Paper - 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, 30 Nov 2010 22:31:59 +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/Nov/30/t1291156463bh1oibmlrm7bnyh.htm/, Retrieved Tue, 30 Nov 2010 23:34:24 +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/Nov/30/t1291156463bh1oibmlrm7bnyh.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:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

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162556 29790 87550 84738 54660 42634 40949 45187 37704 16275 25830 12679 18014 43556 24811 6575 7123 21950 37597 17821 12988 22330 13326 16189 7146 15824 27664 11920 8568 14416 3369 11819 6984 4519 2220 18562 10327 5336 2365 4069 8636 13718 4525 6869 4628 3689 4891 7489 4901 2284 3160 4150 7285 1134 4658 2384 3748 5371 1285 9327 5565 1528 3122 7561 2675 13253 880 2053 1424 4036 3045 5119 1431 554 1975 1765 1012 810 1280 666 1380 4677 876 814 514 5692 3642 540 2099 567 2001 2949 2253 6533 1889 3055 272 1414 2564 1383

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 'George Udny Yule' @ 72.249.76.132 Central Tendency - Ungrouped Data Measure Value S.E. Value/S.E. Arithmetic Mean 12469.4 2186.86721987727 5.701946550143 Geometric Mean 5256.95635608017 Harmonic Mean 2503.78545433251 Quadratic Mean 25078.7234894442 Winsorized Mean ( 1 / 33 ) 11721.76 1752.19951276945 6.68974047451543 Winsorized Mean ( 2 / 33 ) 11666.04 1727.85950752994 6.75172949488073 Winsorized Mean ( 3 / 33 ) 10764.12 1385.38776749298 7.76975244951015 Winsorized Mean ( 4 / 33 ) 10385.72 1271.99947788249 8.16487756527164 Winsorized Mean ( 5 / 33 ) 10309.12 1249.38215415752 8.25137446192487 Winsorized Mean ( 6 / 33 ) 10262.44 1233.9458888583 8.31676663674065 Winsorized Mean ( 7 / 33 ) 10144.77 1203.03815517173 8.43262531316133 Winsorized Mean ( 8 / 33 ) 9890.13 1136.95735202968 8.69876955573072 Winsorized Mean ( 9 / 33 ) 9880.86 1134.55055231647 8.70905221439953 Winsorized Mean ( 10 / 33 ) 9113.36 950.424136954155 9.58872954258694 Winsorized Mean ( 11 / 33 ) 8892.92 898.872521683906 9.89341623586448 Winsorized Mean ( 12 / 33 ) 8690.36 851.737452851463 10.2030971761383 Winsorized Mean ( 13 / 33 ) 8558.54 825.032696924152 10.3735767466036 Winsorized Mean ( 14 / 33 ) 8224.5 756.486511659813 10.8719717711219 Winsorized Mean ( 15 / 33 ) 8167.95 745.756945724248 10.9525630928823 Winsorized Mean ( 16 / 33 ) 7630.83 648.21555883686 11.7720562179849 Winsorized Mean ( 17 / 33 ) 7539.37 632.31999149756 11.9233459346178 Winsorized Mean ( 18 / 33 ) 7505.89 626.400921231923 11.9825653915681 Winsorized Mean ( 19 / 33 ) 7230.58 576.840105146175 12.5348080611831 Winsorized Mean ( 20 / 33 ) 7260.78 569.438625843049 12.7507683365358 Winsorized Mean ( 21 / 33 ) 7210.17 554.827327776377 12.9953404222116 Winsorized Mean ( 22 / 33 ) 6919.33 505.434019523735 13.6898778727241 Winsorized Mean ( 23 / 33 ) 6764.77 481.08725638239 14.0614200651847 Winsorized Mean ( 24 / 33 ) 6683.17 466.197775054382 14.3354824016919 Winsorized Mean ( 25 / 33 ) 6676.42 462.421761480695 14.4379450885309 Winsorized Mean ( 26 / 33 ) 6638.98 449.421110026414 14.7722922931008 Winsorized Mean ( 27 / 33 ) 6564.46 436.680904792162 15.0326243441406 Winsorized Mean ( 28 / 33 ) 6360.62 406.072355541908 15.6637601973956 Winsorized Mean ( 29 / 33 ) 6354.82 399.649092135144 15.9009994644279 Winsorized Mean ( 30 / 33 ) 5912.92 338.562831457389 17.4647641459845 Winsorized Mean ( 31 / 33 ) 5658.72 292.545836546699 19.3430201119840 Winsorized Mean ( 32 / 33 ) 5473.12 261.172643191271 20.9559467374679 Winsorized Mean ( 33 / 33 ) 5541.1 248.767090169731 22.2742485600461 Trimmed Mean ( 1 / 33 ) 