paper univariate

*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: Sat, 04 Dec 2010 09:51:15 +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/Dec/04/t1291456199hpluf9285wk24hv.htm/, Retrieved Sat, 04 Dec 2010 10:50:01 +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/Dec/04/t1291456199hpluf9285wk24hv.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)

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Dataseries X:

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1,79 1,95 2,26 2,04 2,16 2,75 2,79 2,88 3,36 2,97 3,1 2,49 2,2 2,25 2,09 2,79 3,14 2,93 2,65 2,67 2,26 2,35 2,13 2,18 2,9 2,63 2,67 1,81 1,33 0,88 1,28 1,26 1,26 1,29 1,1 1,37 1,21 1,74 1,76 1,48 1,04 1,62 1,49 1,79 1,8 1,58 1,86 1,74 1,59 1,26 1,13 1,92 2,61 2,26 2,41 2,26 2,03 2,86 2,55 2,27 2,26 2,57 3,07 2,76 2,51 2,87 3,14 3,11 3,16 2,47 2,57 2,89

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 2.18888888888889 0.0750780906851099 29.1548289109996 Geometric Mean 2.08657236970492 Harmonic Mean 1.97447077267891 Quadratic Mean 2.27847353872426 Winsorized Mean ( 1 / 24 ) 2.18833333333333 0.0739998519644926 29.5721312305241 Winsorized Mean ( 2 / 24 ) 2.18944444444444 0.0735418740142282 29.7713985915133 Winsorized Mean ( 3 / 24 ) 2.19069444444444 0.073284054716027 29.8931937231544 Winsorized Mean ( 4 / 24 ) 2.19347222222222 0.0721026588716559 30.4215164398673 Winsorized Mean ( 5 / 24 ) 2.19625 0.0713239533289228 30.7926004868464 Winsorized Mean ( 6 / 24 ) 2.19375 0.0708832156587744 30.9487934430136 Winsorized Mean ( 7 / 24 ) 2.18402777777778 0.0692590620281086 31.5341807096867 Winsorized Mean ( 8 / 24 ) 2.18180555555556 0.0681442265912429 32.0174674318768 Winsorized Mean ( 9 / 24 ) 2.17930555555556 0.0673388432613144 32.3632757856942 Winsorized Mean ( 10 / 24 ) 2.18347222222222 0.0661077390777444 33.0289955863473 Winsorized Mean ( 11 / 24 ) 2.18805555555556 0.0647786162796669 33.7774358456365 Winsorized Mean ( 12 / 24 ) 2.20472222222222 0.06136806662922 35.9262128224267 Winsorized Mean ( 13 / 24 ) 2.20472222222222 0.0607935178154806 36.2657451229251 Winsorized Mean ( 14 / 24 ) 2.20861111111111 0.0559255966559899 39.4919543674562 Winsorized Mean ( 15 / 24 ) 2.21069444444444 0.0555968914100741 39.7629145870531 Winsorized Mean ( 16 / 24 ) 2.21069444444444 0.0535844755932576 41.2562485676843 Winsorized Mean ( 17 / 24 ) 2.23666666666667 0.0490105542592103 45.6364287340485 Winsorized Mean ( 18 / 24 ) 2.21666666666667 0.046149040719298 48.0327788425672 Winsorized Mean ( 19 / 24 ) 2.22194444444444 0.0453868086325075 48.9557321034689 Winsorized Mean ( 20 / 24 ) 2.22472222222222 0.0434325211075561 51.2224979229949 Winsorized Mean ( 21 / 24 ) 2.21888888888889 0.0426341001169705 52.0449331122545 Winsorized Mean ( 22 / 24 ) 2.21583333333333 0.0413763359144925 53.5531550670057 Winsorized Mean ( 23 / 24 ) 2.20625 0.039230887066188 56.2375761801599 Winsorized Mean ( 24 / 24 ) 2.22291666666667 0.0368897595856197 60.2583668648576 Trimmed Mean ( 1 / 24 ) 2.19085714285714 0.0729870639597987 30.0170608871713 Trimmed Mean ( 2 / 24 ) 2.19352941176471 0.0717670986518013 30.5645546911022 Trimmed Mean ( 3 / 24 ) 2.19575757575758 0.0705849884018913 31.1079965509882 Trimmed Mean ( 4 / 24 ) 2.19765625 0.069268693464084 31.7265439854079 Trimmed Mean ( 5 / 24 ) 2.19887096774194 0.0680976314238408 32.2899772248484 Trimmed Mean ( 6 / 24 ) 2.1995 0.0669021652677559 32.8763649307486 Trimmed Mean ( 7 / 24 ) 2.20068965517241 0.0655475619972625 33.5739360567571 Trimmed Mean ( 8 / 24 ) 2.20375 0.064281524935645 34.2827896850031 Trimmed Mean ( 9 / 24 ) 2.20740740740741 0.0629616064309 35.0595788852691 Trimmed Mean ( 10 / 24 ) 2.21173076923077 0.0614773629099807 35.9763442109469 Trimmed Mean ( 11 / 24 ) 2.2158 0.0598802137604735 37.0038759190708 Trimmed Mean ( 12 / 24 ) 2.21958333333333 0.0581465792195972 38.1722082902043 Trimmed Mean ( 13 / 24 ) 2.22152173913043 0.0567109523381672 39.1727108704419 Trimmed Mean ( 14 / 24 ) 2.22363636363636 0.0549573070021372 40.4611594878528 Trimmed Mean ( 15 / 24 ) 2.22547619047619 0.0537730879900023 41.3864309018288 Trimmed Mean ( 16 / 24 ) 2.22725 0.0522358893963302 42.6383091345713 Trimmed Mean ( 17 / 24 ) 2.22921052631579 0.0506281756869851 44.0310261246259 Trimmed Mean ( 18 / 24 ) 2.22833333333333 0.04954795656735 44.9732640397467 Trimmed Mean ( 19 / 24 ) 2.22970588235294 0.0486877447348406 45.7960395269116 Trimmed Mean ( 20 / 24 ) 2.230625 0.047574088018652 46.8873938082734 Trimmed Mean ( 21 / 24 ) 2.23133333333333 0.0464256501965086 48.0625112171533 Trimmed Mean ( 22 / 24 ) 2.23285714285714 0.0448597377337287 49.7741907478501 Trimmed Mean ( 23 / 24 ) 2.235 0.042817142505166 52.1987192333151 Trimmed Mean ( 24 / 24 ) 2.23875 0.0403441378210235 55.4913333364972 Median 2.26 Midrange 2.12 Midmean - Weighted Average at Xnp 2.21513513513513 Midmean - Weighted Average at X(n+1)p 2.21513513513513 Midmean - Empirical Distribution Function 2.21513513513513 Midmean - Empirical Distribution Function - Averaging 2.21513513513513 Midmean - Empirical Distribution Function - Interpolation 2.21513513513513 Midmean - Closest Observation 2.21513513513513 Midmean - True Basic - Statistics Graphics Toolkit 2.21513513513513 Midmean - MS Excel (old versions) 2.22921052631579 Number of observations 72

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/04/t1291456199hpluf9285wk24hv/12ajb1291456273.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/04/t1291456199hpluf9285wk24hv/12ajb1291456273.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/04/t1291456199hpluf9285wk24hv/2u1ie1291456273.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/04/t1291456199hpluf9285wk24hv/2u1ie1291456273.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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