paper 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: Mon, 28 Dec 2009 07:07:17 -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/28/t1262009263aervxshguh835ib.htm/, Retrieved Mon, 28 Dec 2009 15:07:45 +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/28/t1262009263aervxshguh835ib.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:

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User-defined keywords:

Dataseries X:

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111.6 104.6 91.6 98.3 97.7 106.3 102.3 106.6 108.1 93.8 88.2 108.9 114.2 102.5 94.2 97.4 98.5 106.5 102.9 97.1 103.7 93.4 85.8 108.6 110.2 101.2 101.2 96.9 99.4 118.7 108.0 101.2 119.9 94.8 95.3 118.0 115.9 111.4 108.2 108.8 109.5 124.8 115.3 109.5 124.2 92.9 98.4 120.9 111.7 116.1 109.4 111.7 114.3 133.7 114.3 126.5 131.0 104.0 108.9 128.5 132.4 128.0 116.4 120.9 118.6 133.1 121.1 127.6 135.4 114.9 114.3 128.9 138.9 129.4 115.0 128.0 127.0 128.8 137.9 128.4 135.9 122.2 113.1 136.2 138.0 115.2 111.0 99.2 102.4 112.7 105.5 98.3 116.4 97.4 93.3 117.4

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 'Gwilym Jenkins' @ 72.249.127.135 Central Tendency - Ungrouped Data Measure Value S.E. Value/S.E. Arithmetic Mean 112.257291666667 1.33730445757966 83.9429578136883 Geometric Mean 111.505166952123 Harmonic Mean 110.759254200663 Quadratic Mean 113.011485865376 Winsorized Mean ( 1 / 32 ) 112.272916666667 1.33038254553747 84.3914534531936 Winsorized Mean ( 2 / 32 ) 112.341666666667 1.31733771485172 85.27932161975 Winsorized Mean ( 3 / 32 ) 112.329166666667 1.30021499290609 86.3927637194843 Winsorized Mean ( 4 / 32 ) 112.333333333333 1.29521019315185 86.729809514527 Winsorized Mean ( 5 / 32 ) 112.3125 1.28945729268135 87.1005970011249 Winsorized Mean ( 6 / 32 ) 112.23125 1.26611222211589 88.6424189259011 Winsorized Mean ( 7 / 32 ) 112.216666666667 1.25393211946816 89.4918193133627 Winsorized Mean ( 8 / 32 ) 112.208333333333 1.23631692221847 90.7601694329193 Winsorized Mean ( 9 / 32 ) 112.123958333333 1.20731664256744 92.8703824498716 Winsorized Mean ( 10 / 32 ) 112.123958333333 1.15670409034174 96.9340034928098 Winsorized Mean ( 11 / 32 ) 112.089583333333 1.14460450455899 97.9286582281284 Winsorized Mean ( 12 / 32 ) 112.114583333333 1.13753540446798 98.559203426085 Winsorized Mean ( 13 / 32 ) 112.073958333333 1.13129452010067 99.0670036334665 Winsorized Mean ( 14 / 32 ) 112.103125 1.12312656205653 99.8134393640644 Winsorized Mean ( 15 / 32 ) 112.134375 1.10108482006423 101.839906387465 Winsorized Mean ( 16 / 32 ) 112.134375 1.10108482006423 101.839906387465 Winsorized Mean ( 17 / 32 ) 112.08125 1.08805245344485 103.010888533124 Winsorized Mean ( 18 / 32 ) 111.9875 1.06880328429806 104.778401830556 Winsorized Mean ( 19 / 32 ) 112.027083333333 1.03597830614550 108.136514701882 Winsorized Mean ( 20 / 32 ) 111.714583333333 0.979532849594985 114.048838055329 Winsorized Mean ( 21 / 32 ) 111.977083333333 0.910390169834017 122.999003112862 Winsorized Mean ( 22 / 32 ) 111.51875 0.847530355120616 131.580832858936 Winsorized Mean ( 23 / 32 ) 111.255208333333 0.813244085453798 136.804202235608 Winsorized Mean ( 24 / 32 ) 111.480208333333 0.771715762986201 144.457601723663 Winsorized Mean ( 25 / 32 ) 111.50625 0.768461113884516 145.103308398188 Winsorized Mean ( 26 / 32 ) 111.2625 0.730775918447274 152.252554020125 Winsorized Mean ( 27 / 32 ) 111.0375 0.675360933391085 164.412086204727 Winsorized Mean ( 28 / 32 ) 111.241666666667 0.642643366943146 173.100155372036 Winsorized Mean ( 29 / 32 ) 111.151041666667 0.609810142282701 182.271552996143 Winsorized Mean ( 30 / 32 ) 111.151041666667 0.564777542744702 196.804995337626 Winsorized Mean ( 31 / 32 ) 111.11875 0.49249479581363 