcriteria and are not arbitrary nor are they related to spec-ification limits. If sample points fall within the controllimits, the process is said to be in-control, or free fromassignable causes. Points beyond the control limits indi-cate an out-of-control process; that is, assignable causesare likely present. This signals the need to find and re-move the assignable causes.

Control limits. See Control chart.

Convolution. A method to derive the probability den-sity function of the sum of two independent randomvariables from an integral (or sum) of probabilitydensity (or mass) functions.

Cook’s distance. In regression, Cook’s distance is ameasure of the influence of each individual observationon the estimates of the regression model parameters. Itexpresses the distance that the vector of model parame-ter estimates with the ith observation removed lies fromthe vector of model parameter estimates based on all ob-servations. Large values of Cook’s distance indicate thatthe observation is influential.

Correction factor. A term used for the quantitythat is subtracted from

to give the corrected sum of squares defined as. The correction factor can also be

written as .nx211�n2 g n

i�11xi � x22g n

i�1 x2

i11�n2 1 g ni�1

xi22


Correlation. In the most general usage, a measure ofthe interdependence among data. The concept mayinclude more than two variables. The term is most com-monly used in a narrow sense to express the relationshipbetween quantitative variables or ranks.

Correlation coefficient. A dimensionless measure ofthe interdependence between two variables, usuallylying in the interval from �1 to �1, with zero indi-cating the absence of correlation (but not necessarilythe independence of the two variables). The mostcommon form of the correlation coefficient used inpractice is

which is also called the product moment correlation co-efficient. It is a measure of the linear association be-tween the two variables y and x.

Correlation matrix. A square matrix that contains thecorrelations among a set of random variables, say X1,X2, p , Xk. The main diagonal elements of the matrix areunity and the off diagonal elements rij are the correla-tions between Xi and Xj.

Counting techniques. Formulas used to determine thenumber of elements in sample spaces and events.

Covariance. A measure of association between tworandom variables obtained as the expected value of theproduct of the two random variables around theirmeans; that is, Cov(X, Y ) � E[(X � X)(Y � Y)].

Covariance matrix. A square matrix that contains thevariances and covariances among a set of random vari-ables, say X1, X2, p , Xk. The main diagonal elements ofthe matrix are the variances of the random variables andthe off diagonal elements are the covariances between Xi and Xj. Also called the variance-covariance matrix.When the random variables are standardized to haveunit variances, the covariance matrix becomes the cor-relation matrix.

Critical region. In hypothesis testing, this is the por-tion of the sample space of a test statistic that will leadto rejection of the null hypothesis.

Critical value(s). The value of a statistic correspondingto a stated significance level as determined from thesampling distribution. For example, if P(Z � z0.05) �P(Z � 1.96) � 0.05, then z0.05 = 1.96 is the criticalvalue of z at the 0.05 level of significance.

B an

i�11yi � y22an

i�11xi � x22

r � an

i�13 1yi � y2 1xi � x2 4 �

692 GLOSSARY

Crossed factors. Another name for factors that arearranged in a factorial experiment.

Cumulative distribution function. For a random vari-able X, the function of X defined as P(X x) that isused to specify the probability distribution.

Cumulative normal distribution function. The cumu-lative distribution of the standard normal distribution,often denoted as �(x) and tabulated in Appendix Table II.

Cumulative sum control chart (CUSUM). A controlchart in which the point plotted at time t is the sum ofthe measured deviations from target for all statistics upto time t.

Curvilinear regression. An expression sometimesused for nonlinear regression models or polynomial re-gression models.

Decision interval. A parameter set in a Tabular CUSUMalgorithm that is determined from a trade-off betweenfalse alarms and the detection of assignable causes.

Defect. Used in statistical quality control, a defect isa particular type of nonconformance to specificationsor requirements. Sometimes defects are classified intotypes, such as appearance defects and functionaldefects.

Defects-per-unit control chart. See U chart.

Degrees of freedom. The number of independent com-parisons that can be made among the elements of a sam-ple. The term is analogous to the number of degrees offreedom for an object in a dynamic system, which is thenumber of independent coordinates required to deter-mine the motion of the object.

Deming. W. Edwards Deming (1900–1993) was aleader in the use of statistical quality control.

Deming’s 14 points. A management philosophypromoted by W. Edwards Deming that emphasizes theimportance of change and quality.

Density function. Another name for a probability den-sity function.

Dependent variable. The response variable in regres-sion or a designed experiment.

Discrete distribution. A probability distribution for adiscrete random variable.

Discrete random variable. A random variable with afinite (or countably infinite) range.

Discrete uniform random variable. A discreterandom variable with a finite range and constant proba-bility mass function.


GLOSSARY 693

Dispersion. The amount of variability exhibited by data.

Distribution free methods(s). Any method of infer-ence (hypothesis testing or confidence interval con-struction) that does not depend on the form of theunderlying distribution of the observations. Sometimescalled nonparametric method(s).

