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: Sun, 28 Nov 2010 17:01:51 +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/28/t1290963620a2erwuk6fcozxti.htm/, Retrieved Sun, 28 Nov 2010 18:00:20 +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/2010/Nov/28/t1290963620a2erwuk6fcozxti.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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376.974 377.632 378.205 370.861 369.167 371.551 382.842 381.903 384.502 392.058 384.359 388.884 386.586 387.495 385.705 378.67 377.367 376.911 389.827 387.82 387.267 380.575 372.402 376.74 377.795 376.126 370.804 367.98 367.866 366.121 379.421 378.519 372.423 355.072 344.693 342.892 344.178 337.606 327.103 323.953 316.532 306.307 327.225 329.573 313.761 307.836 300.074 304.198 306.122 300.414 292.133 290.616 280.244 285.179 305.486 305.957 293.886 289.441 288.776 299.149 306.532 309.914 313.468 314.901 309.16 316.15 336.544 339.196 326.738 320.838 318.62 331.533 335.378

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 'Sir Ronald Aylmer Fisher' @ 193.190.124.24 Central Tendency - Ungrouped Data Measure Value S.E. Value/S.E. Arithmetic Mean 343.845698630137 4.11556722857176 83.5475839740962 Geometric Mean 342.043277298795 Harmonic Mean 340.220904110257 Quadratic Mean 345.614514751009 Winsorized Mean ( 1 / 24 ) 343.882739726027 4.09671714457364 83.941050258137 Winsorized Mean ( 2 / 24 ) 343.955452054795 4.07364815434566 84.4342562299771 Winsorized Mean ( 3 / 24 ) 343.939054794521 4.06190163315554 84.6743928969365 Winsorized Mean ( 4 / 24 ) 343.985630136986 4.04734147920331 84.9905133788458 Winsorized Mean ( 5 / 24 ) 344.073917808219 4.02618836379692 85.4589717913093 Winsorized Mean ( 6 / 24 ) 344.16202739726 3.99237820108796 86.204765696665 Winsorized Mean ( 7 / 24 ) 344.582219178082 3.89489398482961 88.4702434829319 Winsorized Mean ( 8 / 24 ) 344.551753424658 3.85942526575605 89.275404937051 Winsorized Mean ( 9 / 24 ) 344.576041095890 3.85019925577197 89.4956385904767 Winsorized Mean ( 10 / 24 ) 344.886589041096 3.74015921911571 92.2117398848699 Winsorized Mean ( 11 / 24 ) 344.939178082192 3.69127782517471 93.4470918795894 Winsorized Mean ( 12 / 24 ) 344.798301369863 3.64976602029832 94.4713440402078 Winsorized Mean ( 13 / 24 ) 344.622178082192 3.61770473431026 95.2598963684902 Winsorized Mean ( 14 / 24 ) 344.513630136986 3.59333439797951 95.8757499248337 Winsorized Mean ( 15 / 24 ) 344.528835616438 3.58242282204103 96.1720189746191 Winsorized Mean ( 16 / 24 ) 344.745821917808 3.53156265452771 97.6184923339304 Winsorized Mean ( 17 / 24 ) 344.958671232877 3.47457296665787 99.2808827280684 Winsorized Mean ( 18 / 24 ) 345.104397260274 3.44279864518919 100.239494907002 Winsorized Mean ( 19 / 24 ) 345.960438356164 3.30485510899815 104.682482876243 Winsorized Mean ( 20 / 24 ) 345.933041095890 3.27972341484285 105.476284838630 Winsorized Mean ( 21 / 24 ) 346.242863013699 3.23291817948101 107.099172880794 Winsorized Mean ( 22 / 24 ) 346.567739726027 3.17574217365188 109.129683952743 Winsorized Mean ( 23 / 24 ) 346.494643835616 3.13428194328279 110.549928215043 Winsorized Mean ( 24 / 24 ) 345.963684931507 2.88598885236330 119.876999749393 Trimmed Mean ( 1 / 24 ) 344.062450704225 4.07802747208673 84.369821699158 Trimmed Mean ( 2 / 24 ) 344.252579710145 4.0533653003472 84.9300653165056 Trimmed Mean ( 3 / 24 ) 344.414447761194 4.03560342179039 85.3439775329546 Trimmed Mean ( 4 / 24 ) 344.592415384615 4.01646046330766 85.7950472891831 Trimmed Mean ( 5 / 24 ) 344.768190476190 3.99541787185533 86.2908966055389 Trimmed Mean ( 6 / 24 ) 344.934360655738 3.97308579259478 86.8177478821732 Trimmed Mean ( 7 / 24 ) 345.093627118644 3.95149996488311 87.3323118272765 Trimmed Mean ( 8 / 24 ) 345.187192982456 3.94477943574045 87.5048145543943 Trimmed Mean ( 9 / 24 ) 345.292618181818 3.93936173635944 87.6519195977469 Trimmed Mean ( 10 / 24 ) 345.402283018868 3.92921981506965 87.9060727766246 Trimmed Mean ( 11 / 24 ) 345.476098039216 3.93323906626037 87.8350113530439 Trimmed Mean ( 12 / 24 ) 345.548816326531 3.94062307962519 87.6888779627706 Trimmed Mean ( 13 / 24 ) 345.645957446808 3.94974601625600 87.5109326078769 Trimmed Mean ( 14 / 24 ) 345.773711111111 3.95840979136649 87.3516713366218 Trimmed Mean ( 15 / 24 ) 345.926511627907 3.96439335632705 87.258372349913 Trimmed Mean ( 16 / 24 ) 346.092414634146 3.96390102553206 87.3110636226572 Trimmed Mean ( 17 / 24 ) 346.249948717949 3.96278056184131 87.3755039711468 Trimmed Mean ( 18 / 24 ) 346.399810810811 3.96126077710127 87.4468585388856 Trimmed Mean ( 19 / 24 ) 346.549914285714 3.95284613882996 87.670984934489 Trimmed Mean ( 20 / 24 ) 346.618545454545 3.95876950485835 87.5571424477132 Trimmed Mean ( 21 / 24 ) 346.699258064516 3.95515263002736 87.6576179215915 Trimmed Mean ( 22 / 24 ) 346.753965517241 3.9429804034233 87.9420971040558 Trimmed Mean ( 23 / 24 ) 346.776851851852 3.92022689459918 88.4583625324339 Trimmed Mean ( 24 / 24 ) 346.81268 3.87563025830208 89.4854918776331 Median 342.892 Midrange 336.151 Midmean - Weighted Average at Xnp 345.53225 Midmean - Weighted Average at X(n+1)p 346.399810810811 Midmean - Empirical Distribution Function 346.399810810811 Midmean - Empirical Distribution Function - Averaging 346.399810810811 Midmean - Empirical Distribution Function - Interpolation 346.399810810811 Midmean - Closest Observation 345.419815789474 Midmean - True Basic - Statistics Graphics Toolkit 346.399810810811 Midmean - MS Excel (old versions) 346.399810810811 Number of observations 73

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Nov/28/t1290963620a2erwuk6fcozxti/1ng621290963708.png (open in new window) http://www.freestatistics.org/blog/date/2010/Nov/28/t1290963620a2erwuk6fcozxti/1ng621290963708.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Nov/28/t1290963620a2erwuk6fcozxti/2y75m1290963708.png (open in new window) http://www.freestatistics.org/blog/date/2010/Nov/28/t1290963620a2erwuk6fcozxti/2y75m1290963708.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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