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: Wed, 24 Dec 2008 04:41:56 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/Dec/24/t1230118985t7ckqtxa36drz3s.htm/, Retrieved Wed, 24 Dec 2008 12:43:07 +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/2008/Dec/24/t1230118985t7ckqtxa36drz3s.htm/}, year = {2008}, } @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 = {2008}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

Original text written by user:

IsPrivate?

No (this computation is public)

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Dataseries X:

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104.3 103.9 103.9 103.9 108.0 108.0 108.0 108.0 108.0 108.0 108.0 108.0 108.0 108.0 108.0 108.0 108.0 108.0 108.0 108.0 108.0 108.0 108.0 108.0 108.0 108.0 108.0 108.2 112.3 111.3 111.3 115.3 117.2 118.3 118.3 118.3 119.0 120.6 122.6 122.6 127.4 125.9 121.5 118.8 121.6 122.3 122.7 120.8 120.1 120.1 120.1 120.1 128.4 129.8 129.8 128.6 128.6 133.7 130.0 125.9 129.4 129.4 130.6 130.6 130.6 130.8 129.7 125.8 126.0 125.6 125.4 124.7 126.9 129.1

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 'George Udny Yule' @ 72.249.76.132 Central Tendency - Ungrouped Data Measure Value S.E. Value/S.E. Arithmetic Mean 117.65 1.06960505778403 109.993870301757 Geometric Mean 117.294798605810 Harmonic Mean 116.940438509309 Quadratic Mean 118.004400398941 Winsorized Mean ( 1 / 24 ) 117.610810810811 1.06224209373730 110.719403330195 Winsorized Mean ( 2 / 24 ) 117.605405405405 1.06132909328345 110.809555819833 Winsorized Mean ( 3 / 24 ) 117.621621621622 1.05849688798172 111.121367438213 Winsorized Mean ( 4 / 24 ) 117.821621621622 1.02810968091691 114.600245293423 Winsorized Mean ( 5 / 24 ) 117.781081081081 1.02133602216911 115.320598240468 Winsorized Mean ( 6 / 24 ) 117.764864864865 1.01869498667157 115.603656055718 Winsorized Mean ( 7 / 24 ) 117.764864864865 1.01869498667157 115.603656055718 Winsorized Mean ( 8 / 24 ) 117.754054054054 1.01695036184871 115.791348793061 Winsorized Mean ( 9 / 24 ) 117.717567567568 1.01112720430254 116.422114909634 Winsorized Mean ( 10 / 24 ) 117.717567567568 1.01112720430254 116.422114909634 Winsorized Mean ( 11 / 24 ) 117.672972972973 1.00412200674659 117.189915351263 Winsorized Mean ( 12 / 24 ) 117.591891891892 0.991636176050753 118.583705124805 Winsorized Mean ( 13 / 24 ) 117.591891891892 0.991636176050753 118.583705124805 Winsorized Mean ( 14 / 24 ) 117.554054054054 0.98590809374133 119.234292527166 Winsorized Mean ( 15 / 24 ) 117.351351351351 0.956031232388305 122.748449397611 Winsorized Mean ( 16 / 24 ) 117.243243243243 0.940645168168854 124.641307063193 Winsorized Mean ( 17 / 24 ) 117.036486486486 0.91218168832886 128.303920133390 Winsorized Mean ( 18 / 24 ) 117.012162162162 0.908915389421166 128.738234079940 Winsorized Mean ( 19 / 24 ) 117.012162162162 0.908915389421166 128.738234079940 Winsorized Mean ( 20 / 24 ) 116.985135135135 0.90530274274703 129.222115002279 Winsorized Mean ( 21 / 24 ) 116.928378378378 0.897762491318954 130.244223287378 Winsorized Mean ( 22 / 24 ) 116.868918918919 0.889924526781863 131.324528543493 Winsorized Mean ( 23 / 24 ) 116.651351351351 0.861714458297418 135.371236060994 Winsorized Mean ( 24 / 24 ) 116.002702702703 0.782000806210822 148.340899115939 Trimmed Mean ( 1 / 24 ) 117.618055555556 1.05904023633607 111.060988544189 Trimmed Mean ( 2 / 24 ) 117.625714285714 1.05456649854436 111.539399789464 Trimmed Mean ( 3 / 24 ) 117.636764705882 1.04913925255633 112.126931119247 Trimmed Mean ( 4 / 24 ) 117.642424242424 1.04322000309303 112.768566451590 Trimmed Mean ( 5 / 24 ) 117.590625 1.04556347009164 112.466271406453 Trimmed Mean ( 6 / 24 ) 117.545161290323 1.04899709707606 112.054801312572 Trimmed Mean ( 7 / 24 ) 117.5 1.05234202083427 111.655714277045 Trimmed Mean ( 8 / 24 ) 117.451724137931 1.05484863107395 111.344623937515 Trimmed Mean ( 9 / 24 ) 117.401785714286 1.05669016171554 111.103320507577 Trimmed Mean ( 10 / 24 ) 117.353703703704 1.05849765297918 110.868175638754 Trimmed Mean ( 11 / 24 ) 117.301923076923 1.05895054050907 110.771861942232 Trimmed Mean ( 12 / 24 ) 117.252 1.05908023245433 110.711158991495 Trimmed Mean ( 13 / 24 ) 117.208333333333 1.05973548709122 110.601498922196 Trimmed Mean ( 14 / 24 ) 117.160869565217 1.05841682090360 110.694451610466 Trimmed Mean ( 15 / 24 ) 117.113636363636 1.05568325474338 110.936339889284 Trimmed Mean ( 16 / 24 ) 117.085714285714 1.05583699990419 110.893740507615 Trimmed Mean ( 17 / 24 ) 117.0675 1.05635675002971 110.821935862773 Trimmed Mean ( 18 / 24 ) 117.071052631579 1.05966330232397 110.47948190225 Trimmed Mean ( 19 / 24 ) 117.077777777778 1.06105002732851 110.341430434297 Trimmed Mean ( 20 / 24 ) 117.085294117647 1.05900040610178 110.562086136154 Trimmed Mean ( 21 / 24 ) 117.096875 1.05293832699912 111.209623581399 Trimmed Mean ( 22 / 24 ) 117.116666666667 1.04207003684293 112.388479205760 Trimmed Mean ( 23 / 24 ) 117.146428571429 1.02417190891831 114.381606790166 Trimmed Mean ( 24 / 24 ) 117.207692307692 1.0016743379412 117.011775053154 Median 118.9 Midrange 118.8 Midmean - Weighted Average at Xnp 114.628846153846 Midmean - Weighted Average at X(n+1)p 114.628846153846 Midmean - Empirical Distribution Function 114.628846153846 Midmean - Empirical Distribution Function - Averaging 114.628846153846 Midmean - Empirical Distribution Function - Interpolation 114.628846153846 Midmean - Closest Observation 114.628846153846 Midmean - True Basic - Statistics Graphics Toolkit 114.628846153846 Midmean - MS Excel (old versions) 114.628846153846 Number of observations 74

Charts produced by software:

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Dec/24/t1230118985t7ckqtxa36drz3s/18ycy1230118914.png (open in new window) http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Dec/24/t1230118985t7ckqtxa36drz3s/18ycy1230118914.ps (open in new window) http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Dec/24/t1230118985t7ckqtxa36drz3s/2j2h71230118914.png (open in new window) http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Dec/24/t1230118985t7ckqtxa36drz3s/2j2h71230118914.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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