Distribution function. Another name for a cumulativedistribution function.

Efficiency. A concept in parameter estimation that usesthe variances of different estimators; essentially, an esti-mator is more efficient than another estimator if it hassmaller variance. When estimators are biased, the con-cept requires modification.

Enumerative study. A study in which a sample from apopulation is used to make inference to the population.See Analytic study.

Erlang random variable. A continuous random vari-able that is the sum of a fixed number of independent,exponential random variables.

�-error (or �-risk). In hypothesis testing, an error in-curred by failing to reject a null hypothesis when it isactually false (also called a type II error).

�-error (or �-risk). In hypothesis testing, an error in-curred by rejecting a null hypothesis when it is actuallytrue (also called a type I error).

Error mean square. The error sum of squares dividedby its number of degrees of freedom.

Error of estimation. The difference between an esti-mated value and the true value.

Error sum of squares. In analysis of variance, this isthe portion of total variability that is due to the ran-dom component in the data. It is usually based onreplication of observations at certain treatment com-binations in the experiment. It is sometimes called theresidual sum of squares, although this is really a betterterm to use only when the sum of squares is based onthe remnants of a model fitting process and not onreplication.

Error variance. The variance of an error term or com-ponent in a model.

Estimate (or point estimate). The numerical value ofa point estimator.

Estimator (or point estimator). A procedure for pro-ducing an estimate of a parameter of interest. An esti-mator is usually a function of only sample data values,and when these data values are available, it results in anestimate of the parameter of interest.

Event. A subset of a sample space.

Exhaustive. A property of a collection of events thatindicates that their union equals the sample space.

Expected value. The expected value of a random vari-able X is its long-term average or mean value. In thecontinuous case, the expected value of X is

where f(x) is the density functionof the random variable X.

Exponential random variable. A continuous randomvariable that is the time between counts in a Poissonprocess.

Factorial experiment. A type of experimental designin which every level of one factor is tested in combina-tion with every level of another factor. In general, in afactorial experiment, all possible combinations of factorlevels are tested.

F-distribution. The distribution of the random vari-able defined as the ratio of two independent chi-squarerandom variables each divided by their number ofdegrees of freedom.

Finite population correction factor. A term in the for-mula for the variance of a hypergeometric randomvariable.

First-order model. A model that contains only first-order terms. For example, the first-order responsesurface model in two variables is y � 0 � 1x1 � 2x2 � �. A first-order model is also called a maineffects model.

Fixed factor (or fixed effect). In analysis of variance, a factor or effect is considered fixed if all the levels ofinterest for that factor are included in the experiment.Conclusions are then valid about this set of levels only,although when the factor is quantitative, it is customaryto fit a model to the data for interpolating between theselevels.

Forward selection. A method of variable selection inregression, where variables are inserted one at a timeinto the model until no other variables that contributesignificantly to the model can be found.

Fraction defective control chart. See P chart.

Fraction defective. In statistical quality control, thatportion of a number of units or the output of a processthat is defective.

Fractional factorial. A type of factorial experimentin which not all possible treatment combinations arerun. This is usually done to reduce the size of an ex-periment with several factors.

E1X 2 � ��

xf 1x2 dx


694 GLOSSARY

Frequency distribution. An arrangement of the fre-quencies of observations in a sample or populationaccording to the values that the observations take on.

F-test. Any test of significance involving the F-distri-bution. The most common F-tests are (1) testinghypotheses about the variances or standard deviations of two independent normal distributions, (2) testinghypotheses about treatment means or variance compo-nents in the analysis of variance, and (3) testing signifi-cance of regression or tests on subsets of parameters ina regression model.

Gamma function. A function used in the probabilitydensity function of a gamma random variable that canbe considered to extend factorials.

Gamma random variable. A random variable thatgeneralizes an Erlang random variable to nonintegervalues of the parameter r.

Gaussian distribution. Another name for the normaldistribution, based on the strong connection of Karl F.Gauss to the normal distribution; often used in physicsand electrical engineering applications.

Generating function. A function that is used to deter-mine properties of the probability distribution of arandom variable. See Moment generating function.

Geometric mean. The geometric mean of a set of npositive data values is the nth root of the product of thedata values; that is .

Geometric random variable. A discrete random vari-able that is the number of Bernoulli trials until a successoccurs.

Goodness of fit. In general, the agreement of a setof observed values and a set of theoretical valuesthat depend on some hypothesis. The term is oftenused in fitting a theoretical distribution to a set ofobservations.

Harmonic mean. The harmonic mean of a set of datavalues is the reciprocal of the arithmetic mean of the

reciprocals of the data values; that is, .

Hat matrix. In multiple regression, the matrix. This a projection matrix that maps

the vector of observed response values into a vector offitted values by .

