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central tendancy

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

Title produced by software: Central Tendency

Date of computation: Sun, 25 Nov 2007 14:07:23 -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/25/t1196024310owcs90so0ryv4pp.htm/, Retrieved Sun, 25 Nov 2007 21:58:30 +0100

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Dataseries X:

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6,4 6,3 6 5,3 5,2 5,1 5,4 5,8 5,6 5,6 5,4 5,4 5,5 5,5 5,3 5,7 5,6 5,5 5,6 5,9 6 7 6,6 6,6 6,3 6,3 6,3 6,3 6,2 6,2 6,3 6,4 6,4 7,8 7,7 7,7 7,7 7,7 7,6 7,5 7,4 7,4 7,5 7,6 7,6 8,1 7,8 8 7,9 7,9 7,8 6,7 6,6 6,6 7,7 7,9 8 7,7 7,5 7,6 7,8 7,8 7,7 7,4 7,5 7,2 7,5 7,6 7,6 7,8 7,7 7,7 8,2 8,2 8,1 7,8 7,8 7,7 6,7 6,7 6,7 7,2 6,9 6,8 7,2 7,1 6,9 6,9 6,7 6,5 6,6 6,6 6,5

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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 3 seconds R Server 'Herman Ole Andreas Wold' @ 193.190.124.10:1001 Central Tendency - Ungrouped Data Measure Value S.E. Value/S.E. Arithmetic Mean 6.89354838709677 0.0912461959938063 75.5488852112224 Geometric Mean 6.83540089721255 Harmonic Mean 6.77489875999796 Quadratic Mean 6.94888402891513 Winsorized Mean ( 1 / 31 ) 6.89462365591398 0.0910225222630902 75.7463480959784 Winsorized Mean ( 2 / 31 ) 6.89462365591398 0.0902747040235012 76.3738162367069 Winsorized Mean ( 3 / 31 ) 6.89462365591398 0.0902747040235012 76.3738162367068 Winsorized Mean ( 4 / 31 ) 6.89462365591398 0.0888654475517815 77.584976454392 Winsorized Mean ( 5 / 31 ) 6.89462365591398 0.0888654475517815 77.584976454392 Winsorized Mean ( 6 / 31 ) 6.88817204301075 0.0880261110052773 78.2514638480143 Winsorized Mean ( 7 / 31 ) 6.89569892473118 0.0866755755458285 79.5575787216455 Winsorized Mean ( 8 / 31 ) 6.89569892473118 0.0866755755458285 79.5575787216455 Winsorized Mean ( 9 / 31 ) 6.88602150537634 0.0855036422191926 80.5348325130257 Winsorized Mean ( 10 / 31 ) 6.89677419354839 0.083649958745861 82.448028629657 Winsorized Mean ( 11 / 31 ) 6.89677419354839 0.083649958745861 82.448028629657 Winsorized Mean ( 12 / 31 ) 6.89677419354839 0.083649958745861 82.448028629657 Winsorized Mean ( 13 / 31 ) 6.89677419354839 0.083649958745861 82.448028629657 Winsorized Mean ( 14 / 31 ) 6.91182795698925 0.0811593394992305 85.1636792467339 Winsorized Mean ( 15 / 31 ) 6.92795698924731 0.0785915877244603 88.1513809536017 Winsorized Mean ( 16 / 31 ) 6.94516129032258 0.075962210546547 91.429162478978 Winsorized Mean ( 17 / 31 ) 6.94516129032258 0.0710983116085494 97.6839130661921 Winsorized Mean ( 18 / 31 ) 6.94516129032258 0.0710983116085494 97.6839130661921 Winsorized Mean ( 19 / 31 ) 6.98602150537634 0.0654682739892679 106.708502908165 Winsorized Mean ( 20 / 31 ) 6.98602150537634 0.0654682739892679 106.708502908165 Winsorized Mean ( 21 / 31 ) 7.00860215053764 0.0626039278786162 111.951476337503 Winsorized Mean ( 22 / 31 ) 7.00860215053763 0.0626039278786162 111.951476337503 Winsorized Mean ( 23 / 31 ) 7.00860215053764 0.0626039278786162 111.951476337503 Winsorized Mean ( 24 / 31 ) 7.00860215053763 0.0626039278786162 111.951476337503 Winsorized Mean ( 25 / 31 ) 7.00860215053763 0.0626039278786162 111.951476337503 Winsorized Mean ( 26 / 31 ) 7.00860215053764 0.0626039278786162 111.951476337503 Winsorized Mean ( 27 / 31 ) 7.00860215053763 0.0556666410652373 125.903090547965 Winsorized Mean ( 28 / 31 ) 7.00860215053763 0.0556666410652373 125.903090547965 Winsorized Mean ( 29 / 31 ) 7.00860215053763 0.0556666410652373 125.903090547965 Winsorized Mean ( 30 / 31 ) 7.04086021505376 0.051920493375909 135.608499789809 Winsorized Mean ( 31 / 31 ) 7.04086021505376 0.051920493375909 135.608499789809 Trimmed Mean ( 1 / 31 ) 6.8989010989011 0.0899802516048008 76.6712803738483 Trimmed Mean ( 2 / 31 ) 6.90337078651685 0.0887785897239099 77.7594103261322 Trimmed Mean ( 3 / 31 ) 6.9080459770115 0.0878356340681166 78.6474196981865 Trimmed Mean ( 4 / 31 ) 