Part 2: Y[t] - X[t] Central tendensity

*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, 17 Oct 2009 11:11:44 -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/17/t1255799922msosglnlditfv18.htm/, Retrieved Sat, 17 Oct 2009 19:18:44 +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/17/t1255799922msosglnlditfv18.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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-30.07 -12.34 -9.46 -10.49 -16.28 -5.43 -11.40 -8.19 -3.13 -12.06 -15.92 1.68 -25.24 -17.03 -14.97 -19.12 -21.00 -11.95 -18.46 -10.20 -9.92 -13.47 -6.55 0.30 -30.37 -21.13 -12.02 -14.35 -13.56 0.23 -12.59 -9.28 -4.75 -4.82 -7.30 7.65 -25.24 -23.28 -6.33 -16.84 -13.01 4.54 -22.77 -10.35 -3.81 -11.01 -11.20 -8.45 -31.34 -21.70 -17.56 -17.65 -11.11 5.43 -16.05 -11.27 -11.04 -15.73 -24.05 -10.24

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 -12.5508333333333 1.08859595186729 -11.5293771870129 Geometric Mean NaN Harmonic Mean 15.4099903738004 Quadratic Mean 15.0811420323529 Winsorized Mean ( 1 / 20 ) -12.5716666666667 1.07287383135379 -11.7177493748759 Winsorized Mean ( 2 / 20 ) -12.5913333333333 1.06179723097147 -11.8585102372259 Winsorized Mean ( 3 / 20 ) -12.4928333333333 0.963233903681503 -12.9696777548895 Winsorized Mean ( 4 / 20 ) -12.5848333333333 0.941078071884739 -13.3727835227625 Winsorized Mean ( 5 / 20 ) -12.4915 0.917822491392652 -13.6099301522303 Winsorized Mean ( 6 / 20 ) -12.7505 0.82845375081029 -15.3907203480328 Winsorized Mean ( 7 / 20 ) -12.7703333333333 0.800404945769831 -15.9548406101499 Winsorized Mean ( 8 / 20 ) -12.753 0.74752821687763 -17.0602255701709 Winsorized Mean ( 9 / 20 ) -12.678 0.72853408266499 -17.4020684847353 Winsorized Mean ( 10 / 20 ) -12.758 0.706019389628269 -18.0703252451994 Winsorized Mean ( 11 / 20 ) -12.5783333333333 0.611663572399384 -20.564136726325 Winsorized Mean ( 12 / 20 ) -12.4903333333333 0.580560948622893 -21.5142499042706 Winsorized Mean ( 13 / 20 ) -12.4773333333333 0.522768211195492 -23.8678119023334 Winsorized Mean ( 14 / 20 ) -12.664 0.485522736525918 -26.0832275139477 Winsorized Mean ( 15 / 20 ) -12.5965 0.453082027282398 -27.8018090356712 Winsorized Mean ( 16 / 20 ) -12.7671666666667 0.411511139105962 -31.0250815917262 Winsorized Mean ( 17 / 20 ) -12.6595 0.377984128519214 -33.4921470104968 Winsorized Mean ( 18 / 20 ) -12.7285 0.347420179427515 -36.6371925228243 Winsorized Mean ( 19 / 20 ) -12.776 0.328781967146776 -38.8585788657213 Winsorized Mean ( 20 / 20 ) -12.726 0.316787119523577 -40.1720878649956 Trimmed Mean ( 1 / 20 ) -12.5751724137931 1.01916078311414 -12.3387522578807 Trimmed Mean ( 2 / 20 ) -12.5789285714286 0.952280573579198 -13.2092672269371 Trimmed Mean ( 3 / 20 ) -12.5720370370370 0.875567915114705 -14.3587228586260 Trimmed Mean ( 4 / 20 ) -12.6025 0.830784482483178 -15.1693974378670 Trimmed Mean ( 5 / 20 ) -12.6078 0.783564763091848 -16.0903100724581 Trimmed Mean ( 6 / 20 ) -12.636875 0.731891927810243 -17.2660395883972 Trimmed Mean ( 7 / 20 ) -12.6121739130435 0.696706254547046 -18.1025702449637 Trimmed Mean ( 8 / 20 ) -12.5813636363636 0.659968173916255 -19.0635914482145 Trimmed Mean ( 9 / 20 ) -12.5507142857143 0.628197501887055 -19.9789305879329 Trimmed Mean ( 10 / 20 ) -12.5295 0.591700282540894 -21.1754166251121 Trimmed Mean ( 11 / 20 ) -12.4934210526316 0.549128790962022 -22.7513495162843 Trimmed Mean ( 12 / 20 ) -12.4805555555556 0.521036351966489 -23.9533297599133 Trimmed Mean ( 13 / 20 ) -12.4791176470588 0.491585228636638 -25.3854609945632 Trimmed Mean ( 14 / 20 ) -12.479375 0.468328345300705 -26.6466361159225 Trimmed Mean ( 15 / 20 ) -12.453 0.446370329731231 -27.8983596591158 Trimmed Mean ( 16 / 20 ) -12.4325 0.424699901829229 -29.2736116642643 Trimmed Mean ( 17 / 20 ) -12.3842307692308 0.4056046476281 -30.5327634721432 Trimmed Mean ( 18 / 20 ) -12.34375 0.388156555754714 -31.8009571576071 Trimmed Mean ( 19 / 20 ) -12.2854545454545 0.370683659530251 -33.1426925077390 Trimmed Mean ( 20 / 20 ) -12.208 0.347179310564743 -35.1633856871877 Median -11.985 Midrange -11.845 Midmean - Weighted Average at Xnp -12.6177419354839 Midmean - Weighted Average at X(n+1)p -12.453 Midmean - Empirical Distribution Function -12.6177419354839 Midmean - Empirical Distribution Function - Averaging -12.453 Midmean - Empirical Distribution Function - Interpolation -12.453 Midmean - Closest Observation -12.6177419354839 Midmean - True Basic - Statistics Graphics Toolkit -12.453 Midmean - MS Excel (old versions) -12.479375 Number of observations 60

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Oct/17/t1255799922msosglnlditfv18/1z6dk1255799500.png (open in new window) http://www.freestatistics.org/blog/date/2009/Oct/17/t1255799922msosglnlditfv18/1z6dk1255799500.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Oct/17/t1255799922msosglnlditfv18/2juiy1255799500.png (open in new window) http://www.freestatistics.org/blog/date/2009/Oct/17/t1255799922msosglnlditfv18/2juiy1255799500.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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Software written by Ed van Stee & Patrick Wessa

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