Histogram. A univariate data display that uses rectan-gles proportional in area to class frequencies to visuallyexhibit features of data such as location, variability, andshape.

y � X1X¿X2�1X¿y � Hy

H � X1X¿X2�1X¿

h � a1n g n

i�1

1xib�1

g � 1wni�1

xi21�n

Hypergeometric random variable. A discrete randomvariable that is the number of success obtained from a sam-ple drawn without replacement from a finite populations.

Hypothesis (as in statistical hypothesis). A statementabout the parameters of a probability distribution or amodel, or a statement about the form of a probabilitydistribution.

Hypothesis testing. Any procedure used to test a sta-tistical hypothesis.

Independence. A property of a probability model andtwo (or more) events that allows the probability of theintersection to be calculated as the product of the prob-abilities.

Independent random variables. Random variablesfor which P(X A, Y B) � P(X A)P(Y B) for anysets A and B in the range of X and Y, respectively. Thereare several equivalent descriptions of independent ran-dom variables.

Independent variable. The predictor or regressor vari-ables in a regression model.

Indicator variable(s). Variables that are assigned nu-merical values to identify the levels of a qualitative orcategorical response. For example, a response with twocategorical levels (yes and no) could be representedwith an indicator variable taking on the values 0 and 1.

Individuals control chart. A Shewhart control chart inwhich each plotted point is an individual measurement,rather than a summary statistic. See control chart,Shewhart control chart.

Interaction. In factorial experiments, two factors aresaid to interact if the effect of one variable is different atdifferent levels of the other variables. In general, whenvariables operate independently of each other, they donot exhibit interaction.

Intercept. The constant term in a regression model.

Interquartile range. The difference between the thirdand first quartiles if a sample of data. The interquartilerange is less sensitive to extreme data values than theusual sample range.

Interval estimation. The estimation of a parameter bya range of values between lower and upper limits, incontrast to point estimation, where the parameter isestimated by a single numerical value. A confidenceinterval is a typical interval estimation procedure.

Jacobian. A matrix of partial derivatives that is used todetermine the distribution of transformed randomvariables.

PQ220 6234F.Glo 5/16/02 5:58 PM 94 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH 1 14 FIN L: PPEND

GLOSSARY 695

Joint probability density function. A function used tocalculate probabilities for two or more continuousrandom variables.

Joint probability distribution. The probability distri-bution for two or more random variables in a randomexperiment. See Joint probability mass function andJoint probability density function.

Joint probability mass function. A function used tocalculate probabilities for two or more discrete randomvariables.

Kurtosis. A measure of the degree to which a unimodaldistribution is peaked.

Lack of memory property. A property of a Poissonprocess. The probability of a count in an intervaldepends only on the length of the interval (and not onthe starting point of the interval). A similar propertyholds for a series of Bernoulli trials. The probability ofa success in a specified number of trials depends only on the number of trials (and not on the starting trial).

Least significance difference test (or Fisher’s LSDtest). An application of the t-test to compare pairs ofmeans following rejection of the null hypothesis in ananalysis of variance. The error rate is difficult to calculateexactly because the comparisons are not all independent.

Least squares (method of). A method of parameter es-timation in which the parameters of a system are esti-mated by minimizing the sum of the squares of thedifferences between the observed values and the fittedor predicted values from the system.

Least squares estimator. Any estimator obtained bythe method of least squares.

Level of significance. If Z is the test statistic for a hy-pothesis, and the distribution of Z when the hypothe-sis is true are known, then we can find the probabili-ties P(Z zL) and P(Z � zU). Rejection of thehypothesis is usually expressed in terms of theobserved value of Z falling outside the interval fromzL to zU. The probabilities P(Z zL) and P(Z � zU)are usually chosen to have small values, such as 0.01,0.025, 0.05, or 0.10, and are called levels of signifi-cance. The actual levels chosen are somewhat arbi-trary and are often expressed in percentages, such as a5% level of significance.

Likelihood function. Suppose that the random vari-ables X1, X2, p , Xn have a joint distribution given byf(x1, x2, p , xn; �1, �2, p , �p) where the �s are un-known parameters. This joint distribution, considered

as a function of the �s for fixed x’s, is called the like-lihood function.

Likelihood principle. This principle states that theinformation about a model given by a set of data is com-pletely contained in the likelihood.

Likelihood ratio. Let x1, x2, p , xn be a random samplefrom the population f(x; �). The likelihood function forthis sample is We wish to test thehypothesis H0: � �, where � is a subset of the possi-ble values � for �. Let the maximum value of L withrespect to � over the entire set of values that theparameter can take on be denoted by , and let themaximum value of L with � restricted to the set of val-ues given by � be . The null hypothesis is testedby using the likelihood ratio , or a sim-ple function of it. Large values of the likelihood ratioare consistent with the null hypothesis.

Likelihood ratio test. A test of a null hypothesisversus an alternative hypothesis using a test statistic de-rived from a likelihood ratio.

