Paper robustness central tendancy

*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: Sat, 11 Dec 2010 14:54:55 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/11/t1292079488xb62e2oluru2m9c.htm/, Retrieved Sat, 11 Dec 2010 15:58:10 +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/2010/Dec/11/t1292079488xb62e2oluru2m9c.htm/}, year = {2010}, } @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 = {2010}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

Original text written by user:

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Dataseries X:

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0.197299870124694 -7.4891834722532 -2.46357571004573 -11.1352144478093 -8.30666563148338 2.52543893701216 -9.07701600470547 6.13368551812671 -2.75736870709027 6.27193132285353 -4.86579008662195 -16.1915786204687 -6.04877252914152 -16.7022801433001 -12.2750854582395 11.9019305464657 8.86105500160429 -23.4303488204468 4.62806867714669 -9.0040358913486 -4.57543505507405 -1.10614123623148 -20.6385954015612 -14.4684198267564 -3.99999999999997 -4.25613735429343 12.7228091968775 -12.3192977007806 5.36140459843875 6.29824425195156 0.212282472462960 -17.6807022992194 1.63859540156125 5.34035114960969 -14.1070152283177 0.489473275585482 -4.85087787402421 14.8929847716824 -9.17017557480486 14.6807022992194 -30.0421068976581 -4.46841982675639 -6.38245804726782 3.76491609453564 4.02105344882906 0.680702299219377 9.17017557480483 12.4894732755855 -2 -14.5315801732436 -0.212282472462960 -11.6596488503903 3.21228247246296 19.6596488503903 7.04210689765813 -8.170175574804 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 1 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Central Tendency - Ungrouped Data Measure Value S.E. Value/S.E. Arithmetic Mean -2.55426807383316 1.36918870107418 -1.86553399968116 Geometric Mean NaN Harmonic Mean 8.09894378765704 Quadratic Mean 10.5596513458238 Winsorized Mean ( 1 / 19 ) -2.52189800368398 1.30967690547008 -1.9255879012227 Winsorized Mean ( 2 / 19 ) -2.43139025117792 1.28149886846426 -1.89730191029485 Winsorized Mean ( 3 / 19 ) -2.37875867223056 1.22057674826254 -1.94888086768542 Winsorized Mean ( 4 / 19 ) -2.32647191892338 1.20194509396104 -1.93558918008179 Winsorized Mean ( 5 / 19 ) -2.33321237561534 1.18124879426455 -1.97520826005838 Winsorized Mean ( 6 / 19 ) -2.44602885187174 1.08703838729695 -2.25017706868116 Winsorized Mean ( 7 / 19 ) -2.47623449374882 1.07831228215773 -2.29639830197780 Winsorized Mean ( 8 / 19 ) -2.68080200329373 1.02251504378808 -2.62177267667598 Winsorized Mean ( 9 / 19 ) -2.46889161374918 0.95639144311639 -2.58146560335622 Winsorized Mean ( 10 / 19 ) -2.51345863294400 0.946184285157688 -2.65641553381445 Winsorized Mean ( 11 / 19 ) -2.39976809844816 0.923762573932258 -2.59781914332475 Winsorized Mean ( 12 / 19 ) -2.31846523574202 0.899332277061342 -2.5779851283864 Winsorized Mean ( 13 / 19 ) -2.04643272007335 0.794571231891497 -2.57551826436223 Winsorized Mean ( 14 / 19 ) -2.02872244467362 0.790055902818086 -2.56782138762242 Winsorized Mean ( 15 / 19 ) -2.19695990759628 0.756258038855424 -2.90504007193275 Winsorized Mean ( 16 / 19 ) -2.17159709172328 0.697909104249756 -3.11157581768147 Winsorized Mean ( 17 / 19 ) -2.20728137346806 0.679476490840712 -3.24850293309929 Winsorized Mean ( 18 / 19 ) -2.16674711647471 0.618850657235862 -3.5012439449489 Winsorized Mean ( 19 / 19 ) -2.02678648662985 0.527842443801366 -3.83975656075235 Trimmed Mean ( 1 / 19 ) -2.45837858474950 1.26256994521541 -1.94712268739302 Trimmed Mean ( 2 / 19 ) -2.39006524740487 1.20372026005246 -1.98556535660594 Trimmed Mean ( 3 / 19 ) -2.36697186294345 1.14939681983871 -2.05931652331837 Trimmed Mean ( 4 / 19 ) -2.3624014675056 1.11166997245764 -2.12509245192877 Trimmed Mean ( 5 / 19 ) -2.37329500085233 1.07110945166500 -2.21573528005297 Trimmed Mean ( 6 / 19 ) -2.38344926591237 1.02633944816696 -2.32228164879485 Trimmed Mean ( 7 / 19 ) -2.36962354343297 0.999614078769522 -2.37053838452322 Trimmed Mean ( 8 / 19 ) -2.34844994006014 0.96653736007956 -2.42975598984278 Trimmed Mean ( 9 / 19 ) -2.28773177466170 0.938560909114663 -2.43748887519692 Trimmed Mean ( 10 / 19 ) -2.25672243283591 0.919382328074321 -2.45460714647702 Trimmed Mean ( 11 / 19 ) -2.21491110881831 0.894556473920055 -2.47598801572840 Trimmed Mean ( 12 / 19 ) -2.18588397821527 0.865488267982962 -2.52560786677042 Trimmed Mean ( 13 / 19 ) -2.16556910811037 0.830661941510113 -2.60704024091129 Trimmed Mean ( 14 / 19 ) -2.18358177155363 0.812441781650941 -2.68767783842485 Trimmed Mean ( 15 / 19 ) -2.20693357481332 0.784306844815098 -2.81386499353274 Trimmed Mean ( 16 / 19 ) -2.20844957223030 0.751563281975605 -2.93847454392003 Trimmed Mean ( 17 / 19 ) -2.21415770100449 0.720826166995358 -3.07169440065395 Trimmed Mean ( 18 / 19 ) -2.21525560203972 0.675575990600925 -3.27906206386827 Trimmed Mean ( 19 / 19 ) -2.22334034963389 0.626919273886752 -3.54645397301267 Median -2.46357571004573 Midrange -5.1912290236339 Midmean - Weighted Average at Xnp -2.45229366159518 Midmean - Weighted Average at X(n+1)p -2.18358177155363 Midmean - Empirical Distribution Function -2.18358177155363 Midmean - Empirical Distribution Function - Averaging -2.18358177155363 Midmean - Empirical Distribution Function - Interpolation -2.18358177155363 Midmean - Closest Observation -2.41646823166201 Midmean - True Basic - Statistics Graphics Toolkit -2.18358177155363 Midmean - MS Excel (old versions) -2.18358177155363 Number of observations 57

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/11/t1292079488xb62e2oluru2m9c/1w1341292079293.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/11/t1292079488xb62e2oluru2m9c/1w1341292079293.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/11/t1292079488xb62e2oluru2m9c/2os3p1292079293.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/11/t1292079488xb62e2oluru2m9c/2os3p1292079293.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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