*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: Tue, 16 Nov 2010 19:10:53 +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/16/t12899345442bvdzits9jhyqq9.htm/, Retrieved Tue, 16 Nov 2010 20:09:06 +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/16/t12899345442bvdzits9jhyqq9.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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87.48 82.39 83.48 79.31 78.16 72.77 72.45 68.46 67.62 68.76 70.07 68.55 65.3 58.96 59.17 62.37 66.28 55.62 55.23 55.85 56.75 50.89 53.88 52.95 55.08 53.61 58.78 61.85 55.91 53.32 46.41 44.57 50 50 53.36 46.23 50.45 49.07 45.85 48.45 49.96 46.53 50.51 47.58 48.05 46.84 47.67 49.16 55.54 55.82 58.22 56.19 57.77 63.19 54.76 55.74 62.54 61.39 69.6 79.23 80 93.68 107.63 100.18 97.3 90.45 80.64 80.58 75.82 85.59 89.35 89.42 104.73 95.32 89.27 90.44 86.97 79.98 81.22 87.35 83.64 82.22 94.4 102.18

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 67.6230952380952 1.87313448964542 36.1015696480481 Geometric Mean 65.5687534182011 Harmonic Mean 63.6541023117357 Quadratic Mean 69.7430966715225 Winsorized Mean ( 1 / 28 ) 67.6038095238095 1.86234346340732 36.3003983164966 Winsorized Mean ( 2 / 28 ) 67.5521428571429 1.84692708979717 36.5754247854806 Winsorized Mean ( 3 / 28 ) 67.4871428571428 1.83027881127603 36.8726023824165 Winsorized Mean ( 4 / 28 ) 67.3557142857143 1.80098273744877 37.39942248482 Winsorized Mean ( 5 / 28 ) 67.2563095238095 1.77538746572803 37.8826091893297 Winsorized Mean ( 6 / 28 ) 67.2434523809524 1.75577804822245 38.2983785729806 Winsorized Mean ( 7 / 28 ) 67.1909523809524 1.74368456057525 38.5338919092027 Winsorized Mean ( 8 / 28 ) 66.9195238095238 1.68472888452859 39.7212420491317 Winsorized Mean ( 9 / 28 ) 66.9613095238095 1.67880997982442 39.8861755222666 Winsorized Mean ( 10 / 28 ) 66.9136904761905 1.64887770016523 40.5813544991755 Winsorized Mean ( 11 / 28 ) 66.9163095238095 1.64583697237529 40.6579209526659 Winsorized Mean ( 12 / 28 ) 67.0191666666667 1.62932129174236 41.1331804269235 Winsorized Mean ( 13 / 28 ) 66.7483333333333 1.58388964922490 42.1420351891292 Winsorized Mean ( 14 / 28 ) 66.7266666666667 1.58047807292416 42.2192928897841 Winsorized Mean ( 15 / 28 ) 66.7391666666667 1.55966208448977 42.7907861134546 Winsorized Mean ( 16 / 28 ) 66.4877380952381 1.51774068461923 43.8070473889409 Winsorized Mean ( 17 / 28 ) 66.17 1.44937336846525 45.6542126685189 Winsorized Mean ( 18 / 28 ) 66.5771428571428 1.39023169450734 47.8892425774654 Winsorized Mean ( 19 / 28 ) 66.4142857142857 1.34466954359074 49.3907860342669 Winsorized Mean ( 20 / 28 ) 66.3833333333333 1.33776886652974 49.6224235697278 Winsorized Mean ( 21 / 28 ) 66.1958333333333 1.29513984262048 51.1109543193405 Winsorized Mean ( 22 / 28 ) 66.1146428571429 1.26576451000627 52.232972511463 Winsorized Mean ( 23 / 28 ) 66.3391666666667 1.23585335352810 53.6788337202648 Winsorized Mean ( 24 / 28 ) 66.264880952381 1.20266462254508 55.0983871232123 Winsorized Mean ( 25 / 28 ) 66.3035714285714 1.19685600811267 55.398118887439 Winsorized Mean ( 26 / 28 ) 66.1921428571429 1.15775686078413 57.1727493908452 Winsorized Mean ( 27 / 28 ) 66.1921428571429 1.15140006908485 57.4883957665152 