Linear combination. A random variable that isdefined as a linear function of several random variables.

Linear model. A model in which the observationsare expressed as a linear function of the unknownparameters. For example, y � 0 � 1x � � and y � 0 � 1x � 2 x2 � � are linear models.

Location parameter. A parameter that defines acentral value in a sample or a probability distribution.The mean and the median are location parameters.

Lognormal random variable. A continuous randomvariable with probability distribution equal to that ofexp(W) for a normal random variable W.

Main effect. An estimate of the effect of a factor (orvariable) that independently expresses the change inresponse due to a change in that factor, regardless ofother factors that may be present in the system.

Marginal probability density function. The probabil-ity density function of a continuous random variableobtained from the joint probability distribution of two ormore random variables.

Marginal probability distribution. The probabilitydistribution of a random variable obtained from the jointprobability distribution of two or more random variables.

Marginal probability mass function. The probabilitymass function of a discrete random variable obtainedfrom the joint probability distribution of two or morerandom variables.

� � L1�2�L1�2L1�2

L1�2

L � wni�1 f 1xi; �2.


696 GLOSSARY

Maximum likelihood estimation. A method of param-eter estimation that maximizes the likelihood function ofa sample.

Mean. The mean usually refers either to the expectedvalue of a random variable or to the arithmetic averageof a set of data.

Mean square. In general, a mean square is deter-mined by dividing a sum of squares by the number ofdegrees of freedom associated with the sum ofsquares.

Mean square(d) error. The expected squared deviationof an estimator from the true value of the parameter it es-timates. The mean square error can be decomposed intothe variance of the estimator plus the square of the bias;that is, .

Median. The median of a set of data is that value thatdivides the data into two equal halves. When the numberof observations is even, say 2n, it is customary to definethe median as the average of the nth and (n � 1)st rank-ordered values. The median can also be defined for arandom variable. For example, in the case of a continu-ous random variable X, the median M can be defined as .

Method of steepest ascent. A technique that allows anexperimenter to move efficiently towards a set of opti-mal operating conditions by following the gradientdirection. The method of steepest ascent is usually em-ployed in conjunction with fitting a first-order responsesurface and deciding that the current region of operationis inappropriate.

Mixed model. In an analysis of variance context, amixed model contains both random and fixed factors.

Mode. The mode of a sample is that observed valuethat occurs most frequently. In a probability distribu-tion f (x) with continuous first derivative, the mode is avalue of x for which df (x)�dx � 0 and d2f (x)�dx2 � 0.There may be more than one mode of either a sampleor a distribution.

Moment (or population moment). The expected valueof a function of a random variable such as E(X � c)r forconstants c and r. When c � 0, it is said that the momentis about the origin. See Moment generating function.

Moment estimator. A method of estimating parame-ters by equating sample moments to population mo-ments. Since the population moments will be functionsof the unknown parameters, this results in equations thatmay be solved for estimates of the parameters.

�M��

f 1x2 dx � ��M f 1x2 dx � 1�2

V1�2 � 3E1�2 � � 42MSE1�2 � E1� � �22 �

Moment generating function. A function that is usedto determine properties (such as moments) of theprobability distribution of a random variable. It is theexpected value of exp(tX). See generating functionand moment.

Moving range. The absolute value of the differencebetween successive observations in time-ordered data.Used to estimate chance variation in an individuals con-trol chart.

Multicollinearity. A condition occurring in multipleregression where some of the predictor or regressorvariables are nearly linearly dependent. This conditioncan lead to instability in the estimates of the regressionmodel parameters.

Multinomial distribution. The joint probability distri-bution of the random variables that count the number ofresults in each of k classes in a random experiment witha series of independent trials with constant probabilityof each class on each trial. It generalizes a binomialdistribution.

Multiplication rule. For probability, A formula usedto determine the probability of the intersection of two(or more) events. For counting techniques, a formulaused to determine the numbers of ways to completean operation from the number of ways to completesuccessive steps.

Mutually exclusive events. A collection of eventswhose intersections are empty.

Natural tolerance limits. A set of symmetric limitsthat are three times the process standard deviation fromthe process mean.

Negative binomial random variable. A discrete ran-dom variable that is the number of trials until a specifiednumber of successes occur in Bernoulli trials.

Nonlinear regression model. A regression model thatis nonlinear in the parameters. It is sometimes applied toregression models that are nonlinear in the regressors orpredictors, but this is an incorrect usage.

Nonparametric statistical method(s). See Distributionfree method(s).

Normal approximation. A method to approximateprobabilities for binomial and Poisson random variables.

Normal equations. The set of simultaneous linearequations arrived at in parameter estimation using themethod of least squares.

Normal probability plot. A specially constructedplot for a variable x (usually on the abscissa) in which


GLOSSARY 697

y (usually on the ordinate) is scaled so that thegraph of the normal cumulative distribution is astraight line.

