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: Thu, 17 Dec 2009 15:54:55 -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/17/t1261090601hpu515ghtmvkox4.htm/, Retrieved Thu, 17 Dec 2009 23:56:43 +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/17/t1261090601hpu515ghtmvkox4.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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220206 220115 218444 214912 210705 209673 237041 242081 241878 242621 238545 240337 244752 244576 241572 240541 236089 236997 264579 270349 269645 267037 258113 262813 267413 267366 264777 258863 254844 254868 277267 285351 286602 283042 276687 277915 277128 277103 275037 270150 267140 264993 287259 291186 292300 288186 281477 282656 280190 280408 276836 275216 274352 271311 289802 290726 292300 278506 269826 265861 269034 264176 255198 253353 246057 235372 258556 260993 254663 250643 243422 247105 248541 245039 237080 237085 225554 226839 247934 248333 246969 245098 246263 255765 264319 268347 273046 273963 267430 271993 292710 295881 293299 288576

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 260289.372340426 2174.25851601937 119.714086628927 Geometric Mean 259422.061146662 Harmonic Mean 258531.653956245 Quadratic Mean 261132.544050257 Winsorized Mean ( 1 / 31 ) 260272.882978723 2166.86562074373 120.114916442945 Winsorized Mean ( 2 / 31 ) 260349.861702128 2143.60784126253 121.454053624281 Winsorized Mean ( 3 / 31 ) 260449.5 2116.60856552734 123.050385527997 Winsorized Mean ( 4 / 31 ) 260520.606382979 2101.6711311249 123.958788092377 Winsorized Mean ( 5 / 31 ) 260466.191489362 2091.17227138415 124.555109616559 Winsorized Mean ( 6 / 31 ) 260778.191489362 2019.10326796276 129.155450158070 Winsorized Mean ( 7 / 31 ) 260805.074468085 1990.47067342642 131.026835989064 Winsorized Mean ( 8 / 31 ) 261426.946808511 1852.10490098615 141.151263445885 Winsorized Mean ( 9 / 31 ) 261458.255319149 1835.94694194300 142.410572629318 Winsorized Mean ( 10 / 31 ) 261456.234042553 1806.44481425452 144.735245704392 Winsorized Mean ( 11 / 31 ) 261384.5 1793.97952075162 145.700938598501 Winsorized Mean ( 12 / 31 ) 261229.776595745 1769.47468277314 147.631259796462 Winsorized Mean ( 13 / 31 ) 260911.138297872 1723.91133605477 151.348351183174 Winsorized Mean ( 14 / 31 ) 261071.095744681 1684.1819447935 155.013593722317 Winsorized Mean ( 15 / 31 ) 261168.914893617 1618.14768263686 161.399925171247 Winsorized Mean ( 16 / 31 ) 261021.680851064 1589.12355775593 164.255120111405 Winsorized Mean ( 17 / 31 ) 261168.712765957 1558.42215355749 167.585344041583 Winsorized Mean ( 18 / 31 ) 260904.840425532 1509.09236217070 172.888583207885 Winsorized Mean ( 19 / 31 ) 260826.414893617 1488.66917964548 175.207775145672 Winsorized Mean ( 20 / 31 ) 260803.436170213 1456.28764100125 179.087859312533 Winsorized Mean ( 21 / 31 ) 260951.329787234 1428.69164712162 182.650560261181 Winsorized Mean ( 22 / 31 ) 261215.563829787 1392.80696351003 187.546135734053 Winsorized Mean ( 23 / 31 ) 261193.297872340 1379.2911506292 189.367776160378 Winsorized Mean ( 24 / 31 ) 261228.531914894 1365.28800863937 191.335842885803 Winsorized Mean ( 25 / 31 ) 260853 1316.49968322181 198.141331383860 Winsorized Mean ( 26 / 31 ) 261068.744680851 1276.84591430976 204.463782007698 Winsorized Mean ( 27 / 31 ) 260931.159574468 1246.39731985949 209.348299628793 Winsorized Mean ( 28 / 31 ) 261025.585106383 1206.52856032286 216.344306873708 Winsorized Mean ( 29 / 31 ) 260784.638297872 1168.99256449445 223.084941870998 