Q6: the mean of the error component

*The author of this computation has been verified*

R Software Module: rwasp_centraltendency.wasp (opens new window with default values)

Title produced by software: Central Tendency

Date of computation: Mon, 03 Nov 2008 03:13:55 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/Nov/03/t1225707290pyy8qhbkhx639xl.htm/, Retrieved Mon, 03 Nov 2008 10:14:50 +0000

BibTeX entries for LaTeX users:

@Manual{KEY, author = {{YOUR NAME}}, publisher = {Office for Research Development and Education}, title = {Statistical Computations at FreeStatistics.org, URL http://www.freestatistics.org/blog/date/2008/Nov/03/t1225707290pyy8qhbkhx639xl.htm/}, year = {2008}, } @Manual{R, title = {R: A Language and Environment for Statistical Computing}, author = {{R Development Core Team}}, organization = {R Foundation for Statistical Computing}, address = {Vienna, Austria}, year = {2008}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

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Dataseries X:

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0.989130435 0.919087137 0.925417076 0.925612053 1.066666667 0.851108765 1.030693069 0.989031079 0.913000978 0.792723264 0.978170478 0.987513007 0.909433962 0.883608147 0.82745098 0.8252149 1.023255814 0.815418024 1.026192703 0.914742451 0.807276303 0.739130435 0.98973306 0.972164948 0.853889943 0.856864654 0.775739042 0.789473684 0.931350114 0.73971079 0.885245902 0.842435094 0.818458418 0.72755418 0.923238696 0.922680412 0.883762201 0.818270165 0.771047228 0.825852783 0.924485126 0.755165289 0.874671341 0.815956482 0.799807507 0.712598425 0.832980973 0.910323253 0.869149952 0.779182879 0.750254842 0.75856014 0.920889988 0.743991641 0.816254417 0.769593957 0.784007353 0.683284457 0.850505051 0.900695134 0.868398268

Output produced by software:

Summary of computational transaction Raw Input view raw input (R code) Raw Output view raw output of R engine Computing time 3 seconds R Server 'Herman Ole Andreas Wold' @ 193.190.124.10:1001 Central Tendency - Ungrouped Data Measure Value S.E. Value/S.E. Arithmetic Mean -5.2600761993146e-16 0.0114291533021248 -4.60233235154585e-14 Geometric Mean NaN Harmonic Mean -0.441033274704734 Quadratic Mean 0.0885298408007616 Winsorized Mean ( 1 / 20 ) -0.000109174262295607 0.0111502480278886 -0.00979119585703782 Winsorized Mean ( 2 / 20 ) 0.000233625377048655 0.0110085322801376 0.0212222093830055 Winsorized Mean ( 3 / 20 ) 0.000658512229507673 0.0108607091364145 0.0606325260382651 Winsorized Mean ( 4 / 20 ) -0.00150164508196774 0.0103531978250052 -0.145041668028495 Winsorized Mean ( 5 / 20 ) -0.00120015114754151 0.0102755125989696 -0.116797204614577 Winsorized Mean ( 6 / 20 ) -0.000593871311475937 0.0101588317676167 -0.0584586225129765 Winsorized Mean ( 7 / 20 ) -0.000204582377049707 0.0100209628685834 -0.0204154410841188 Winsorized Mean ( 8 / 20 ) -0.000984605721311996 0.00969077564784127 -0.101602364670503 Winsorized Mean ( 9 / 20 ) -0.000242727311475944 0.00923404695015402 -0.0262861249012705 Winsorized Mean ( 10 / 20 ) -0.00669544255737757 0.00798724816389706 -0.838266499298401 Winsorized Mean ( 11 / 20 ) -0.00688411004918085 0.00767868173199379 -0.89652238358802 Winsorized Mean ( 12 / 20 ) -0.00624499004918085 0.00755708675194955 -0.826375328769355 Winsorized Mean ( 13 / 20 ) -0.00541543575409889 0.00735497059804504 -0.73629604386703 Winsorized Mean ( 14 / 20 ) -0.00444693388524645 0.00710734829977794 -0.625681153882017 Winsorized Mean ( 15 / 20 ) -0.00378513978688578 0.00695956233210063 -0.543876124138872 Winsorized Mean ( 16 / 20 ) -0.00239659709836118 0.00660049543348655 -0.363093516617318 Winsorized Mean ( 17 / 20 ) -0.000817563245902178 0.00621439653638094 -0.131559555479911 Winsorized Mean ( 18 / 20 ) 0.000302873311474856 0.00567639224490088 0.0533566565536285 Winsorized Mean ( 19 / 20 ) -7.18362786890354e-05 0.00557021747750842 -0.0128964944329549 Winsorized Mean ( 20 / 20 ) -0.000852095295082476 0.00542357106444244 -0.157109639563665 Trimmed Mean ( 1 / 20 ) -0.000436456663518169 0.0108689626323786 -0.04015623921808 Trimmed Mean ( 2 / 20 ) -0.000786706250791438 0.0105229395313002 -0.074761073030155 Trimmed Mean ( 3 / 20 ) -0.00135252651713913 0.0101895014217329 -0.132737261732390 Trimmed Mean ( 4 / 20 ) -0.00212405710547532 0.00984516991178467 -0.215746109463568 Trimmed Mean ( 5 / 20 ) -0.00231017050466141 0.00961676338850195 -0.24022328629022 Trimmed Mean ( 6 / 20 ) -0.00258654267112800 0.00935158638958732 -0.276588651740204 Trimmed Mean ( 7 / 20 ) -0.00301758151133642 0.00904727309070538 -0.333534920531635 Trimmed Mean ( 8 / 20 ) -0.00356232102623004 0.00869401211366102 -0.409744198611423 Trimmed Mean ( 9 / 20 ) -0.00401941589134632 0.00833090155268804 -0.482470698510345 Trimmed Mean ( 10 / 20 ) -0.00464374652379101 0.0079817772173857 -0.581793552653428 Trimmed Mean ( 11 / 20 ) -0.00432284022110183 0.00785974978204419 -0.549997180696194 Trimmed Mean ( 12 / 20 ) -0.00393896439920301 0.00776021353221818 -0.507584537828703 Trimmed Mean ( 13 / 20 ) -0.00360404162623004 0.00763942698591083 -0.471768580664083 Trimmed Mean ( 14 / 20 ) -0.00334647742623004 0.00750909119782298 -0.445656782967324 Trimmed Mean ( 15 / 20 ) -0.00319180497461714 0.00737514831120705 -0.432778412030978 Trimmed Mean ( 16 / 20 ) -0.00310860170209211 0.00720373584145335 -0.431526331685276 Trimmed Mean ( 17 / 20 ) -0.00320913938919300 0.00704112188384609 -0.455771032249206 Trimmed Mean ( 18 / 20 ) -0.00355240090623004 0.00689356397359014 -0.515321380905378 Trimmed Mean ( 19 / 20 ) -0.00412044855666482 0.00681304458128153 -0.604788139503083 Trimmed Mean ( 20 / 20 ) -0.00473940933099194 0.0066701101111457 -0.710544391624424 Median -0.00821014742623005 Midrange 0.0128754715737699 Midmean - Weighted Average at Xnp -0.00531754252623004 Midmean - Weighted Average at X(n+1)p -0.00319180497461714 Midmean - Empirical Distribution Function -0.00319180497461714 Midmean - Empirical Distribution Function - Averaging -0.00319180497461714 Midmean - Empirical Distribution Function - Interpolation -0.00319180497461714 Midmean - Closest Observation -0.00536163626998004 Midmean - True Basic - Statistics Graphics Toolkit -0.00319180497461714 Midmean - MS Excel (old versions) -0.00319180497461714 Number of observations 61

Charts produced by software:

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/03/t1225707290pyy8qhbkhx639xl/1fd4f1225707229.png (open in new window) http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/03/t1225707290pyy8qhbkhx639xl/1fd4f1225707229.ps (open in new window) http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/03/t1225707290pyy8qhbkhx639xl/2690r1225707229.png (open in new window) http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/03/t1225707290pyy8qhbkhx639xl/2690r1225707229.ps (open in new window)

Parameters (Session):

Parameters (R input):