Normal random variable. A continuous random vari-able that is the most important one in statistics becauseit results from the central limit theorem. See Centrallimit theorem.

NP chart. An attribute control chart that plots the totalof defective units in a subgroup. Similar to a fraction-defective chart or P chart.

Nuisance factor. A factor that probably influences theresponse variable, but which is of no interest in the cur-rent study. When the levels of the nuisance factor can becontrolled, blocking is the design technique that iscustomarily used to remove its effect.

Null hypothesis. This term generally relates to a par-ticular hypothesis that is under test, as distinct from thealternative hypothesis (which defines other conditionsthat are feasible but not being tested). The null hypothe-sis determines the probability of type I error for the testprocedure.

One-way model. In an analysis of variance context,this involves a single variable or factor with a differentlevels.

Operating characteristic curves (OC curves). A plotof the probability of type II error versus some measureof the extent to which the null hypothesis is false.Typically, one OC curve is used to represent each sam-ple size of interest.

Orthogonal. There are several related meanings, in-cluding the mathematical sense of perpendicular, twovariables being said to be orthogonal if they are statisti-cally independent, or in experimental design where adesign is orthogonal if it admits statistically independ-ent estimates of effects.

Orthogonal design. See Orthogonal.

Outcome. An element of a sample space.

Outlier(s). One or more observations in a sample that areso far from the main body of data that they give rise to thequestion that they may be from another population.

Overcontrol. Unnecessary adjustments made toprocesses that increase the deviations from target.

Overfitting. Adding more parameters to a model thanis necessary.

P chart. An attribute control chart that plots the propor-tion of defective units in a subgroup. Also called a frac-tion-defective control chart. Similar to an NP chart.

Parameter estimation. The process of estimatingthe parameters of a population or probability distribu-tion. Parameter estimation, along with hypothesistesting, is one of the two major techniques of statisti-cal inference.

Parameter. An unknown quantity that may vary over aset of values. Parameters occur in probability distri-butions and in statistical models, such as regressionmodels.

Pareto diagram. A bar chart used to rank the causes ofa problem.

PCR. A process capability ratio with numerator equal tothe difference between the product specification limitsand denominator equal to six times the process standarddeviation. Said to measure the potential capability of theprocess because the process mean is not considered. Seeprocess capability, process capability ratio, process capa-bility study and PCRk. Sometimes denoted as Cp in otherreferences.

PCRk. A process capability ratio with numerator equalto the difference between the product target and thenearest specification limit and denominator equal tothree times the process standard deviation. Said tomeasure the actual capability of the process because theprocess mean is considered. See process capability,process capability ratio, process capability study, andPCR. Sometimes denoted as Cpk in other references.

Percentage point. A particular value of a random vari-able determined from a probability (expressed as apercentage). For example, the upper 5 percentage pointof the standard normal random variable is Z0.05 � 1.645.

Percentile. The set of values that divide the sample into100 equal parts.

Poisson process. A random experiment with countsthat occur in an interval and satisfy the followingassumptions. The interval can be partitioned into subin-tervals such that the probability of more than one countin a subinterval is zero, the probability of a count in asubinterval is proportional to the length of the subinter-val, and the count in each subinterval is independent ofother subintervals.

Poisson random variable. A discrete random variablethat is the number of counts that occur in a Poissonprocess.

Pooling. When several sets of data can be thought of ashaving been generated from the same model, it is possi-ble to combine them, usually for purposes of estimating


698 GLOSSARY

one or more parameters. Combining the samples for thispurpose is usually called pooling.

Population standard deviation. See standard deviation.

Population variance. See variance.

Population. Any finite or infinite collection of individ-ual units or objects.

Power. The power of a statistical test is the probabilitythat the test rejects the null hypothesis when the null hy-pothesis is indeed false. Thus the power is equal to oneminus the probability of type II error.

Prediction. The process of determining the value ofone or more statistical quantities at some future point intime. In a regression model, predicting the response yfor some specified set of regressors or predictor vari-ables also leads to a predicted value, although there maybe no temporal element to the problem.

Prediction interval. The interval between a set ofupper and lower limits associated with a predicted valuedesigned to show on a probability basis the range oferror associated with the prediction.

Predictor variable(s). The independent or regressorvariables in a regression model.

Probability density function. A function used tocalculate probabilities and to specify the probability dis-tribution of a continuous random variable.

Probability distribution. For a sample space, adescription of the set of possible outcomes along with amethod to determine probabilities. For a randomvariable, a probability distribution is a description of therange along with a method to determine probabilities.

Probability mass function. A function that providesprobabilities for the values in the range of a discrete ran-dom variable.

Probability. A numerical measure between 0 and 1 as-signed to events in a sample space. Higher numbers in-dicate the event is more likely to occur. See axioms ofprobability.

Process capability ratio. A ratio that relates the width ofthe product specification limits to measures of processperformance. Used to quantify the capability of theprocess to produce product within specifications. Seeprocess capability, process capability study, PCR andPCRk.

