WS 3 Part 2 Y[t]/X[t]=c+e[t] 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, 19 Oct 2009 11:32:48 -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/Oct/19/t1255973766xkn5pjp6yeg6q65.htm/, Retrieved Mon, 19 Oct 2009 19:36:06 +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/Oct/19/t1255973766xkn5pjp6yeg6q65.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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1.049659864 1.039117043 1.076570916 1.084210526 0.984090909 1.033829974 1.050407978 1.072569779 1.064566929 1.149788584 1.141293014 1.19226361 1.121692607 1.116106014 1.090781387 1.147713226 1.028422153 1.067844523 1.054957507 1.067095588 1.069110459 1.127777778 1.134955313 1.157402597 1.134756703 1.150955414 1.143106457 1.217798595 1.083383085 1.119163763 1.136824034 1.134100974 1.140686275 1.140849057 1.126210826 1.12037037 1.140904807 1.167612077 1.155631399 1.241945946 1.09731286 1.145333333 1.134803922 1.142453663 1.1496 1.156097561 1.126152482 1.172586521 1.175703325 1.176626948 1.172562554 1.2728125 1.133066132 1.16609589 1.192739844 1.159356137 1.176564157 1.09043956 1.09860515 1.092822502

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 'Sir Ronald Aylmer Fisher' @ 193.190.124.24 Central Tendency - Ungrouped Data Measure Value S.E. Value/S.E. Arithmetic Mean 1.12513771835 0.0068536236503665 164.166837245263 Geometric Mean 1.12390031671505 Harmonic Mean 1.12265584905947 Quadratic Mean 1.12636860863802 Winsorized Mean ( 1 / 20 ) 1.12536212985 0.0064566007417953 174.296378985498 Winsorized Mean ( 2 / 20 ) 1.12473747885 0.00618495539054613 181.850540194549 Winsorized Mean ( 3 / 20 ) 1.12374889475 0.00583316230407758 192.648315985390 Winsorized Mean ( 4 / 20 ) 1.12442000055 0.00566156743098985 198.605777332128 Winsorized Mean ( 5 / 20 ) 1.12317928821667 0.00540542830946913 207.787287873026 Winsorized Mean ( 6 / 20 ) 1.12362796201667 0.0053025151070727 211.9047167858 Winsorized Mean ( 7 / 20 ) 1.12464863085 0.005049073635857 222.743558910111 Winsorized Mean ( 8 / 20 ) 1.12457021151667 0.00491157326129711 228.963338565712 Winsorized Mean ( 9 / 20 ) 1.12467895671667 0.00488876832628569 230.053641664619 Winsorized Mean ( 10 / 20 ) 1.12406486655 0.00471398811133631 238.453055035676 Winsorized Mean ( 11 / 20 ) 1.1244211076 0.004546656945007 247.307224010997 Winsorized Mean ( 12 / 20 ) 1.1238733844 0.00419099749945942 268.163697197377 Winsorized Mean ( 13 / 20 ) 1.12492608735 0.00385413048144600 291.875454856928 Winsorized Mean ( 14 / 20 ) 1.12481464851667 0.00377576930065194 297.903436081874 Winsorized Mean ( 15 / 20 ) 1.12625536651667 0.00348131766175441 323.514104699395 Winsorized Mean ( 16 / 20 ) 1.12509959105 0.00329301155283274 341.662813202752 Winsorized Mean ( 17 / 20 ) 1.12534730513333 0.00314752227303108 357.534342100022 Winsorized Mean ( 18 / 20 ) 1.12663783733333 0.00290676223556971 387.592016831235 Winsorized Mean ( 19 / 20 ) 1.12644958406667 0.00275769373245786 408.475230881673 Winsorized Mean ( 20 / 20 ) 1.13148990773333 0.00169075030405231 669.22354236544 Trimmed Mean ( 1 / 20 ) 1.12502344296552 0.00613861448622246 183.269929312311 Trimmed Mean ( 2 / 20 ) 1.12466056416071 0.00574254401475255 195.847095167485 Trimmed Mean ( 3 / 20 ) 1.12461783377778 0.00543828833268242 206.796286805754 Trimmed Mean ( 4 / 20 ) 1.12495204109615 0.00523529295026842 214.878527674077 Trimmed Mean ( 5 / 20 ) 1.12511165326 0.0050501872291551 222.786126970630 Trimmed Mean ( 6 / 20 ) 1.12559474452083 0.0049033247548502 229.557453523231 Trimmed Mean ( 7 / 20 ) 1.12602230593478 0.00474693807956907 237.210236801109 Trimmed Mean ( 8 / 20 ) 1.12628990497727 0.004620712624839 243.748096110287 Trimmed Mean ( 9 / 20 ) 1.12659699309524 0.00449187932760049 250.807493018085 Trimmed Mean ( 10 / 20 ) 1.126916665825 0.00432464510321341 260.580149105796 Trimmed Mean ( 11 / 20 ) 1.12736694992105 0.00414620835390717 271.903110913054 Trimmed Mean ( 12 / 20 ) 1.12781328966667 0.00395016323062783 285.510553316404 Trimmed Mean ( 13 / 20 ) 1.1283926875 0.0037796515051557 298.5441080906 Trimmed Mean ( 14 / 20 ) 1.12889267790625 0.00364132725049951 310.02230786903 Trimmed Mean ( 15 / 20 ) 1.12947525353333 0.00345852400883232 326.57724816971 Trimmed Mean ( 16 / 20 ) 1.12993523739286 0.00329191613251635 343.245450949302 Trimmed Mean ( 17 / 20 ) 1.13063268638462 0.00308969378639615 365.93680945431 Trimmed Mean ( 18 / 20 ) 1.13140994833333 0.0028213655847159 401.015010058422 Trimmed Mean ( 19 / 20 ) 1.13213299545455 0.00250064334901688 452.736691099768 Trimmed Mean ( 20 / 20 ) 1.1330303762 0.00199078170227982 569.138431854416 Median 1.1347803125 Midrange 1.1284517045 Midmean - Weighted Average at Xnp 1.12801510103226 Midmean - Weighted Average at X(n+1)p 1.12947525353333 Midmean - Empirical Distribution Function 1.12801510103226 Midmean - Empirical Distribution Function - Averaging 1.12947525353333 Midmean - Empirical Distribution Function - Interpolation 1.12947525353333 Midmean - Closest Observation 1.12801510103226 Midmean - True Basic - Statistics Graphics Toolkit 1.12947525353333 Midmean - MS Excel (old versions) 1.12889267790625 Number of observations 60

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Oct/19/t1255973766xkn5pjp6yeg6q65/1ndod1255973562.png (open in new window) http://www.freestatistics.org/blog/date/2009/Oct/19/t1255973766xkn5pjp6yeg6q65/1ndod1255973562.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Oct/19/t1255973766xkn5pjp6yeg6q65/2alh71255973563.png (open in new window) http://www.freestatistics.org/blog/date/2009/Oct/19/t1255973766xkn5pjp6yeg6q65/2alh71255973563.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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