R code (references can be found in the software module):

x <- x - 0.86210009042623 geomean <- function(x) { return(exp(mean(log(x)))) } harmean <- function(x) { return(1/mean(1/x)) } quamean <- function(x) { return(sqrt(mean(x*x))) } winmean <- function(x) { x <-sort(x[!is.na(x)]) n<-length(x) denom <- 3 nodenom <- n/denom if (nodenom>40) denom <- n/40 sqrtn = sqrt(n) roundnodenom = floor(nodenom) win <- array(NA,dim=c(roundnodenom,2)) for (j in 1:roundnodenom) { win[j,1] <- (j*x[j+1]+sum(x[(j+1):(n-j)])+j*x[n-j])/n win[j,2] <- sd(c(rep(x[j+1],j),x[(j+1):(n-j)],rep(x[n-j],j)))/sqrtn } return(win) } trimean <- function(x) { x <-sort(x[!is.na(x)]) n<-length(x) denom <- 3 nodenom <- n/denom if (nodenom>40) denom <- n/40 sqrtn = sqrt(n) roundnodenom = floor(nodenom) tri <- array(NA,dim=c(roundnodenom,2)) for (j in 1:roundnodenom) { tri[j,1] <- mean(x,trim=j/n) tri[j,2] <- sd(x[(j+1):(n-j)]) / sqrt(n-j*2) } return(tri) } midrange <- function(x) { return((max(x)+min(x))/2) } q1 <- function(data,n,p,i,f) { np <- n*p; i <<- floor(np) f <<- np - i qvalue <- (1-f)*data[i] + f*data[i+1] } q2 <- function(data,n,p,i,f) { np <- (n+1)*p i <<- floor(np) f <<- np - i qvalue <- (1-f)*data[i] + f*data[i+1] } q3 <- function(data,n,p,i,f) { np <- n*p i <<- floor(np) f <<- np - i if (f==0) { qvalue <- data[i] } else { qvalue <- data[i+1] } } q4 <- function(data,n,p,i,f) { np <- n*p i <<- floor(np) f <<- np - i if (f==0) { qvalue <- (data[i]+data[i+1])/2 } else { qvalue <- data[i+1] } } q5 <- function(data,n,p,i,f) { np <- (n-1)*p i <<- floor(np) f <<- np - i if (f==0) { qvalue <- data[i+1] } else { qvalue <- data[i+1] + f*(data[i+2]-data[i+1]) } } q6 <- function(data,n,p,i,f) { np <- n*p+0.5 i <<- floor(np) f <<- np - i qvalue <- data[i] } q7 <- function(data,n,p,i,f) { np <- (n+1)*p i <<- floor(np) f <<- np - i if (f==0) { qvalue <- data[i] } else { qvalue <- f*data[i] + (1-f)*data[i+1] } } q8 <- function(data,n,p,i,f) { np <- (n+1)*p i <<- floor(np) f <<- np - i if (f==0) { qvalue <- data[i] } else { if (f == 0.5) { qvalue <- (data[i]+data[i+1])/2 } else { if (f < 0.5) { qvalue <- data[i] } else { qvalue <- data[i+1] } } } } midmean <- function(x,def) { x <-sort(x[!is.na(x)]) n<-length(x) if (def==1) { qvalue1 <- q1(x,n,0.25,i,f) qvalue3 <- q1(x,n,0.75,i,f) } if (def==2) { qvalue1 <- q2(x,n,0.25,i,f) qvalue3 <- q2(x,n,0.75,i,f) } if (def==3) { qvalue1 <- q3(x,n,0.25,i,f) qvalue3 <- q3(x,n,0.75,i,f) } if (def==4) { qvalue1 <- q4(x,n,0.25,i,f) qvalue3 <- q4(x,n,0.75,i,f) } if (def==5) { qvalue1 <- q5(x,n,0.25,i,f) qvalue3 <- q5(x,n,0.75,i,f) } if (def==6) { qvalue1 <- q6(x,n,0.25,i,f) qvalue3 <- q6(x,n,0.75,i,f) } if (def==7) { qvalue1 <- q7(x,n,0.25,i,f) qvalue3 <- q7(x,n,0.75,i,f) } if (def==8) { qvalue1 <- q8(x,n,0.25,i,f) qvalue3 <- q8(x,n,0.75,i,f) } midm <- 0 myn <- 0 roundno4 <- round(n/4) round3no4 <- round(3*n/4) for (i in 1:n) { if ((x[i]>=qvalue1) & (x[i]<=qvalue3)){ midm = midm + x[i] myn = myn + 1 } } midm = midm / myn return(midm) } (arm <- mean(x)) sqrtn <- sqrt(length(x)) (armse <- sd(x) / sqrtn) (armose <- arm / armse) (geo <- geomean(x)) (har <- harmean(x)) (qua <- quamean(x)) (win <- winmean(x)) (tri <- trimean(x)) (midr <- midrange(x)) midm <- array(NA,dim=8) for (j in 1:8) midm[j] <- midmean(x,j) midm bitmap(file='test1.png') lb <- win[,1] - 2*win[,2] ub <- win[,1] + 2*win[,2] if ((ylimmin == '') | (ylimmax == '')) plot(win[,1],type='b',main=main, xlab='j', pch=19, ylab='Winsorized Mean(j/n)', ylim=c(min(lb),max(ub))) else