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central tendency paper

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

Title produced by software: Central Tendency

Date of computation: Tue, 20 Nov 2007 13:28:53 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2007/Nov/20/t11955901590u983nl43hbp0h9.htm/, Retrieved Tue, 20 Nov 2007 21:22:42 +0100

User-defined keywords:

central tendency paper, Q3 W6

Dataseries X:

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Text written by user:

Output produced by software:

Summary of compuational transaction Raw Input view raw input (R code) Raw Output view raw output of R engine Computing time 2 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Central Tendency - Ungrouped Data Measure Value S.E. Value/S.E. Arithmetic Mean -0.0253063118129726 0.0381842097175612 -0.662742845803459 Geometric Mean NaN Harmonic Mean -1.14020047434508 Quadratic Mean 0.412034389998117 Winsorized Mean ( 1 / 39 ) -0.0259373212764409 0.0379445881872909 -0.683557854111284 Winsorized Mean ( 2 / 39 ) -0.0260419190713519 0.0379247803011145 -0.686672905276832 Winsorized Mean ( 3 / 39 ) -0.0270980856431934 0.037632367232178 -0.720073905423172 Winsorized Mean ( 4 / 39 ) -0.0264133698875179 0.0371895494324359 -0.71023635108848 Winsorized Mean ( 5 / 39 ) -0.0279865459285603 0.0369108680299532 -0.758219663266906 Winsorized Mean ( 6 / 39 ) -0.0259437014154883 0.036296238199305 -0.714776591255264 Winsorized Mean ( 7 / 39 ) -0.0261591620718738 0.035987634125891 -0.726893076115104 Winsorized Mean ( 8 / 39 ) -0.0234249110613941 0.0355038580111352 -0.659784946583757 Winsorized Mean ( 9 / 39 ) -0.021743800976523 0.0349957448415426 -0.621326994895432 Winsorized Mean ( 10 / 39 ) -0.0183903951528354 0.0344075888194399 -0.534486599724913 Winsorized Mean ( 11 / 39 ) -0.0188245210357860 0.0341687589862950 -0.550927853228045 Winsorized Mean ( 12 / 39 ) -0.0174338010849945 0.0337047905258344 -0.517249946165119 Winsorized Mean ( 13 / 39 ) -0.0157994412437291 0.0322708006404979 -0.489589378947781 Winsorized Mean ( 14 / 39 ) -0.0162671457002361 0.0319841196316491 -0.508600702085274 Winsorized Mean ( 15 / 39 ) -0.0177834746051338 0.0313433596681028 -0.567376145807099 Winsorized Mean ( 16 / 39 ) -0.0171866094260796 0.0307040507332151 -0.559750554590096 Winsorized Mean ( 17 / 39 ) -0.0162678803407526 0.0303988240711881 -0.535148343325925 Winsorized Mean ( 18 / 39 ) -0.0149120427452701 0.0301478173639465 -0.494630923534228 Winsorized Mean ( 19 / 39 ) -0.0115916580312488 0.029548964498099 -0.392286438057567 Winsorized Mean ( 20 / 39 ) -0.0120832293666301 0.0293418778266755 -0.411808318404383 Winsorized Mean ( 21 / 39 ) -0.0163604109747120 0.0277435597583705 -0.589701217767336 Winsorized Mean ( 22 / 39 ) -0.0165622294415488 0.0271376122035579 -0.610305332588451 Winsorized Mean ( 23 / 39 ) -0.0188085526625504 0.0265923772063951 -0.707291135221533 Winsorized Mean ( 24 / 39 ) -0.0183621553577356 0.026431807922919 -0.694699182563815 Winsorized Mean ( 25 / 39 ) -0.0267398628804093 0.0253942681794464 -1.05298812674791 Winsorized Mean ( 26 / 39 ) -0.0273499155092651 0.0253123826957135 -1.08049549653406 Winsorized Mean ( 27 / 39 ) -0.0278818361640297 0.0244108977500313 -1.14218806901495 Winsorized Mean ( 28 / 39 ) -0.0258581926293407 0.023660687512972 -1.09287579302858 Winsorized Mean ( 29 / 39 ) -0.0321266357988009 0.0227572066791875 -1.41171261709295 Winsorized Mean ( 30 / 39 ) -0.0293135969469509 0.0222047893641691 -1.32014749008396 Winsorized Mean ( 31 / 39 ) -0.0295321423767393 0.0212367230534111 -1.39061673039032 Winsorized Mean ( 32 / 39 ) -0.0306481148073962 0.0206946546364115 -1.48096768686693 Winsorized Mean ( 33 / 39 ) -0.0345679942502686 0.0201093002798081 -1.71900532436619 Winsorized Mean ( 34 / 39 ) -0.0301466098980595 0.0193785287820036 -1.55567072388158 Winsorized Mean ( 35 / 39 ) -0.0292322949395511 0.0187244849996081 -1.56118018413659 Winsorized Mean ( 36 / 39 ) -0.0328582672351034 0.0171108203537428 -1.92032097560512 Winsorized Mean ( 37 / 39 ) -0.0299764216744292 0.0166957798088596 -1.79544903068991 Winsorized Mean ( 38 / 39 ) -0.027244016847711 0.0152113893351341 -1.79102751546729 Winsorized Mean ( 39 / 39 ) -0.027419044780123 0.0149563000282044 -1.83327726298726 Trimmed Mean ( 1 / 39 ) -0.0255757942387629 0.0373331562018681 -0.685069167483975 Trimmed Mean ( 2 / 39 ) -0.025201469783822 0.0366540933262485 -0.687548579077111 Trimmed Mean ( 3 / 39 ) -0.0247585302944481 0.0359100024447498 -0.689460557195478 Trimmed Mean ( 4 / 39 ) -0.0239214416834292 0.0352004647140709 -0.679577439608828 Trimmed Mean ( 5 / 39 ) -0.0232402370108162 0.0345512996776723 -0.67262989316244 Trimmed Mean ( 6 / 39 ) -0.0221824881662903 0.0338997857979359 -0.654354818007169 Trimmed Mean ( 7 / 39 ) -0.0214704138132868 0.0333148986435944 -0.644468831887472 Trimmed Mean ( 8 / 39 ) -0.0206944823475801 0.0327225648285529 -0.63242238058071 Trimmed Mean ( 9 / 39 ) -0.0202911235603121 0.0321496345432743 -0.631146320901399 Trimmed Mean ( 10 / 39 ) -0.0200964348344281 0.0315944339245524 -0.636075166987277 Trimmed Mean ( 11 / 39 ) -0.0203065470899506 0.0310655191576548 -0.653668364172403 Trimmed Mean ( 12 / 39 ) -0.0204760456709255 0.0305033899715898 -0.671271150190075 Trimmed Mean ( 13 / 39 ) -0.0208020004479895 0.0299357749131877 -0.69488765559984 Trimmed Mean ( 14 / 39 ) -0.0213078772214540 0.0295016794669436 -0.72225980372844 Trimmed Mean ( 15 / 39 ) -0.0213078772214540 0.0290441360051386 -0.733637840618986 Trimmed Mean ( 16 / 39 ) -0.0221599344143533 0.0286074571041655 -0.774620908585638 Trimmed Mean ( 17 / 39 ) -0.0225980963297359 0.0281901169898789 -0.801631874669029 Trimmed Mean ( 18 / 39 ) -0.0231359578189959 0.0277464155929270 -0.8338359144629 Trimmed Mean ( 19 / 39 ) -0.0238126090592392 0.0272631500511454 -0.873435718710675 Trimmed Mean ( 20 / 39 ) -0.024789951007479 0.0267786703394913 -0.925734948494457 Trimmed Mean ( 21 / 39 ) -0.0257810752954652 0.0262387490353534 -0.982557333839676 Trimmed Mean ( 22 / 39 ) -0.0265000692455814 0.0258235399676455 -1.02619816178508 Trimmed Mean ( 23 / 39 ) -0.0272444528928745 0.0254117858115026 -1.07211878358200 Trimmed Mean ( 24 / 39 ) -0.0278663812652425 0.0249948427346678 -1.11488524096980 Trimmed Mean ( 25 / 39 ) -0.0285579200906021 0.0245164623895581 -1.16484669104484 Trimmed Mean ( 26 / 39 ) -0.0286888202097360 0.0240896048540936 -1.19092116219834 Trimmed Mean ( 27 / 39 ) -0.0287844562597696 0.0235836495236794 -1.22052595086559 Trimmed Mean ( 28 / 39 ) -0.0288485768130189 0.0231022644971825 -1.24873372549852 Trimmed Mean ( 29 / 39 ) -0.0290603655233641 0.0226254729025075 -1.28440919880810 Trimmed Mean ( 30 / 39 ) -0.0290603655233641 0.0221760670425442 -1.31043820654098 Trimmed Mean ( 31 / 39 ) -0.0288099877488764 0.0217037168587443 -1.32742183914314 Trimmed Mean ( 32 / 39 ) -0.028758562251944 0.0212687215612994 -1.35215283951401 Trimmed Mean ( 33 / 39 ) -0.0286230980062406 0.0208072949518157 -1.37562802240869 Trimmed Mean ( 34 / 39 ) -0.0281929478141866 0.0203186169352992 -1.38754266119400 Trimmed Mean ( 35 / 39 ) -0.028049907473878 0.0198241307990147 -1.41493757069400 Trimmed Mean ( 36 / 39 ) -0.0279620729764280 0.0193057992947099 -1.44837686073377 Trimmed Mean ( 37 / 39 ) -0.0275920117824584 0.018942280363392 -1.45663622611050 Trimmed Mean ( 38 / 39 ) -0.0274081120083248 0.0185323893320262 -1.4789302942692 Trimmed Mean ( 39 / 39 ) -0.0274210668894259 0.0182839742353856 -1.49973230854576 Median -0.0340370907426796 Midrange -0.00981107233002698 Midmean - Weighted Average at Xnp -0.0341537692790481 Midmean - Weighted Average at X(n+1)p -0.0290603655233642 Midmean - Empirical Distribution Function -0.0290603655233642 Midmean - Empirical Distribution Function - Averaging -0.0290603655233642 Midmean - Empirical Distribution Function - Interpolation -0.0290603655233642 Midmean - Closest Observation -0.0342435771366661 Midmean - True Basic - Statistics Graphics Toolkit -0.0290603655233642 Midmean - MS Excel (old versions) -0.0290603655233642 Number of observations 117

Charts produced by software:

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2007/Nov/20/t11955901590u983nl43hbp0h9/1fftd1195590531.png (open in new window) http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2007/Nov/20/t11955901590u983nl43hbp0h9/1fftd1195590531.ps (open in new window) http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2007/Nov/20/t11955901590u983nl43hbp0h9/26zfz1195590531.png (open in new window) http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2007/Nov/20/t11955901590u983nl43hbp0h9/26zfz1195590531.ps (open in new window)

Parameters:

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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