Process capability study. A study that collects data toestimate process capability. See process capability,process capability ratio, PCR and PCRk.

Process capability. The capability of a process toproduce product within specification limits. Seeprocess capability ratio, process capability study, PCR,and PCRk.

P-Value. The exact significance level of a statisticaltest; that is, the probability of obtaining a value of thetest statistic that is at least as extreme as that observedwhen the null hypothesis is true.

Qualitative (data). Data derived from nonnumericattributes, such as sex, ethnic origin or nationality, orother classification variable.

Quality control. Systems and procedures used by anorganization to assure that the outputs from processessatisfy customers.

Quantiles. The set of n � 1 values of a variable thatpartition it into a number n of equal proportions. Forexample, n � 1 � 3 values partition data into fourquantiles with the central value usually called themedian and the lower and upper values usually calledthe lower and upper quartiles, respectively.

Quantitative (data). Data in the form of numericalmeasurements or counts.

Quartile(s). The three values of a variable that parti-tion it into four equal parts. The central value is usuallycalled the median and the lower and upper values areusually called the lower and upper quartiles, respec-tively. Also see Quantiles.

R2. A quantity used in regression models to measure theproportion of total variability in the response accountedfor by the model. Computationally, R2 � SSRegression�SSTotal, and large values of R2 (near unity) are consid-ered good. However, it is possible to have large valuesof R2 and find that the model is unsatisfactory. R2

is also called the coefficient of determination (or thecoefficient of multiple determination in multipleregression).

Random. Nondeterministic, occurring purely bychance, or independent of the occurrence of other events.

Random effects model. In an analysis of variance con-text, this refers to a model that involves only randomfactors.

Random error. An error (usually a term in a statisticalmodel) that behaves as if it were drawn at random froma particular probability distribution.

Random experiment. An experiment that can result indifferent outcomes, even though it is repeated in thesame manner each time.


GLOSSARY 699

Random factor. In analysis of variance, a factor whoselevels are chosen at random from some population offactor levels.

Random order. A sequence or order for a set of objectsthat is carried out in such a way that every possible or-dering is equally likely. In experimental design the runsof the experiment are typically arranged and carried outin random order.

Random sample. A sample is said to be random if it isselected in such a way so that every possible sample hasthe same probability of being selected.

Random variable. A function that assigns a real num-ber to each outcome in the sample space of a randomexperiment.

Randomization. A set of objects is said to be random-ized when they are arranged in random order.

Randomized block design. A type of experimental de-sign in which treatment (or factor levels) are assigned toblocks in a random manner.

Range. The largest minus the smallest of a set of datavalues. The range is a simple measure of variability andis widely used in quality control.

Range (control) chart. A control chart used to monitorthe variability (dispersion) in a process. See Control chart.

Rank. In the context of data, the rank of a single ob-servation is its ordinal number when all data values areordered according to some criterion, such as their mag-nitude.

Rational subgroup. A sample of data selected in a man-ner to include chance sources of variation and to excludeassignable sources of variation, to the extent possible.

Reference distribution. The distribution of a teststatistic when the null hypothesis is true. Sometimes areference distribution is called the null distribution ofthe test statistic.

Reference value. A parameter set in a TabularCUSUM algorithm that is determined from the magni-tude of the process shift that should be detected.

Regression. The statistical methods used to investigatethe relationship between a dependent or responsevariable y and one or more independent variables x. Theindependent variables are usually called regressor vari-ables or predictor variables.

Regression coefficient(s). The parameter(s) in a re-gression model.

Regression diagnostics. Techniques for examining a fit-ted regression model to investigate the adequacy of the fit

and to determine if any of the underlying assumptionshave been violated.

Regression line (or curve). A graphical display of aregression model, usually with the response y on theordinate and the regressor x on the abcissa.

Regression sum of squares. The portion of the totalsum of squares attributable to the model that has been fitto the data.

Regressor variable. The independent or predictorvariable in a regression model.

Rejection region. In hypothesis testing, this is theregion in the sample space of the test statistic that leadsto rejection of the null hypothesis when the test statisticfalls in this region.

Relative frequency. The relative frequency of an eventis the proportion of times the event occurred in a seriesof trial of a random experiment.

Reliability. The probability that a specified missionwill be completed. It usually refers to the probabilitythat a lifetime of a continuous random variable exceedsa specified time limit.

Replicates. One of the independent repetitions of oneor more treatment combinations in an experiment.

Replication. The independent execution of an experi-ment more than once.

Reproductive property of the normal distribution.A linear combination of independent, normal randomvariables is a normal random variable.

Residual. Generally this is the difference between theobserved and the predicted value of some variable. Forexample, in regression a residual is the differencebetween the observed value of the response and thecorresponding predicted value obtained from the regres-sion model.

Residual analysis. Any technique that uses the residu-als, usually to investigate the adequacy of the model thatwas used to generate the residuals.