plot(win[,1],type='l',main=main, xlab='j', pch=19, ylab='Winsorized Mean(j/n)', ylim=c(ylimmin,ylimmax)) lines(ub,lty=3) lines(lb,lty=3) grid() dev.off() bitmap(file='test2.png') lb <- tri[,1] - 2*tri[,2] ub <- tri[,1] + 2*tri[,2] if ((ylimmin == '') | (ylimmax == '')) plot(tri[,1],type='b',main=main, xlab='j', pch=19, ylab='Trimmed Mean(j/n)', ylim=c(min(lb),max(ub))) else plot(tri[,1],type='l',main=main, xlab='j', pch=19, ylab='Trimmed Mean(j/n)', ylim=c(ylimmin,ylimmax)) lines(ub,lty=3) lines(lb,lty=3) grid() dev.off() load(file='createtable') a<-table.start() a<-table.row.start(a) a<-table.element(a,'Central Tendency - Ungrouped Data',4,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'Measure',header=TRUE) a<-table.element(a,'Value',header=TRUE) a<-table.element(a,'S.E.',header=TRUE) a<-table.element(a,'Value/S.E.',header=TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,hyperlink('http://www.xycoon.com/arithmetic_mean.htm', 'Arithmetic Mean', 'click to view the definition of the Arithmetic Mean'),header=TRUE) a<-table.element(a,arm) a<-table.element(a,hyperlink('http://www.xycoon.com/arithmetic_mean_standard_error.htm', armse, 'click to view the definition of the Standard Error of the Arithmetic Mean')) a<-table.element(a,armose) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,hyperlink('http://www.xycoon.com/geometric_mean.htm', 'Geometric Mean', 'click to view the definition of the Geometric Mean'),header=TRUE) a<-table.element(a,geo) a<-table.element(a,'') a<-table.element(a,'') a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,hyperlink('http://www.xycoon.com/harmonic_mean.htm', 'Harmonic Mean', 'click to view the definition of the Harmonic Mean'),header=TRUE) a<-table.element(a,har) a<-table.element(a,'') a<-table.element(a,'') a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,hyperlink('http://www.xycoon.com/quadratic_mean.htm', 'Quadratic Mean', 'click to view the definition of the Quadratic Mean'),header=TRUE) a<-table.element(a,qua) a<-table.element(a,'') a<-table.element(a,'') a<-table.row.end(a) for (j in 1:length(win[,1])) { a<-table.row.start(a) mylabel <- paste('Winsorized Mean (',j) mylabel <- paste(mylabel,'/') mylabel <- paste(mylabel,length(win[,1])) mylabel <- paste(mylabel,')') a<-table.element(a,hyperlink('http://www.xycoon.com/winsorized_mean.htm', mylabel, 'click to view the definition of the Winsorized Mean'),header=TRUE) a<-table.element(a,win[j,1]) a<-table.element(a,win[j,2]) a<-table.element(a,win[j,1]/win[j,2]) a<-table.row.end(a) } for (j in 1:length(tri[,1])) { a<-table.row.start(a) mylabel <- paste('Trimmed Mean (',j) mylabel <- paste(mylabel,'/') mylabel <- paste(mylabel,length(tri[,1])) mylabel <- paste(mylabel,')') a<-table.element(a,hyperlink('http://www.xycoon.com/arithmetic_mean.htm', mylabel, 'click to view the definition of the Trimmed Mean'),header=TRUE) a<-table.element(a,tri[j,1]) a<-table.element(a,tri[j,2]) a<-table.element(a,tri[j,1]/tri[j,2]) a<-table.row.end(a) } a<-table.row.start(a) a<-table.element(a,hyperlink('http://www.xycoon.com/median_1.htm', 'Median', 'click to view the definition of the Median'),header=TRUE) a<-table.element(a,median(x)) a<-table.element(a,'') a<-table.element(a,'') a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,hyperlink('http://www.xycoon.com/midrange.htm', 'Midrange', 'click to view the definition of the Midrange'),header=TRUE) a<-table.element(a,midr) a<-table.element(a,'') a<-table.element(a,'') a<-table.row.end(a) a<-table.row.start(a) mymid <- hyperlink('http://www.xycoon.com/midmean.htm', 'Midmean', 'click to view the definition of the Midmean') mylabel <- paste(mymid,hyperlink('http://www.xycoon.com/method_1.htm','Weighted Average at Xnp',''),sep=' - ') a<-table.element(a,mylabel,header=TRUE) a<-table.element(a,midm[1]) a<-table.element(a,'') a<-table.element(a,'') a<-table.row.end(a) a<-table.row.start(a) mymid <- hyperlink('http://www.xycoon.com/midmean.htm', 'Midmean', 'click to view the definition of the Midmean') mylabel <- paste(mymid,hyperlink('http://www.xycoon.com/method_2.htm','Weighted Average at X(n+1)p',''),sep=' - ') a<-table.element(a,mylabel,header=TRUE) a<-table.element(a,midm[2]) a<-table.element(a,'') a<-table.element(a,'') a<-table.row.end(a) a<-table.row.start(a) mymid <- hyperlink('http://www.xycoon.com/midmean.htm', 'Midmean', 'click to view the definition of the Midmean') mylabel <- paste(mymid,hyperlink('http://www.xycoon.com/method_3.htm','Empirical Distribution Function',''),sep=' - ') a<-table.element(a,mylabel,header=TRUE) a<-table.element(a,midm[3]) a<-table.element(a,'') a<-table.element(a,'') a<-table.row.end(a) a<-table.row.start(a) mymid <- hyperlink('http://www.xycoon.com/midmean.htm', 'Midmean', 'click to view the definition of the Midmean') mylabel <- paste(mymid,hyperlink('http://www.xycoon.com/method_4.htm','Empirical Distribution Function - Averaging',''),sep=' - ') a<-table.element(a,mylabel,header=TRUE) a<-table.element(a,midm[4]) a<-table.element(a,'') a<-table.element(a,'') a<-table.row.end(a) a<-table.row.start(a) mymid <- hyperlink('http://www.xycoon.com/midmean.htm', 'Midmean', 'click to view the definition of the Midmean') mylabel <- paste(mymid,hyperlink('http://www.xycoon.com/method_5.htm','Empirical Distribution Function - Interpolation',''),sep=' - ') a<-table.element(a,mylabel,header=TRUE) a<-table.element(a,midm[5]) a<-table.element(a,'') a<-table.element(a,'') a<-table.row.end(a) a<-table.row.start(a) mymid <- hyperlink('http://www.xycoon.com/midmean.htm', 'Midmean', 'click to view the definition of the Midmean') mylabel <- paste(mymid,hyperlink('http://www.xycoon.com/method_6.htm','Closest Observation',''),sep=' - ') a<-table.element(a,mylabel,header=TRUE) a<-table.element(a,midm[6]) a<-table.element(a,'') a<-table.element(a,'') a<-table.row.end(a) a<-table.row.start(a) mymid <- hyperlink('http://www.xycoon.com/midmean.htm', 'Midmean', 'click to view the definition of the Midmean') mylabel <- paste(mymid,hyperlink('http://www.xycoon.com/method_7.htm','True Basic - Statistics Graphics Toolkit',''),sep=' - ') a<-table.element(a,mylabel,header=TRUE) a<-table.element(a,midm[7]) a<-table.element(a,'') a<-table.element(a,'') a<-table.row.end(a) a<-table.row.start(a) mymid <- hyperlink('http://www.xycoon.com/midmean.htm', 'Midmean', 'click to view the definition of the Midmean') mylabel <- paste(mymid,hyperlink('http://www.xycoon.com/method_8.htm','MS Excel (old versions)',''),sep=' - ') a<-table.element(a,mylabel,header=TRUE) a<-table.element(a,midm[8]) a<-table.element(a,'') a<-table.element(a,'') a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'Number of observations',header=TRUE) a<-table.element(a,length(x)) a<-table.element(a,'') a<-table.element(a,'') a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable.tab')

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Software written by Ed van Stee & Patrick Wessa

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