Central tendency of Yt (Paper)

*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: Sun, 28 Nov 2010 13:17:17 +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/28/t12909501219mcc241c50xb8y9.htm/, Retrieved Sun, 28 Nov 2010 14:15:23 +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/28/t12909501219mcc241c50xb8y9.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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9769 9321 9939 9336 10195 9464 10010 10213 9563 9890 9305 9391 9928 8686 9843 9627 10074 9503 10119 10000 9313 9866 9172 9241 9659 8904 9755 9080 9435 8971 10063 9793 9454 9759 8820 9403 9676 8642 9402 9610 9294 9448 10319 9548 9801 9596 8923 9746 9829 9125 9782 9441 9162 9915 10444 10209 9985 9842 9429 10132 9849 9172 10313 9819 9955 10048 10082 10541 10208 10233 9439 9963 10158 9225 10474 9757 10490 10281 10444 10640 10695 10786 9832 9747 10411 9511 10402 9701 10540 10112 10915 11183 10384 10834 9886 10216

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 'Gwilym Jenkins' @ 72.249.127.135 Central Tendency - Ungrouped Data Measure Value S.E. Value/S.E. Arithmetic Mean 9816.77083333333 52.3438959040473 187.543755843636 Geometric Mean 9803.50905583586 Harmonic Mean 9790.23794622693 Quadratic Mean 9830.01925332974 Winsorized Mean ( 1 / 32 ) 9814.4375 51.5391347911655 190.426896760446 Winsorized Mean ( 2 / 32 ) 9815.54166666667 50.559140974814 194.139802959791 Winsorized Mean ( 3 / 32 ) 9816.66666666667 49.7203554159668 197.437580333833 Winsorized Mean ( 4 / 32 ) 9813.66666666667 48.8182510689076 201.024544136465 Winsorized Mean ( 5 / 32 ) 9813.30208333333 47.8133251443342 205.241991719043 Winsorized Mean ( 6 / 32 ) 9813.92708333333 45.517933724242 215.605724609301 Winsorized Mean ( 7 / 32 ) 9817.13541666667 44.9612494455915 218.346588178039 Winsorized Mean ( 8 / 32 ) 9816.05208333333 43.7785129387666 224.220774631225 Winsorized Mean ( 9 / 32 ) 9815.48958333333 43.3902603338615 226.214120583955 Winsorized Mean ( 10 / 32 ) 9812.36458333333 42.8984345188125 228.734794017490 Winsorized Mean ( 11 / 32 ) 9818.4375 41.9690879677709 233.944504763597 Winsorized Mean ( 12 / 32 ) 9816.3125 41.034309330302 239.222071973588 Winsorized Mean ( 13 / 32 ) 9822.27083333333 39.8167581288008 246.686854855432 Winsorized Mean ( 14 / 32 ) 9821.25 39.1939789056449 250.580580850022 Winsorized Mean ( 15 / 32 ) 9812.34375 37.5296807339514 261.455561521023 Winsorized Mean ( 16 / 32 ) 9812.67708333333 37.2017216128916 263.769434797688 Winsorized Mean ( 17 / 32 ) 9809.66666666667 36.0418940769623 272.174005220689 Winsorized Mean ( 18 / 32 ) 9810.97916666667 33.4271890699098 293.502966885667 Winsorized Mean ( 19 / 32 ) 9809.79166666667 32.6966986794105 300.023918709689 Winsorized Mean ( 20 / 32 ) 9809.375 32.5877400326963 301.014276846383 Winsorized Mean ( 21 / 32 ) 9814.1875 31.7366849843184 309.237953013976 Winsorized Mean ( 22 / 32 ) 9815.33333333333 31.5315341554900 311.286259809987 Winsorized Mean ( 23 / 32 ) 9813.17708333333 31.0037844138523 316.515459930397 Winsorized Mean ( 24 / 32 ) 9804.42708333333 29.7615219645652 329.432987163988 Winsorized Mean ( 25 / 32 ) 9799.47916666667 28.6902582157066 341.561204956389 Winsorized Mean ( 26 / 32 ) 9797.58333333333 28.0536245065271 349.244830415898 Winsorized Mean ( 27 / 32 ) 9798.42708333333 27.4549421332851 356.891194152407 Winsorized Mean ( 28 / 32 ) 9801.05208333333 24.9656622507992 392.581297658931 Winsorized Mean ( 29 / 32 ) 9801.05208333333 24.3769032985734 402.063049735564 Winsorized Mean ( 30 / 32 ) 9809.17708333333 22.5406795749358 435.176634791468 Winsorized Mean ( 31 / 