225.624211554206 Winsorized Mean ( 32 / 32 ) 111.385416666667 0.460981930532834 241.626426740675 Trimmed Mean ( 1 / 32 ) 112.255319148936 1.30554281548306 85.9836367047075 Trimmed Mean ( 2 / 32 ) 112.236956521739 1.27729419181323 87.8708736335903 Trimmed Mean ( 3 / 32 ) 112.181111111111 1.25291160970515 89.5363330039787 Trimmed Mean ( 4 / 32 ) 112.127272727273 1.23224456369943 90.9943334549157 Trimmed Mean ( 5 / 32 ) 112.069767441860 1.21014755266741 92.6083494486568 Trimmed Mean ( 6 / 32 ) 112.014285714286 1.18631896048667 94.4217275835614 Trimmed Mean ( 7 / 32 ) 111.971951219512 1.16468960677393 96.1388773182782 Trimmed Mean ( 8 / 32 ) 111.93 1.14235959445263 97.9814066809954 Trimmed Mean ( 9 / 32 ) 111.887179487179 1.12007289939000 99.8927655049188 Trimmed Mean ( 10 / 32 ) 111.853947368421 1.09979175188166 101.704661066104 Trimmed Mean ( 11 / 32 ) 111.818918918919 1.08529376364588 103.031015808365 Trimmed Mean ( 12 / 32 ) 111.786111111111 1.07014600081013 104.458747709646 Trimmed Mean ( 13 / 32 ) 111.748571428571 1.05322091583158 106.101739671909 Trimmed Mean ( 14 / 32 ) 111.713235294118 1.0340152148272 108.038289661711 Trimmed Mean ( 15 / 32 ) 111.672727272727 1.01231241999964 110.314488952696 Trimmed Mean ( 16 / 32 ) 111.6265625 0.989872431925253 112.768634522826 Trimmed Mean ( 17 / 32 ) 111.577419354839 0.962670299492679 115.904084101939 Trimmed Mean ( 18 / 32 ) 111.53 0.93192074297466 119.677559321193 Trimmed Mean ( 19 / 32 ) 111.487931034483 0.897870295909779 124.169305457996 Trimmed Mean ( 20 / 32 ) 111.439285714286 0.862025998385428 129.27601478727 Trimmed Mean ( 21 / 32 ) 111.414814814815 0.828651027609574 134.453239183466 Trimmed Mean ( 22 / 32 ) 111.365384615385 0.80000416955637 139.206005235124 Trimmed Mean ( 23 / 32 ) 111.352 0.776324358888784 143.434891260384 Trimmed Mean ( 24 / 32 ) 111.360416666667 0.752794672537726 147.929336815393 Trimmed Mean ( 25 / 32 ) 111.35 0.730751383536879 152.377405652056 Trimmed Mean ( 26 / 32 ) 111.336363636364 0.702630897942179 158.456401451229 Trimmed Mean ( 27 / 32 ) 111.342857142857 0.67416350592377 165.15705190878 Trimmed Mean ( 28 / 32 ) 111.37 0.649064415435491 171.585434899056 Trimmed Mean ( 29 / 32 ) 111.381578947368 0.62322128416395 178.719151251046 Trimmed Mean ( 30 / 32 ) 111.402777777778 0.596144838466572 186.871999201289 Trimmed Mean ( 31 / 32 ) 111.426470588235 0.570772201447535 195.220563134726 Trimmed Mean ( 32 / 32 ) 111.45625 0.555070787043448 200.796461643505 Median 111.5 Midrange 112.35 Midmean - Weighted Average at Xnp 110.762745098039 Midmean - Weighted Average at X(n+1)p 111.360416666667 Midmean - Empirical Distribution Function 110.762745098039 Midmean - Empirical Distribution Function - Averaging 111.360416666667 Midmean - Empirical Distribution Function - Interpolation 111.360416666667 Midmean - Closest Observation 110.762745098039 Midmean - True Basic - Statistics Graphics Toolkit 111.360416666667 Midmean - MS Excel (old versions) 110.961538461538 Number of observations 96

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/28/t1262009263aervxshguh835ib/1lepq1262009234.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/28/t1262009263aervxshguh835ib/1lepq1262009234.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/28/t1262009263aervxshguh835ib/2ebjl1262009234.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/28/t1262009263aervxshguh835ib/2ebjl1262009234.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')

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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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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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