central tendency

*Unverified author*

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

Title produced by software: Central Tendency

Date of computation: Wed, 03 Jun 2009 04:08:31 -0600

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2009/Jun/03/t124402391794d3fgydjr601ky.htm/, Retrieved Wed, 03 Jun 2009 12:12:00 +0200

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/Jun/03/t124402391794d3fgydjr601ky.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:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

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62,9 64,92 64 65,01 47 61,35 61 61,71 59,8 57,55 40 45,01 38 40,5 44,5 40,98 36,51 37,49 34,95 38,21 33,63 33,49 28 28 26,75 26,62 29,5 31 34 35 35,9 32,8 33 33,5 34,35 32,6 28,6 27,45 26,51 26,9 24 19,61 20 19,2 20,44 20,3 20,4 23,15 23,78 22 23,4 23,75 24,78 25,28 29,57 22,05 22,75 20,85 20,35 16,8 12,99 9,66 11 12,51 12,7 12,82 12,9 14,5 14,13 14 13,59 13,87

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 30.7794444444444 1.75163406706027 17.5718462110645 Geometric Mean 27.5081595543625 Harmonic Mean 24.5329605073631 Quadratic Mean 34.1352890064747 Winsorized Mean ( 1 / 24 ) 30.7968055555556 1.74822572504124 17.616034997328 Winsorized Mean ( 2 / 24 ) 30.8131944444444 1.73480765019989 17.7617353952146 Winsorized Mean ( 3 / 24 ) 30.7752777777778 1.72143547675615 17.8776830112566 Winsorized Mean ( 4 / 24 ) 30.7158333333333 1.70328795486810 18.033259288628 Winsorized Mean ( 5 / 24 ) 30.6963888888889 1.69608079835610 18.0984236827874 Winsorized Mean ( 6 / 24 ) 30.6747222222222 1.68756986723993 18.1768605956396 Winsorized Mean ( 7 / 24 ) 30.6163888888889 1.64972390995906 18.5584925477917 Winsorized Mean ( 8 / 24 ) 30.3975 1.58383066997645 19.1923925809897 Winsorized Mean ( 9 / 24 ) 29.095 1.29011107296801 22.5523217416190 Winsorized Mean ( 10 / 24 ) 28.8366666666667 1.23456378159633 23.3577779427322 Winsorized Mean ( 11 / 24 ) 28.8152777777778 1.21081784354354 23.7981938665917 Winsorized Mean ( 12 / 24 ) 28.6119444444445 1.04557150995479 27.3648853015147 Winsorized Mean ( 13 / 24 ) 28.9586111111111 0.965599951864391 29.9902781221121 Winsorized Mean ( 14 / 24 ) 28.9411111111111 0.938083809991358 30.8513064641610 Winsorized Mean ( 15 / 24 ) 28.6494444444444 0.866674338375419 33.0567586645612 Winsorized Mean ( 16 / 24 ) 28.6694444444444 0.850133835751127 33.723448284015 Winsorized Mean ( 17 / 24 ) 28.5608333333333 0.83006143478917 34.408095758102 Winsorized Mean ( 18 / 24 ) 28.3283333333333 0.791906659941352 35.772313539365 Winsorized Mean ( 19 / 24 ) 28.1779166666667 0.767287331372659 36.7240739088668 Winsorized Mean ( 20 / 24 ) 28.0418055555556 0.716303306902881 39.1479493188457 Winsorized Mean ( 21 / 24 ) 28.3626388888889 0.668037850251294 42.4566345727563 Winsorized Mean ( 22 / 24 ) 28.1945833333333 0.640842710344285 43.9961052505164 Winsorized Mean ( 23 / 24 ) 28.3063888888889 0.595960361243544 47.4971000249483 Winsorized Mean ( 24 / 24 ) 28.3163888888889 0.562060097699073 50.3796462421168 Trimmed Mean ( 1 / 24 ) 30.5921428571429 1.70643240368861 17.9275445022113 Trimmed Mean ( 2 / 24 ) 30.3754411764706 1.65633151392266 18.3389864415083 Trimmed Mean ( 3 / 24 ) 30.1366666666667 1.60459261229679 18.7815065554424 Trimmed Mean ( 4 / 24 ) 29.8971875 1.54819996716537 19.3109340744525 Trimmed Mean ( 5 / 24 ) 29.6595161290323 1.48651673931274 19.9523593274468 Trimmed Mean ( 6 / 24 ) 29.4106666666667 1.41304683163696 20.8136531700054 Trimmed Mean ( 7 / 24 ) 29.1491379310345 1.32417051127146 22.0131302448698 Trimmed Mean ( 8 / 24 ) 28.8796428571429 1.22357818267141 23.6026134383098 Trimmed Mean ( 9 / 24 ) 28.6266666666667 1.1153693455887 25.6656387230701 Trimmed Mean ( 10 / 24 ) 28.5546153846154 1.06512642596240 26.8086629799037 Trimmed Mean ( 11 / 24 ) 28.514 1.01614461448704 28.0609665135056 Trimmed Mean ( 12 / 24 ) 28.4729166666667 0.959674930110947 29.6693346604094 Trimmed Mean ( 13 / 24 ) 28.4547826086957 0.928617107988711 30.6421046563807 Trimmed Mean ( 14 / 24 ) 28.3913636363636 0.906316604516615 31.3260989535839 Trimmed Mean ( 15 / 24 ) 28.3240476190476 0.882809174982714 32.0839977898986 Trimmed Mean ( 16 / 24 ) 28.285 0.86785965721157 32.5916751227723 Trimmed Mean ( 17 / 24 ) 28.2394736842105 0.850379042535814 33.2081016484143 Trimmed Mean ( 18 / 24 ) 28.2016666666667 0.830410923330914 33.9610979026446 Trimmed Mean ( 19 / 24 ) 28.1867647058824 0.811554121914538 34.7318360473445 Trimmed Mean ( 20 / 24 ) 28.1878125 0.790228470226333 35.6704593190962 Trimmed Mean ( 21 / 24 ) 28.2053333333333 0.772521432702468 36.5107453843244 Trimmed Mean ( 22 / 24 ) 28.1860714285714 0.75872322774446 37.1493456347228 Trimmed Mean ( 23 / 24 ) 28.185 0.743747527468418 37.8959243010012 Trimmed Mean ( 24 / 24 ) 28.1691666666667 0.732519455315324 38.4551788519266 Median 27.725 Midrange 37.335 Midmean - Weighted Average at Xnp 27.9894594594595 Midmean - Weighted Average at X(n+1)p 28.2016666666667 Midmean - Empirical Distribution Function 27.9894594594595 Midmean - Empirical Distribution Function - Averaging 28.2016666666667 Midmean - Empirical Distribution Function - Interpolation 28.2016666666667 Midmean - Closest Observation 27.9894594594595 Midmean - True Basic - Statistics Graphics Toolkit 28.2016666666667 Midmean - MS Excel (old versions) 28.2394736842105 Number of observations 72

Charts produced by software:

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2009/Jun/03/t124402391794d3fgydjr601ky/1bnxg1244023709.png (open in new window) http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2009/Jun/03/t124402391794d3fgydjr601ky/1bnxg1244023709.ps (open in new window) http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2009/Jun/03/t124402391794d3fgydjr601ky/2ipy71244023709.png (open in new window) http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2009/Jun/03/t124402391794d3fgydjr601ky/2ipy71244023709.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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