Winsorized Mean ( 30 / 31 ) 260713.14893617 1098.27277197650 237.384696760696 Winsorized Mean ( 31 / 31 ) 260619.819148936 1057.16686820271 246.526661956420 Trimmed Mean ( 1 / 31 ) 260452.684782609 2116.29450584799 123.070151183068 Trimmed Mean ( 2 / 31 ) 260640.477777778 2058.41078616519 126.622188112097 Trimmed Mean ( 3 / 31 ) 260640.477777778 2006.29697869182 129.911214813135 Trimmed Mean ( 4 / 31 ) 260921.825581395 1958.16669265467 133.248015380992 Trimmed Mean ( 5 / 31 ) 261034.071428571 1907.90327198053 136.817246063843 Trimmed Mean ( 6 / 31 ) 261034.071428571 1852.80703714855 140.885729703564 Trimmed Mean ( 7 / 31 ) 261239.875 1807.98321520079 144.492422719194 Trimmed Mean ( 8 / 31 ) 261314.730769231 1762.59960017597 148.255299015807 Trimmed Mean ( 9 / 31 ) 261297.381578947 1739.27589752395 150.233428722224 Trimmed Mean ( 10 / 31 ) 261274.675675676 1714.78215237475 152.366103947283 Trimmed Mean ( 11 / 31 ) 261250.972222222 1691.12512076703 154.483526389655 Trimmed Mean ( 12 / 31 ) 261250.972222222 1665.07258177419 156.900651107864 Trimmed Mean ( 13 / 31 ) 261235.235294118 1638.13534921627 159.471093410627 Trimmed Mean ( 14 / 31 ) 261270.742424242 1613.39935867958 161.938047773905 Trimmed Mean ( 15 / 31 ) 261291.6875 1590.00005831883 164.334388626548 Trimmed Mean ( 16 / 31 ) 261304.096774194 1572.06111792535 166.217517750860 Trimmed Mean ( 17 / 31 ) 261331.75 1554.16639265534 168.149144927466 Trimmed Mean ( 18 / 31 ) 261347.293103448 1536.46325499294 170.096676411991 Trimmed Mean ( 19 / 31 ) 261388.553571429 1521.59436479858 171.785963209735 Trimmed Mean ( 20 / 31 ) 261440.055555556 1505.28069970930 173.681928962515 Trimmed Mean ( 21 / 31 ) 261497.596153846 1489.03720228494 175.615220192334 Trimmed Mean ( 22 / 31 ) 261546.5 1472.04459039883 177.675663975055 Trimmed Mean ( 23 / 31 ) 261575.958333333 1455.44012041444 179.722926875789 Trimmed Mean ( 24 / 31 ) 261575.958333333 1435.05636138253 182.275738690383 Trimmed Mean ( 25 / 31 ) 261643.909090909 1409.90269100973 185.575863326800 Trimmed Mean ( 26 / 31 ) 261714.714285714 1384.91682257173 188.975041692195 Trimmed Mean ( 27 / 31 ) 261773.1 1358.73315845954 192.659683301455 Trimmed Mean ( 28 / 31 ) 261850.236842105 1328.37107353716 197.121302969099 Trimmed Mean ( 29 / 31 ) 261927.138888889 1294.86221253416 202.281861616978 Trimmed Mean ( 30 / 31 ) 262036.058823529 1255.60193356868 208.693577015104 Trimmed Mean ( 31 / 31 ) 262036.058823529 1217.45038471086 215.233460118182 Median 264449 Midrange 252777 Midmean - Weighted Average at Xnp 261251.276595745 Midmean - Weighted Average at X(n+1)p 261575.958333333 Midmean - Empirical Distribution Function 261575.958333333 Midmean - Empirical Distribution Function - Averaging 261575.958333333 Midmean - Empirical Distribution Function - Interpolation 261609.956521739 Midmean - Closest Observation 261575.958333333 Midmean - True Basic - Statistics Graphics Toolkit 261575.958333333 Midmean - MS Excel (old versions) 261575.958333333 Number of observations 94

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/17/t1261090601hpu515ghtmvkox4/14ru21261090493.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/17/t1261090601hpu515ghtmvkox4/14ru21261090493.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/17/t1261090601hpu515ghtmvkox4/287i81261090493.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/17/t1261090601hpu515ghtmvkox4/287i81261090493.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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