11062.3673469388 1604.64508493259 6.89396518321279 Trimmed Mean ( 2 / 33 ) 10375.5 1423.09997005857 7.29077381652463 Trimmed Mean ( 3 / 33 ) 9689.0425531915 1209.99865945034 8.00748205588331 Trimmed Mean ( 4 / 33 ) 9299.52173913043 1129.42546718413 8.23385164345184 Trimmed Mean ( 5 / 33 ) 8997.8 1077.92384461126 8.34734294540532 Trimmed Mean ( 6 / 33 ) 8699.77272727273 1024.48303956745 8.4918660351331 Trimmed Mean ( 7 / 33 ) 8396.93023255814 965.46422802325 8.69729813786112 Trimmed Mean ( 8 / 33 ) 8099.67857142857 903.334614020922 8.96642113089776 Trimmed Mean ( 9 / 33 ) 7826.74390243902 846.06709788263 9.25073664018641 Trimmed Mean ( 10 / 33 ) 7541.45 776.78978285023 9.70848248328988 Trimmed Mean ( 11 / 33 ) 7339.92307692308 738.177260501718 9.94330693949355 Trimmed Mean ( 12 / 33 ) 7154.15789473684 703.188127869322 10.1738889085260 Trimmed Mean ( 13 / 33 ) 6981.16216216216 671.108946227925 10.4024275065932 Trimmed Mean ( 14 / 33 ) 6812.63888888889 637.967595644904 10.6786597554413 Trimmed Mean ( 15 / 33 ) 6668.57142857143 611.965340420818 10.8969756751024 Trimmed Mean ( 16 / 33 ) 6521.57352941176 582.506528473168 11.1957089073417 Trimmed Mean ( 17 / 33 ) 6416.5303030303 565.958662414165 11.3374540035483 Trimmed Mean ( 18 / 33 ) 6313.328125 548.721953514884 11.5055140122597 Trimmed Mean ( 19 / 33 ) 6206.46774193548 528.671985826212 11.7397325909675 Trimmed Mean ( 20 / 33 ) 6116.63333333333 513.462531965654 11.9125212698918 Trimmed Mean ( 21 / 33 ) 6018 495.70002625727 12.1404068614607 Trimmed Mean ( 22 / 33 ) 5916.625 476.24192774459 12.4235701548166 Trimmed Mean ( 23 / 33 ) 5832.22222222222 461.820888397909 12.6287536331556 Trimmed Mean ( 24 / 33 ) 5754.25 448.402251434191 12.8327856998829 Trimmed Mean ( 25 / 33 ) 5676.84 434.197828851771 13.0743168730537 Trimmed Mean ( 26 / 33 ) 5593.54166666667 416.50394300358 13.4297448094473 Trimmed Mean ( 27 / 33 ) 5506.13043478261 396.208196391277 13.8970634250711 Trimmed Mean ( 28 / 33 ) 5417.04545454545 372.529515487924 14.5412517111575 Trimmed Mean ( 29 / 33 ) 5417.04545454545 349.376124305485 15.5049102605791 Trimmed Mean ( 30 / 33 ) 5249.05 319.289210490892 16.4397976114815 Trimmed Mean ( 31 / 33 ) 5190.81578947368 298.064144946198 17.4150963055642 Trimmed Mean ( 32 / 33 ) 5148.88888888889 282.899833015649 18.2003956453522 Trimmed Mean ( 33 / 33 ) 5119.08823529412 271.213114063484 18.8747813798409 Median 4784 Midrange 81414 Midmean - Weighted Average at Xnp 5605.78431372549 Midmean - Weighted Average at X(n+1)p 5676.84 Midmean - Empirical Distribution Function 5605.78431372549 Midmean - Empirical Distribution Function - Averaging 5676.84 Midmean - Empirical Distribution Function - Interpolation 5676.84 Midmean - Closest Observation 5605.78431372549 Midmean - True Basic - Statistics Graphics Toolkit 5676.84 Midmean - MS Excel (old versions) 5754.25 Number of observations 100

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Nov/30/t1291156463bh1oibmlrm7bnyh/1xsek1291156316.png (open in new window) http://www.freestatistics.org/blog/date/2010/Nov/30/t1291156463bh1oibmlrm7bnyh/1xsek1291156316.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Nov/30/t1291156463bh1oibmlrm7bnyh/28jv41291156316.png (open in new window) http://www.freestatistics.org/blog/date/2010/Nov/30/t1291156463bh1oibmlrm7bnyh/28jv41291156316.ps (open in new window)

Parameters (Session):

par1 = 0 ;

Parameters (R input):

par1 = 0 ;

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')

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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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