Winsorized Mean ( 28 / 28 ) 65.8754761904762 1.09860251818620 59.9629757805736 Trimmed Mean ( 1 / 28 ) 67.4163414634146 1.83342305366563 36.7707503888024 Trimmed Mean ( 2 / 28 ) 67.2195 1.79953269162520 37.3538643186818 Trimmed Mean ( 3 / 28 ) 67.0403846153846 1.76927899625158 37.8913584332472 Trimmed Mean ( 4 / 28 ) 66.8757894736842 1.74088932972674 38.4147276519763 Trimmed Mean ( 5 / 28 ) 66.7395945945946 1.71745448955771 38.8596000653163 Trimmed Mean ( 6 / 28 ) 66.6190277777778 1.69691614978303 39.2588801669993 Trimmed Mean ( 7 / 28 ) 66.4941428571429 1.67707537127777 39.6488696906215 Trimmed Mean ( 8 / 28 ) 66.3711764705882 1.65575079359551 40.0852451512109 Trimmed Mean ( 9 / 28 ) 66.2839393939394 1.64255508901720 40.3541651888214 Trimmed Mean ( 10 / 28 ) 66.18515625 1.62692011089405 40.6812576762781 Trimmed Mean ( 11 / 28 ) 66.0864516129032 1.61290943414987 40.9734422861355 Trimmed Mean ( 12 / 28 ) 65.9808333333333 1.59540465248755 41.3568013797982 Trimmed Mean ( 13 / 28 ) 65.8555172413793 1.57591285969239 41.7888063012786 Trimmed Mean ( 14 / 28 ) 65.7525 1.55951680932635 42.1620976489523 Trimmed Mean ( 15 / 28 ) 65.6442592592593 1.53864946639022 42.6635570304816 Trimmed Mean ( 16 / 28 ) 65.5263461538461 1.51528708480772 43.2435192055775 Trimmed Mean ( 17 / 28 ) 65.4254 1.49293484992562 43.8233456759747 Trimmed Mean ( 18 / 28 ) 65.34875 1.47637451357844 44.262989775953 Trimmed Mean ( 19 / 28 ) 65.2241304347826 1.46359631112570 44.5642899882799 Trimmed Mean ( 20 / 28 ) 65.1045454545455 1.45350267523233 44.7914864994238 Trimmed Mean ( 21 / 28 ) 64.9766666666667 1.43896573172571 45.155117480624 Trimmed Mean ( 22 / 28 ) 64.85475 1.42594327698077 45.4819985107125 Trimmed Mean ( 23 / 28 ) 64.7281578947368 1.41163171705277 45.8534312546316 Trimmed Mean ( 24 / 28 ) 64.5647222222222 1.39447187967067 46.300483475845 Trimmed Mean ( 25 / 28 ) 64.389705882353 1.3748719263182 46.8332392638081 Trimmed Mean ( 26 / 28 ) 64.18875 1.34428162776929 47.7494809674034 Trimmed Mean ( 27 / 28 ) 63.973 1.30770784374710 48.9199482177088 Trimmed Mean ( 28 / 28 ) 63.7264285714286 1.25120885613190 50.9318873976311 Median 62.455 Midrange 76.1 Midmean - Weighted Average at Xnp 64.706511627907 Midmean - Weighted Average at X(n+1)p 64.9766666666666 Midmean - Empirical Distribution Function 64.706511627907 Midmean - Empirical Distribution Function - Averaging 64.9766666666666 Midmean - Empirical Distribution Function - Interpolation 64.9766666666666 Midmean - Closest Observation 64.706511627907 Midmean - True Basic - Statistics Graphics Toolkit 64.9766666666666 Midmean - MS Excel (old versions) 65.1045454545454 Number of observations 84

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Nov/16/t12899345442bvdzits9jhyqq9/177go1289934652.png (open in new window) http://www.freestatistics.org/blog/date/2010/Nov/16/t12899345442bvdzits9jhyqq9/177go1289934652.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Nov/16/t12899345442bvdzits9jhyqq9/20hfr1289934652.png (open in new window) http://www.freestatistics.org/blog/date/2010/Nov/16/t12899345442bvdzits9jhyqq9/20hfr1289934652.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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