Robustness of tendency

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

Title produced by software: Central Tendency

Date of computation: Mon, 11 Aug 2008 09:38:40 -0600

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/Aug/11/t1218469201zggm9rdq9uxn0pq.htm/, Retrieved Mon, 11 Aug 2008 15:40:06 +0000

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

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81,6 113,8 97,6 120,1 101,9 97,6 150 143,8 103,7 115,3 87,4 97,6 84 96,1 88 120,6 121,1 109,7 149,2 156,5 109,3 102 90 85,9 88,4 88,4 100 120,6 103,3 107,1 168,1 172,1 104,9 112,8 88,8 109 89,5 90,2 91,2 116,9 106,1 106,1 170,4 163,9 108 118,6 94,7 101,4 94,86 101,61 96,04 115,96 123,87 115,59 178,58 184,31 115,59 117,91 95,22 113,49 97,84 97,84 98,82 142,58 122,57 122,29 167,48 176,09 121,78 122,6 107,8 116,77 99,56 103,41 107,77 139,17 130,56 125,18 181,69 194,7 125,18 123,54 112,25 123,21 109,02 115,97 118,71 126,4 129,39 128,97 192,64 188,1 128,03 132,42 114,73 123,82

Text written by user:

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 'George Udny Yule' @ 72.249.76.132 Central Tendency - Ungrouped Data Measure Value S.E. Value/S.E. Arithmetic Mean 119.429270833333 2.77909040340190 42.9742302327191 Geometric Mean 116.706998464811 Harmonic Mean 114.298298258285 Quadratic Mean 122.462518190091 Winsorized Mean ( 1 / 32 ) 119.4328125 2.76957148303306 43.1232099375911 Winsorized Mean ( 2 / 32 ) 119.3778125 2.73870527101257 43.5891418341131 Winsorized Mean ( 3 / 32 ) 119.30625 2.70213381508120 44.1526061122976 Winsorized Mean ( 4 / 32 ) 119.222083333333 2.67178476601163 44.6226375904168 Winsorized Mean ( 5 / 32 ) 119.0809375 2.62999991694235 45.2779244337178 Winsorized Mean ( 6 / 32 ) 118.9253125 2.59341202280407 45.8566982239154 Winsorized Mean ( 7 / 32 ) 118.663541666667 2.52359455404931 47.0216348645473 Winsorized Mean ( 8 / 32 ) 118.580208333333 2.485030154337 47.7178146616777 Winsorized Mean ( 9 / 32 ) 118.411458333333 2.43240828508139 48.6807494693972 Winsorized Mean ( 10 / 32 ) 118.367708333333 2.41599567883661 48.993344388071 Winsorized Mean ( 11 / 32 ) 118.072083333333 2.31531839118478 50.9960460655755 Winsorized Mean ( 12 / 32 ) 117.584583333333 2.07519258204112 56.6620102398783 Winsorized Mean ( 13 / 32 ) 116.726041666667 1.90456243811629 61.287589910738 Winsorized Mean ( 14 / 32 ) 116.661875 1.87685985134477 62.158010847966 Winsorized Mean ( 15 / 32 ) 115.94625 1.71185926193582 67.7311812823239 Winsorized Mean ( 16 / 32 ) 115.752916666667 1.67608527237661 69.061472333406 Winsorized Mean ( 17 / 32 ) 115.4146875 1.54369056070757 74.7654293144723 Winsorized Mean ( 18 / 32 ) 114.1490625 1.35021900496267 84.5411463477036 Winsorized Mean ( 19 / 32 ) 113.7809375 1.29895058080067 87.5945083529397 Winsorized Mean ( 20 / 32 ) 113.5871875 1.25971032509014 90.169291493163 Winsorized Mean ( 21 / 32 ) 113.4953125 1.24764637148123 90.9675330240058 Winsorized Mean ( 22 / 32 ) 113.504479166667 1.19032363163206 95.3559823146916 Winsorized Mean ( 23 / 32 ) 113.29125 1.11814250388366 101.320940404737 Winsorized Mean ( 24 / 32 ) 113.09625 1.06691051814483 106.003500834029 Winsorized Mean ( 25 / 32 ) 113.460833333333 1.02066264237120 111.163893556191 Winsorized Mean ( 26 / 32 ) 113.162916666667 0.971525025599185 116.479672355196 Winsorized Mean ( 27 / 32 ) 113.230416666667 0.959719678972773 117.982801798815 Winsorized Mean ( 28 / 32 ) 113.177916666667 0.946646886182954 119.556635445155 Winsorized Mean ( 29 / 32 ) 113.4709375 0.887015250345137 127.924449388946 Winsorized Mean ( 30 / 32 ) 113.3146875 0.861021450121035 131.604953028837 