Robustness of Central Tendency

*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: Tue, 29 Dec 2009 14:57:55 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2009/Dec/29/t1262123958pomsqb91al2dmea.htm/, Retrieved Tue, 29 Dec 2009 22:59:20 +0100

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/2009/Dec/29/t1262123958pomsqb91al2dmea.htm/}, year = {2009}, } @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 = {2009}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

Robustness of Central Tendency

Dataseries X:

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-706.026542296181 -59.4405579520267 -969.833720383066 1994.56903063601 102.064814516947 1282.18411025837 -307.754606959943 225.069461418321 -862.471322666482 -2850.16172979440 -3875.76341994283 2996.98129088558 1931.51753614196 488.179241950267 -601.581627830989 -3495.10356420977 265.163825175723 640.313274555833 -3418.45097379814 2072.03398243007 2662.9984145817 -1064.81719671349 -9.57987333730985 -2208.8867746199 -4275.21498681124 381.837227886395 1914.99171248840 1019.08609621188 884.745394297871 -1702.08983953302 -2064.71644375347 -2452.89205796152 1313.39939683182 -569.737683944086 -187.9171774845 -128.334928612711 -2601.14827891639 581.657297679053 -299.792761503996 2868.57371158154 2723.0124492194 -1846.18418996859 -6526.92686969168 -3652.78139505582 2790.5729463211 -7703.28365973859 -1373.42544129638 -2976.48819823291 5585.1945937738 -2568.07437379205 -4750.48266905558 1324.51875012340 -1689.9683295 etc...

