paper : 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: Sat, 18 Dec 2010 10:58:46 +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/18/t1292669810e1wx4n9jvr16xju.htm/, Retrieved Sat, 18 Dec 2010 11:56:50 +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/18/t1292669810e1wx4n9jvr16xju.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:

» Textbox « » Textfile « » CSV «

627 696 825 677 656 785 412 352 839 729 696 641 695 638 762 635 721 854 418 367 824 687 601 676 740 691 683 594 729 731 386 331 707 715 657 653 642 643 718 654 632 731 392 344 792 852 649 629 685 617 715 715 629 916 531 357 917 828 708 858 775 785 1006 789 734 906 532 387 991 841 892 782

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 'RServer@AstonUniversity' @ vre.aston.ac.uk Central Tendency - Ungrouped Data Measure Value S.E. Value/S.E. Arithmetic Mean 680.611111111111 18.4470921700082 36.8953060373205 Geometric Mean 659.728212263828 Harmonic Mean 635.054753946218 Quadratic Mean 698.134975806573 Winsorized Mean ( 1 / 24 ) 680.583333333333 18.348966339489 37.0910993426722 Winsorized Mean ( 2 / 24 ) 678.75 17.8528342108017 38.0191734256585 Winsorized Mean ( 3 / 24 ) 678.916666666667 17.7916096303689 38.1593729163103 Winsorized Mean ( 4 / 24 ) 678.916666666667 17.5484991934827 38.6880187975738 Winsorized Mean ( 5 / 24 ) 679.263888888889 17.0493621129 39.8410148362641 Winsorized Mean ( 6 / 24 ) 676.513888888889 16.5609439423825 40.8499594734793 Winsorized Mean ( 7 / 24 ) 676.611111111111 16.3819525917436 41.3022261737052 Winsorized Mean ( 8 / 24 ) 678.611111111111 15.811951903143 42.9176053195698 Winsorized Mean ( 9 / 24 ) 677.986111111111 15.4253147049357 43.952821973491 Winsorized Mean ( 10 / 24 ) 693.402777777778 11.9734915170493 57.9114936349543 Winsorized Mean ( 11 / 24 ) 691.875 11.6620201595555 59.3271997933481 Winsorized Mean ( 12 / 24 ) 701.708333333333 9.75867732120844 71.9060903682426 Winsorized Mean ( 13 / 24 ) 702.791666666667 9.53313424190483 73.720945161708 Winsorized Mean ( 14 / 24 ) 699.680555555556 8.00527570538852 87.4024307600781 Winsorized Mean ( 15 / 24 ) 701.138888888889 7.60899685631756 92.1460347702405 Winsorized Mean ( 16 / 24 ) 700.694444444444 7.40462674399427 94.6292728411682 Winsorized Mean ( 17 / 24 ) 700.694444444444 7.40462674399427 94.6292728411682 Winsorized Mean ( 18 / 24 ) 700.694444444444 7.1830427293507 97.548416575796 Winsorized Mean ( 19 / 24 ) 699.638888888889 6.78478994741171 103.118725017536 Winsorized Mean ( 20 / 24 ) 696.861111111111 6.11865287229608 113.891264246473 Winsorized Mean ( 21 / 24 ) 691.319444444444 5.0799487073092 136.087878889358 Winsorized Mean ( 22 / 24 ) 689.791666666667 4.79427873283095 143.878089928942 Winsorized Mean ( 23 / 24 ) 689.152777777778 4.62610895736108 148.970286720375 Winsorized Mean ( 24 / 24 ) 691.152777777778 4.34895431652159 158.923899281283 Trimmed Mean ( 1 / 24 ) 680.957142857143 17.6899620120947 38.4939855942919 Trimmed Mean ( 2 / 24 ) 681.35294117647 16.9036255560306 40.3080947881852 Trimmed Mean ( 3 / 24 ) 682.772727272727 16.2846302392806 41.9274320165891 Trimmed Mean ( 4 / 24 ) 684.21875 15.5663975445471 43.9548551963895 Trimmed Mean ( 5 / 24 ) 685.758064516129 14.7877935979097 46.3732510178573 Trimmed Mean ( 6 / 24 ) 687.316666666667 14.0067274325946 49.0704677430386 Trimmed Mean ( 7 / 24 ) 689.551724137931 13.1868963818251 52.2906758476018 Trimmed Mean ( 8 / 24 ) 691.928571428571 12.2075891113252 56.6801982863803 Trimmed Mean ( 9 / 24 ) 694.148148148148 11.1333448639582 62.3485714877391 Trimmed Mean ( 10 / 24 ) 696.634615384615 9.83659974046484 70.8206731762062 Trimmed Mean ( 11 / 24 ) 697.1 9.23371152741023 75.4951026930679 Trimmed Mean ( 12 / 24 ) 697.8125 8.5384329417677 81.7260620021375 Trimmed Mean ( 13 / 24 ) 697.304347826087 8.15039014301167 85.5547201533624 Trimmed Mean ( 14 / 24 ) 696.613636363636 7.69944232825067 90.4758561289086 Trimmed Mean ( 15 / 24 ) 696.238095238095 7.4943888823487 92.9012500109147 Trimmed Mean ( 16 / 24 ) 695.65 7.307578609102 95.1956916527082 Trimmed Mean ( 17 / 24 ) 695.052631578947 7.09947079633976 97.9020340413671 Trimmed Mean ( 18 / 24 ) 694.388888888889 6.80782437076188 101.998649064911 Trimmed Mean ( 19 / 24 ) 693.647058823529 6.4590969100133 107.390718623248 Trimmed Mean ( 20 / 24 ) 692.9375 6.08540319588206 113.868790233802 Trimmed Mean ( 21 / 24 ) 692.466666666667 5.76767315646845 120.05997009211 Trimmed Mean ( 22 / 24 ) 692.607142857143 5.63819628690014 122.841970661141 Trimmed Mean ( 23 / 24 ) 692.961538461538 5.512916516755 125.697811014455 Trimmed Mean ( 24 / 24 ) 693.458333333333 5.35107302210713 129.592388380502 Median 695.5 Midrange 668.5 Midmean - Weighted Average at Xnp 690.947368421053 Midmean - Weighted Average at X(n+1)p 694.388888888889 Midmean - Empirical Distribution Function 690.947368421053 Midmean - Empirical Distribution Function - Averaging 694.388888888889 Midmean - Empirical Distribution Function - Interpolation 694.388888888889 Midmean - Closest Observation 690.947368421053 Midmean - True Basic - Statistics Graphics Toolkit 694.388888888889 Midmean - MS Excel (old versions) 695.65 Number of observations 72

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/18/t1292669810e1wx4n9jvr16xju/1woui1292669922.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/18/t1292669810e1wx4n9jvr16xju/1woui1292669922.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/18/t1292669810e1wx4n9jvr16xju/2woui1292669922.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/18/t1292669810e1wx4n9jvr16xju/2woui1292669922.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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