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I3I STATISTICS REVIEW
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the variance [Eq (135)]; :0009435 s, : (0097133)2 and the coefficient variation[Eq (137)]: of cv:_ 0097133 x1007a:147Vo 66
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The validity of Eq (136)can alsobe verifiedby computing
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1045651 (158400)' /24 : 0009435 24-l
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1312 The Normol Distribution
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Another characteristic that bearson the presentdiscussion the datadistribution-that is, is the shapewith which the data is spreadaroundthe meanA histogramprovidesa simple visual representation the distributionA histograntis constructed sorting the meaof by surenrents into intervals,or bins The units of measurement plotted on the abscissa are and the frequencyof occurrence eachinterval is plotted on the ordinate of As an example,a histogramcan be createdfor the data from Table 132The result (Fig l3a) suggests that most of the datais groupedcloseto the meanvalue of 66 Notice also,that now that we have groupedthe data,we can seethat the bin with the most values is from 66 to 664Although we could say that the mode is the midpoint of this b|n,662, it is more common to report the most frequent range as the modal class interval If we have a very large set of data, the histogramoften can be approximatedby a smooth curve The symmetric,bell-shaped curve superimposed Fig 133 is one such on characteristic shape-the normal distributionGiven enoughadditionalmeasurements, the histogramfor this particularcasecould eventuallyapproach normal distribution the
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FIGURE I33
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A histogrom of used to depict the distribution doto a rh^ ^"-h^, ^{ l^r^ ^^inis increcses, ihe q-66+\ hcllt-cned n:srr-a,i- ofronnnnrnfpq 1l^p r',"" i rli 'i"ll]r[io ,r brion
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LINEAR EGRESSION R
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The conceptsof the rrrean, standarddeviation,residualsum of the squares, norand mal distribution all have greatrelevance engineering to and science very simpleexamA ple is their useto quantify the confidencethat can be ascribed a particularmeasurement to If a quantity is norrnallydistributed,the rangedefinedbyv - s, to y * 1, will encompass approximately68Vcof the total rneasurements Similarly, the rangedefined by tr - 2s, to approximately95Vo )r + 2s, will encornpass For example, the datain Table 132,we calculated Example131that) = 66 for in and s, :0097133 Basedon our analysis can tentativelymake the statement we that approximately95Vo thereadingsshouldfall between6405'734 of and6794266 Becauseit is so far outsidethesebounds,ifsomeonetold us that they had rneasured valueof735,we a would suspect that the measurement might be erroneous
1313 DescriptiveStotisticsin MATLAB
StandardMATLAB has severalfunctionsto computedescriptivestatisticsr example, For the arithmeticmeanis computedasmean(x) If x is a vector,the function returnsthemean of the vector's valuesIf it is a matrix, it returns a row vector containingthe arithmetic mean of eachcolumn of >r The following is the result of using meanand the otherstatistical functionsto analyzea column vector s that holds the datafrom Table 132:
g >> format short (s),mode >> mean (s),median (s)
661, 6555 >> min(s),max(s)
o, t/a
>> range=max(s)
min(s)
O3B >> var(s),std(s)
anc -
I MATI-AB also ofiers a Statistics Toolbox that provides a wide range of common statistical tasks,from random
rrrrmher oeneration fn 'rrrve fiffinq fo desiqn
I3I STATISTICS REVIEW
Theseresultsare consistent with thoseobtainedpreviouslyin Example 131Note that althoughthere are four valuesthat occur twice, the mode function only returnsone of the values: 655,5 MATLAB can also be usedto generate histogrambasedon the hist function The a hist functionhasthe syntax ln, xl = hisL (Y, x) where n : the number of elementsin each bin, x : a vector specifyingthe midpoint of eachbin, and y is the vector being analyzed For the datafiom TabIe 132,the result is
>> hist(s)
3 64I4 6452 649 6528 6566 6604 6642 668 6718 67s5 The resulting histogram depicted Fig 134is similarto the one we generated handin in by Fig 133Note that all the arguments and outputswith the exceptionof y are optionalFor just producesa histogrambar plot with example,his'* (y) without output arguments l0 bins determinedautomaticallybasedon the ranqeof valuesin r'
FIGURE I34
Hisfogrcmgeneroiedwith the MATLAB hist funciion
LINEAR EGRESSION R
TINEAR TEASTSQUARES REGRESSION
Where substantial with data,the best curve-fitting strategy to derive is error is associated an approximatingfunction that fits the shapeor generaltrend of the datawithoutnecessarily matching the individual points One approachto do this is to visually inspect the plotted data and then sketch a "best" line through the points Although such"eyeball" calculaapproaches have commonsense appealand are valid for "back-ot'-the-envelope" tions, they are deficient becausethey are arbitraryThat is, unlessthe points defineaperfert would straightline (in which case,interpolationwould be appropriate), differentanalysts draw different lines forthe To removethis subjectivitysomecriterion must be devisedto establish basis a fit One way to do this is to derivea curve that minimizesthe discrepancy between data the points and the curve To do this, we must first quantify the discrepancy The simplest examrr p l e i s f i t t i n g a s t r a i g h t l i n e t o a s e t op a i r e d o b s e r v a t i o(n s,:) l ) ( x z , y ) , , ( r , , ) n ) , f for The mathematical expression the straightline is y : a o* z t l x1 e (138)
where a6 and a 1 are coefficients representingthe intercept and the slope,respectively, and e is the error, or rcsidual, between the model and the observations,which can be represented rearranging (138)as by Eq e:!-as-a1x (139)
Thus, the residualis the discrepancy value, betweenthe true valueof y andthe approximate a0 + atx, predictedby the linear equation
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