Winsorized Mean ( 31 / 32 ) 113.398645833333 0.848617666217528 133.627486614526 Winsorized Mean ( 32 / 32 ) 113.7053125 0.79044038848225 143.850585264664 Trimmed Mean ( 1 / 32 ) 119.030957446809 2.69142963521897 44.2259221230298 Trimmed Mean ( 2 / 32 ) 118.611630434783 2.60158186873154 45.592119110445 Trimmed Mean ( 3 / 32 ) 118.203 2.51710488756056 46.9599024594306 Trimmed Mean ( 4 / 32 ) 117.801818181818 2.43560671746685 48.366518837794 Trimmed Mean ( 5 / 32 ) 117.405465116279 2.35186023329670 49.9202560824402 Trimmed Mean ( 6 / 32 ) 117.0225 2.26720020808989 51.6154239852471 Trimmed Mean ( 7 / 32 ) 116.651219512195 2.17835881020541 53.5500483050336 Trimmed Mean ( 8 / 32 ) 116.30625 2.09195843055215 55.5968265436814 Trimmed Mean ( 9 / 32 ) 115.956410256410 1.99916308641871 58.002476658437 Trimmed Mean ( 10 / 32 ) 115.611842105263 1.9011624210589 60.8111336646715 Trimmed Mean ( 11 / 32 ) 115.254324324324 1.78731490246491 64.4846211293686 Trimmed Mean ( 12 / 32 ) 114.912777777778 1.67271486174247 68.6983659953094 Trimmed Mean ( 13 / 32 ) 114.607428571429 1.58660725943182 72.234277191238 Trimmed Mean ( 14 / 32 ) 114.377352941176 1.51900988409935 75.297306580064 Trimmed Mean ( 15 / 32 ) 114.14 1.44270000693953 79.1155468572645 Trimmed Mean ( 16 / 32 ) 113.959375 1.38369551833140 82.3587078878624 Trimmed Mean ( 17 / 32 ) 113.785806451613 1.31899793600160 86.2668570934558 Trimmed Mean ( 18 / 32 ) 113.6325 1.26580490961642 89.7709426916618 Trimmed Mean ( 19 / 32 ) 113.585 1.23738225163472 91.7945928591923 Trimmed Mean ( 20 / 32 ) 113.567321428571 1.21117326639850 93.7663706583215 Trimmed Mean ( 21 / 32 ) 113.565555555556 1.18523282355342 95.817086144372 Trimmed Mean ( 22 / 32 ) 113.571730769231 1.15468083780372 98.3576820978966 Trimmed Mean ( 23 / 32 ) 113.5776 1.12653025394848 100.820727718507 Trimmed Mean ( 24 / 32 ) 113.6025 1.10372630051743 102.926332322372 Trimmed Mean ( 25 / 32 ) 113.646521739130 1.08327871511776 104.909770821793 Trimmed Mean ( 26 / 32 ) 113.662727272727 1.06504128316478 106.721428614464 Trimmed Mean ( 27 / 32 ) 113.706666666667 1.04946818381249 108.34694030799 Trimmed Mean ( 28 / 32 ) 113.749 1.02994702378077 110.441602697627 Trimmed Mean ( 29 / 32 ) 113.800526315789 1.00526005972429 113.205060934184 Trimmed Mean ( 30 / 32 ) 113.830833333333 0.984804337588606 115.587258289357 Trimmed Mean ( 31 / 32 ) 113.879411764706 0.96110897966773 118.487512003140 Trimmed Mean ( 32 / 32 ) 113.9259375 0.929581923389943 122.556102516002 Median 115.015 Midrange 138.15 Midmean - Weighted Average at Xnp 113.315918367347 Midmean - Weighted Average at X(n+1)p 113.6025 Midmean - Empirical Distribution Function 113.315918367347 Midmean - Empirical Distribution Function - Averaging 113.6025 Midmean - Empirical Distribution Function - Interpolation 113.6025 Midmean - Closest Observation 113.315918367347 Midmean - True Basic - Statistics Graphics Toolkit 113.6025 Midmean - MS Excel (old versions) 113.5776 Number of observations 96

Charts produced by software:

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Aug/11/t1218469201zggm9rdq9uxn0pq/19dyt1218469117.png (open in new window) http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Aug/11/t1218469201zggm9rdq9uxn0pq/19dyt1218469117.ps (open in new window) http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Aug/11/t1218469201zggm9rdq9uxn0pq/2n69y1218469117.png (open in new window) http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Aug/11/t1218469201zggm9rdq9uxn0pq/2n69y1218469117.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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