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 2 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Central Tendency - Ungrouped Data Measure Value S.E. Value/S.E. Arithmetic Mean -241.03271601053 336.628696282865 -0.716019515484186 Geometric Mean NaN Harmonic Mean -671.500373256785 Quadratic Mean 3039.23114985297 Winsorized Mean ( 1 / 27 ) -257.464744398881 328.144540480607 -0.784607734206983 Winsorized Mean ( 2 / 27 ) -252.444081876043 323.006691877249 -0.78154443305459 Winsorized Mean ( 3 / 27 ) -188.987533277275 308.900588288296 -0.611806971053398 Winsorized Mean ( 4 / 27 ) -188.768248900336 307.817573988939 -0.613247146529453 Winsorized Mean ( 5 / 27 ) -229.335562560781 298.177791946404 -0.769123552306682 Winsorized Mean ( 6 / 27 ) -233.164059540506 286.617138224641 -0.813503550362572 Winsorized Mean ( 7 / 27 ) -279.596082396468 265.899723319948 -1.05150948976370 Winsorized Mean ( 8 / 27 ) -272.523597049491 259.845117396767 -1.04879244905481 Winsorized Mean ( 9 / 27 ) -300.40688086362 254.922408245584 -1.17842477219271 Winsorized Mean ( 10 / 27 ) -298.049863935498 249.562054373717 -1.1942915948639 Winsorized Mean ( 11 / 27 ) -303.674484793883 247.169890757169 -1.22860629934989 Winsorized Mean ( 12 / 27 ) -305.916158276273 244.977234937487 -1.24875341316648 Winsorized Mean ( 13 / 27 ) -259.521147245662 237.323233246812 -1.09353451701783 Winsorized Mean ( 14 / 27 ) -246.265245748855 232.285541711319 -1.06018327242644 Winsorized Mean ( 15 / 27 ) -314.550166562759 220.977796956624 -1.42344692948723 Winsorized Mean ( 16 / 27 ) -315.132181534442 214.887303549307 -1.46649977141220 Winsorized Mean ( 17 / 27 ) -281.826856609444 204.915658161519 -1.37533099782595 Winsorized Mean ( 18 / 27 ) -291.571232707675 201.505365989708 -1.44696510326463 Winsorized Mean ( 19 / 27 ) -296.330686461759 198.096766999850 -1.49588855461727 Winsorized Mean ( 20 / 27 ) -282.636589365349 195.088170885498 -1.44876333650817 Winsorized Mean ( 21 / 27 ) -236.551736946391 184.700758330599 -1.28072964661565 Winsorized Mean ( 22 / 27 ) -298.687029904208 175.846182292762 -1.69856988653260 Winsorized Mean ( 23 / 27 ) -381.381250181437 164.977112342507 -2.31172218234525 Winsorized Mean ( 24 / 27 ) -346.303022299931 159.367171828832 -2.17298844125737 Winsorized Mean ( 25 / 27 ) -351.795376241113 157.606381875856 -2.23211377644731 Winsorized Mean ( 26 / 27 ) -329.258795329001 142.438219887065 -2.31159021497222 Winsorized Mean ( 27 / 27 ) -319.890558854938 131.426156032812 -2.43399463631177 Trimmed Mean ( 1 / 27 ) -249.556620016925 315.888739829589 -0.790014294753122 Trimmed Mean ( 2 / 27 ) -241.242950794868 301.532832274047 -0.800055333860346 Trimmed Mean ( 3 / 27 ) -235.200235343182 287.950365497764 -0.816808254216328 Trimmed Mean ( 4 / 27 ) -252.269791962121 278.442496179480 -0.906003197872181 Trimmed Mean ( 5 / 27 ) -270.350092417213 267.604536238916 -1.01025975200901 Trimmed Mean ( 6 / 27 ) -279.95921084072 257.860168569972 -1.08570165137681 Trimmed Mean ( 7 / 27 ) -289.364118700077 249.541439887162 -1.15958342963366 Trimmed Mean ( 8 / 27 ) -291.097839429288 244.828942255065 -1.18898458959979 Trimmed Mean ( 9 / 27 ) -294.072620435428 240.508882176985 -1.22271002124166 Trimmed Mean ( 10 / 27 ) -293.141779297235 236.303159395032 -1.24053262786548 Trimmed Mean ( 11 / 27 ) -292.471007730006 232.24205636886 -1.25933697067118 Trimmed Mean ( 12 / 27 ) -291.031062402046 227.696832573908 -1.27815156281360 Trimmed Mean ( 13 / 27 ) -289.214726298346 222.490727012042 -1.29989564141562 Trimmed Mean ( 14 / 27 ) -292.683207042392 217.517402084283 -1.34556225956112 Trimmed Mean ( 15 / 27 ) -297.911603781499 212.249203330206 -1.40359350757149 Trimmed Mean ( 16 / 27 ) -296.092454250748 207.819124438115 -1.42476037781075 Trimmed Mean ( 17 / 27 ) -294.059566702228 203.313331539242 -1.44633686574297 Trimmed Mean ( 18 / 27 ) -295.342280548275 199.435900648481 -1.48088824322976 Trimmed Mean ( 19 / 27 ) -295.73271731965 194.974618529645 -1.51677546313386 Trimmed Mean ( 20 / 27 ) -295.671271868957 189.736327173632 -1.55832715997702 Trimmed Mean ( 21 / 27 ) -297.007326825577 183.364354630426 -1.61976588865486 Trimmed Mean ( 22 / 27 ) -303.219555359529 177.189862096432 -1.71126921016794 Trimmed Mean ( 23 / 27 ) -303.688831984953 170.931036161888 -1.77667460985453 Trimmed Mean ( 24 / 27 ) -295.542056675348 164.957868510098 -1.79162145670703 Trimmed Mean ( 25 / 27 ) -290.122266074806 158.000229857812 -1.83621420257359 Trimmed Mean ( 26 / 27 ) -283.37933936329 148.330417919351 -1.91046006165348 Trimmed Mean ( 27 / 27 ) -278.211598444075 139.384165426935 -1.99600577003788 Median -243.854969494248 Midrange 99.9234442452548 Midmean - Weighted Average at Xnp -349.589881243527 Midmean - Weighted Average at X(n+1)p -295.671271868958 Midmean - Empirical Distribution Function -295.671271868958 Midmean - Empirical Distribution Function - Averaging -295.671271868958 Midmean - Empirical Distribution Function - Interpolation -297.007326825577 Midmean - Closest Observation -295.671271868958 Midmean - True Basic - Statistics Graphics Toolkit -295.671271868958 Midmean - MS Excel (old versions) -295.671271868958 Number of observations 82

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/29/t1262123958pomsqb91al2dmea/1z58w1262123872.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/29/t1262123958pomsqb91al2dmea/1z58w1262123872.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/29/t1262123958pomsqb91al2dmea/21i7q1262123872.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/29/t1262123958pomsqb91al2dmea/21i7q1262123872.ps (open in new window)

Parameters (Session):

Parameters (R input):

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

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')

Copyright

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License.

Software written by Ed van Stee & Patrick Wessa

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As a general rule, we do not log on-line behavior of individuals (other than normal logging of webserver 'hits'). However, in cases of abuse, hacking, unauthorized access, Denial of Service attacks, illegal copying, hotlinking, non-compliance with international webstandards (such as robots.txt), or any other harmful behavior, our system engineers are empowered to log, track, identify, publish, and ban misbehaving individuals - even if this leads to ban entire blocks of IP addresses, or disclosing user's identity.

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