Residual sum of squares. See Error sum of squares.

Response (variable). The dependent variable in aregression model or the observed output variable in adesigned experiment.

Response surface. When a response y depends on afunction of k quantitative variables x1, x2, p , xk, thevalues of the response may be viewed as a surface ink � 1 dimensions. This surface is called a responsesurface. Response surface methodology is a subset ofexperimental design concerned with approximating this


700 GLOSSARY

surface with a model and using the resulting model tooptimize the system or process.

Response surface designs. Experimental designs thathave been developed to work well in fitting responsesurfaces. These are usually designs for fitting a first- orsecond-order model. The central composite design is awidely used second-order response surface design.

Ridge regression. A method for fitting a regressionmodel that is intended to overcome the problems associ-ated with using standard (or ordinary) least squares whenthere is a problem with multicollinearity in the data.

Rotatable design. In a rotatable design, the variance ofthe predicted response is the same at all points that arethe same distance from the center of the design.

Run rules. A set of rules applied to the points plottedon a Shewhart control chart that are used to make thechart more sensitized to assignable causes. See controlchart, Shewhart control chart.

Sample. Any subset of the elements of a population.

Sample mean. The arithmetic average or mean of theobservations in a sample. If the observations are x1, x2, p , xn then the sample mean is . Thesample mean is usually denoted by .

Sample moment. The quantity is calledthe kth sample moment.

Sample range. See range.

Sample size. The number of observations in a sample.

Sample space. The set of all possible outcomes of arandom experiment.

Sample standard deviation. The positive square rootof the sample variance. The sample standard deviationis the most widely used measure of variability ofsample data.

Sample variance. A measure of variability of sampledata, defined as , where

is the sample mean.

Sampling distribution. The probability distribution ofa statistic. For example, the sampling distribution of thesample mean is the normal distribution.

Scatter diagram. A diagram displaying observations ontwo variables, x and y. Each observation is represented bya point showing its x-y coordinates. The scatter diagramcan be very effective in revealing the joint variability of xand y or the nature of the relationship between them.

Screening experiment. An experiment designed andconducted for the purpose of screening out or isolating

X

xs2 � 31� 1n � 12 4 g n

i�11xi � x22

11�n2 g ni�1

xki

x11�n2 g n

i�1 xi

a promising set of factors for future experimentation.Many screening experiments are fractional factorials,such as two-level fractional factorial designs.

Second-order model. A model that contains second-order terms. For example, the second-order responsesurface model in two variables is y � 0 � 1x1 � 2x2 � 12x1x2 � 11x

21 � 22x

22 � �. The second order

terms in this model are 12x1x2, 11x21, and 22x

22.

Shewhart control chart. A specific type of controlchart developed by Walter A. Shewhart. Typically, eachplotted point is a summary statistic calculated from thedata in a rational subgroup. See control chart.

Sign test. A statistical test based on the signs of certainfunctions of the observations and not their magnitudes.

Signed-rank test. A statistical test based on the differ-ences within a set of paired observations. Each differ-ence has a sign and a rank, and the test uses the sum ofthe differences with regard to sign.

Significance. In hypothesis testing, an effect is said tobe significant if the value of the test statistic lies in thecritical region.

Significance level. See Level of significance.

Skewness. A term for asymmetry usually employedwith respect to a histogram of data or a probability dis-tribution.

Standard deviation. The positive square root of thevariance. The standard deviation is the most widely used measure of variability.

Standard error. The standard deviation of the estima-tor of a parameter. The standard error is also the stan-dard deviation of the sampling distribution of the esti-mator of a parameter.

Standard normal random variable. A normal ran-dom variable with mean zero and variance one that hasits cumulative distribution function tabulated inAppendix Table II.

Standardize. The transformation of a normal randomvariable that subtracts its mean and divides by its standarddeviation to generate a standard normal random variable.

Standardized residual. In regression, the standardizedresidual is computed by dividing the ordinary residual bythe square root of the residual mean square. This producesscaled residuals that have, approximately, a unit variance.

Statistic. A summary value calculated from a sampleof observations. Usually, a statistic is an estimator ofsome population parameter.


GLOSSARY 701

Statistical Process Control. A set of problem-solvingtools based on data that are used to improve a process.

Statistical quality control. Statistical and engineeringmethods used to measure, monitor, control, and improve quality.

Statistics. The science of collecting, analyzing, inter-preting, and drawing conclusions from data.

Stem and leaf display. A method of displaying data inwhich the stem corresponds to a range of data valuesand the leaf represents the next digit. It is an alternativeto the histogram but displays the individual observa-tions rather than sorting them into bins.

Stepwise regression. A method of selecting variablesfor inclusion in a regression model. It operates by intro-ducing the candidate variables one at a time (as in for-ward selection) and then attempting to remove variablesfollowing each forward step.