6.91294117647059 0.0867364764407782 79.7005073314293 Trimmed Mean ( 5 / 31 ) 6.91807228915663 0.0859194207699971 80.51814394415 Trimmed Mean ( 6 / 31 ) 6.92345679012346 0.0849522670066112 81.4981993309802 Trimmed Mean ( 7 / 31 ) 6.93037974683544 0.0840077435415473 82.496915815956 Trimmed Mean ( 8 / 31 ) 6.93636363636364 0.0831889426185277 83.3808366596402 Trimmed Mean ( 9 / 31 ) 6.94266666666667 0.0822032930081355 84.4572816076792 Trimmed Mean ( 10 / 31 ) 6.95068493150685 0.0812320296874996 85.565816319575 Trimmed Mean ( 11 / 31 ) 6.95774647887324 0.0804079631038306 86.5305650124319 Trimmed Mean ( 12 / 31 ) 6.96521739130435 0.0793952549512002 87.7283837124205 Trimmed Mean ( 13 / 31 ) 6.96521739130435 0.0781566834132178 89.118640749876 Trimmed Mean ( 14 / 31 ) 6.98153846153846 0.0766451710711165 91.0890844650934 Trimmed Mean ( 15 / 31 ) 6.98888888888889 0.0752572346195108 92.866671546246 Trimmed Mean ( 16 / 31 ) 6.99508196721311 0.0740117487907237 94.513129084309 Trimmed Mean ( 17 / 31 ) 7 0.0729282330588739 95.9847744336409 Trimmed Mean ( 18 / 31 ) 7.00526315789474 0.0723496216183483 96.8251526573038 Trimmed Mean ( 19 / 31 ) 7.01090909090909 0.07157077104398 97.957713583956 Trimmed Mean ( 20 / 31 ) 7.01320754716981 0.0714781592921829 98.1167900323477 Trimmed Mean ( 21 / 31 ) 7.0156862745098 0.0712560997881342 98.4573432361517 Trimmed Mean ( 22 / 31 ) 7.01632653061225 0.0713301070604912 98.3641665455805 Trimmed Mean ( 23 / 31 ) 7.01702127659574 0.0712852246009161 98.435844396646 Trimmed Mean ( 24 / 31 ) 7.01777777777778 0.0710874329265714 98.7203713633418 Trimmed Mean ( 25 / 31 ) 7.01860465116279 0.0706915542026542 99.2849107694291 Trimmed Mean ( 26 / 31 ) 7.01860465116279 0.0700365333406754 100.213478828578 Trimmed Mean ( 27 / 31 ) 7.02051282051282 0.0690380683949977 101.690458376462 Trimmed Mean ( 28 / 31 ) 7.02162162162162 0.0690057158200889 101.754203085558 Trimmed Mean ( 29 / 31 ) 7.02285714285714 0.0687323042816167 102.176948907204 Trimmed Mean ( 30 / 31 ) 7.02424242424242 0.0681228701405041 103.111369350041 Trimmed Mean ( 31 / 31 ) 7.02258064516129 0.0679515446301299 103.346887600366 Median 6.9 Midrange 6.65 Midmean - Weighted Average at Xnp 7.06545454545454 Midmean - Weighted Average at X(n+1)p 7.06545454545454 Midmean - Empirical Distribution Function 7.06545454545454 Midmean - Empirical Distribution Function - Averaging 7.06545454545454 Midmean - Empirical Distribution Function - Interpolation 7.06545454545454 Midmean - Closest Observation 7.06545454545454 Midmean - True Basic - Statistics Graphics Toolkit 7.06545454545454 Midmean - MS Excel (old versions) 7.06545454545454 Number of observations 93

Charts produced by software:

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2007/Nov/25/t1196024310owcs90so0ryv4pp/1269v1196024838.png (open in new window) http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2007/Nov/25/t1196024310owcs90so0ryv4pp/1269v1196024838.ps (open in new window) http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2007/Nov/25/t1196024310owcs90so0ryv4pp/28a341196024838.png (open in new window) http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2007/Nov/25/t1196024310owcs90so0ryv4pp/28a341196024838.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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Software written by Ed van Stee & Patrick Wessa

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As a general rule, we do not log on-line behavior of individuals (other than normal logging of webserver 'hits'). However, in cases of abuse, hacking, unauthorized access, Denial of Service attacks, illegal copying, hotlinking, non-compliance with international webstandards (such as robots.txt), or any other harmful behavior, our system engineers are empowered to log, track, identify, publish, and ban misbehaving individuals - even if this leads to ban entire blocks of IP addresses, or disclosing user's identity.

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