32 ) 9809.17708333333 21.3797893720837 458.806067385373 Winsorized Mean ( 32 / 32 ) 9807.51041666667 18.5811283885332 527.821034954964 Trimmed Mean ( 1 / 32 ) 9814.73404255319 49.8690556589525 196.810104239286 Trimmed Mean ( 2 / 32 ) 9815.04347826087 47.9588679302319 204.655445423343 Trimmed Mean ( 3 / 32 ) 9814.77777777778 46.3763758195027 211.633134421219 Trimmed Mean ( 4 / 32 ) 9814.0909090909 44.9294455498837 218.433385700134 Trimmed Mean ( 5 / 32 ) 9814.20930232558 43.5823641721536 225.187630105579 Trimmed Mean ( 6 / 32 ) 9814.41666666667 42.3286638081786 231.862189440772 Trimmed Mean ( 7 / 32 ) 9814.51219512195 41.4734462774848 236.645687205649 Trimmed Mean ( 8 / 32 ) 9814.0625 40.6100136671667 241.666072324784 Trimmed Mean ( 9 / 32 ) 9813.7564102564 39.8563236567346 246.228339943696 Trimmed Mean ( 10 / 32 ) 9813.51315789474 39.0521760447731 251.292351715398 Trimmed Mean ( 11 / 32 ) 9813.66216216216 38.2025296997927 256.885139263838 Trimmed Mean ( 12 / 32 ) 9813.08333333333 37.3725939501160 262.574317063236 Trimmed Mean ( 13 / 32 ) 9812.71428571429 36.5578819695881 268.415831471783 Trimmed Mean ( 14 / 32 ) 9811.67647058824 35.7989118559306 274.077505765383 Trimmed Mean ( 15 / 32 ) 9810.68181818182 34.9973569371056 280.326363954077 Trimmed Mean ( 16 / 32 ) 9810.515625 34.3172494124436 285.877096590458 Trimmed Mean ( 17 / 32 ) 9810.3064516129 33.5469755860772 292.434900023727 Trimmed Mean ( 18 / 32 ) 9810.36666666667 32.8054612260269 299.046753193746 Trimmed Mean ( 19 / 32 ) 9810.31034482759 32.3287260133808 303.454900782886 Trimmed Mean ( 20 / 32 ) 9810.35714285714 31.8476122835491 308.040585759225 Trimmed Mean ( 21 / 32 ) 9810.44444444445 31.2554192831883 313.879790111192 Trimmed Mean ( 22 / 32 ) 9810.11538461538 30.6511738749782 320.056759477776 Trimmed Mean ( 23 / 32 ) 9809.66 29.9148807417967 327.919074278443 Trimmed Mean ( 24 / 32 ) 9809.35416666667 29.0712088809672 337.425051941641 Trimmed Mean ( 25 / 32 ) 9809.78260869565 28.2276616489502 347.523742160927 Trimmed Mean ( 26 / 32 ) 9810.68181818182 27.3439406757307 358.78814741905 Trimmed Mean ( 27 / 32 ) 9811.83333333333 26.3132853455355 372.885149250209 Trimmed Mean ( 28 / 32 ) 9813.025 25.0787091103395 391.289079387035 Trimmed Mean ( 29 / 32 ) 9814.1052631579 24.0412397424752 408.21959966643 Trimmed Mean ( 30 / 32 ) 9815.30555555555 22.7681837131194 431.097433121106 Trimmed Mean ( 31 / 32 ) 9815.88235294118 21.5354354878283 455.801432875089 Trimmed Mean ( 32 / 32 ) 9816.53125 20.1434865190499 487.330296109187 Median 9824 Midrange 9912.5 Midmean - Weighted Average at Xnp 9801.79591836735 Midmean - Weighted Average at X(n+1)p 9809.35416666667 Midmean - Empirical Distribution Function 9801.79591836735 Midmean - Empirical Distribution Function - Averaging 9809.35416666667 Midmean - Empirical Distribution Function - Interpolation 9809.35416666667 Midmean - Closest Observation 9801.79591836735 Midmean - True Basic - Statistics Graphics Toolkit 9809.35416666667 Midmean - MS Excel (old versions) 9809.66 Number of observations 96

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Nov/28/t12909501219mcc241c50xb8y9/14jfv1290950234.png (open in new window) http://www.freestatistics.org/blog/date/2010/Nov/28/t12909501219mcc241c50xb8y9/14jfv1290950234.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Nov/28/t12909501219mcc241c50xb8y9/24jfv1290950234.png (open in new window) http://www.freestatistics.org/blog/date/2010/Nov/28/t12909501219mcc241c50xb8y9/24jfv1290950234.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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