Studentized range. The range of a sample divided bythe sample standard deviation.

Studentized residual. In regression, the studentizedresidual is calculated by dividing the ordinary residualby its exact standard deviation, producing a set of scaledresiduals that have, exactly, unit standard deviation.

Sufficient statistic. An estimator is said to be asufficient statistic for an unknown parameter if thedistribution of the sample given the statistic does notdepend on the unknown parameter. This means that thedistribution of the estimator contains all of the useful in-formation about the unknown parameter.

Tabular CUSUM. A numerical algorithm used to de-tect assignable causes on a cumulative sum controlchart. See V mask.

Tampering. Another name for overcontrol.

t-distribution. The distribution of the random variabledefined as the ratio of two independent random vari-ables. The numerator is a standard normal randomvariable and the denominator is the square root of a chi-square random variable divided by its number ofdegrees of freedom.

Test statistic. A function of a sample of observationsthat provides the basis for testing a statistical hypothesis.

Time series. A set of ordered observations taken atdifference points in time.

Tolerance interval. An interval that contains a speci-fied proportion of a population with a stated level ofconfidence.

Tolerance limits. A set of limits between which somestated proportion of the values of a population must fallwith specified level of confidence.

Total probability rule. Given a collection of mutuallyexclusive events whose union is the sample space, theprobability of an event can be written as the sum of theprobabilities of the intersections of the event with themembers of this collection.

Treatment. In experimental design, a treatment is aspecific level of a factor of interest. Thus if the factor istemperature, the treatments are the specific temperaturelevels used in the experiment.

Treatment sum of squares. In analysis of variance,this is the sum of squares that accounts for the variabil-ity in the response variable due to the different treat-ments that have been applied.

t-test. Any test of significance based on the t distribu-tion. The most common t-tests are (1) testing hypothe-ses about the mean of a normal distribution withunknown variance, (2) testing hypotheses about themeans of two normal distributions and (3) testinghypotheses about individual regression coefficients.

Type I error. In hypothesis testing, an error incurredby rejecting a null hypothesis when it is actually true(also called an �-error).

Type II error. In hypothesis testing, an error incurredby failing to reject a null hypothesis when it is actuallyfalse (also called a -error).

U chart. An attribute control chart that plots the aver-age number of defects per unit in a subgroup. Also calleda defects-per-unit control chart. Similar to a C chart.

Unbiased estimator. An estimator that has its expectedvalue equal to the parameter that is being estimated issaid to be unbiased.

Uniform random variable. Refers to either a discreteor continuous uniform random variable.

Uniqueness property of moment generating function.Refers to the fact that random variables with the samemoment generating function have the same distribution.

Universe. Another name for population.

V mask. A geometrical figure used to detect assignablecauses on a cumulative sum control chart. With appro-priate values for parameters, identical conclusions canbe made from a V mask and a tabular CUSUM.

Variable selection. The problem of selecting a subsetof variables for a model from a candidate list that


contains all or most of the useful information about theresponse in the data.

Variables control chart. Any control chart for a con-tinuous random variable. See attributes control charts.

Variance. A measure of variability defined as theexpected value of the square of the random variablearound its mean.

Variance component. In analysis of variance modelsinvolving random effects, one of the objectives is todetermine how much variability can be associated witheach of the potential sources of variability defined bythe experimenters. It is customary to define a varianceassociated with each of these sources. These variancesin some sense sum to the total variance of the response,and are usually called variance components.

Variance inflation factors. Quantities used in multipleregression to assess the extent of multicollinearity (ornear linear dependence) in the regressors. The varianceinflation factor for the ith regressor VIFi can be definedas VIFi = [1�(1 � R2

i)], where R2i is the coefficient of

determination obtained when xi is regressed on the otherregressor variables. Thus when xi is nearly linearly de-pendent on a subset of the other regressors R2

i will beclose to unity and the value of the corresponding vari-

702 GLOSSARY

ance inflation factor will be large. Values of the varianceinflation factors that exceed 10 are usually taken as asignal that multicollinearity is present.

Warning limits. Horizontal lines added to a controlchart (in addition to the control limits) that are used tomake the chart more sensitive to assignable causes.

Weibull random variable. A continuous random vari-able that is often used to model the time until failure ofa physical system. The parameters of the distribution are flexible enough that the probability density function can assume many different shapes.

Western Electric rules. A specific set of run rules thatwere developed at Western Electric Corporation. Seerun rules.

Wilcoxon signed rank test. A distribution-free test ofthe equality of the location parameters of two otherwiseidentical distributions. It is an alternative to the two-sample t-test for nonnormal populations.

With replacement. A method to select samples inwhich items are replaced between successive selections.

Without replacement. A method to select samples inwhich items are not replaced between successiveselections.


glossary - remoclicmeteo.edu.vn/giaotrinhxs/e-book/pq220-6234f.glo.pdf · 2009-04-02 · 689...

Documents