central tendency workshop 8

*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, 29 Nov 2010 13:52:56 +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/29/t12910387111c8qnjs6v4eempk.htm/, Retrieved Mon, 29 Nov 2010 14:51:53 +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/29/t12910387111c8qnjs6v4eempk.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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13328 12873 14000 13477 14237 13674 13529 14058 12975 14326 14008 16193 14483 14011 15057 14884 15414 14440 14900 15074 14442 15307 14938 17193 15528 14765 15838 15723 16150 15486 15986 15983 15692 16490 15686 18897 16316 15636 17163 16534 16518 16375 16290 16352 15943 16362 16393 19051 16747 16320 17910 16961 17480 17049 16879 17473 16998 17307 17418 20169 17871 17226 19062 17804 19100 18522 18060 18869 18127 18871 18890 21263 19547 18450 20254 19240 20216 19420 19415 20018 18652 19978 19509 21971

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 16750.2857142857 227.180590759787 73.7311477986115 Geometric Mean 16623.1754136431 Harmonic Mean 16497.0390487161 Quadratic Mean 16877.6712326700 Winsorized Mean ( 1 / 28 ) 16743.0714285714 224.744336027652 74.4983020462478 Winsorized Mean ( 2 / 28 ) 16727.4523809524 217.842184322602 76.7870209939723 Winsorized Mean ( 3 / 28 ) 16731.4166666667 216.596338739475 77.2469967130491 Winsorized Mean ( 4 / 28 ) 16731.6547619048 215.718998502654 77.5622679413605 Winsorized Mean ( 5 / 28 ) 16731.2976190476 212.498426155801 78.7361013524893 Winsorized Mean ( 6 / 28 ) 16751.7261904762 208.090499911759 80.5021190183108 Winsorized Mean ( 7 / 28 ) 16716.4761904762 201.585801776728 82.9248689299606 Winsorized Mean ( 8 / 28 ) 16713.1428571429 200.930045757858 83.1789133083856 Winsorized Mean ( 9 / 28 ) 16708.6428571429 198.533144678927 84.160470455271 Winsorized Mean ( 10 / 28 ) 16729.3571428571 195.08020008291 85.7563050260718 Winsorized Mean ( 11 / 28 ) 16718.0952380952 189.558931551628 88.1947112765928 Winsorized Mean ( 12 / 28 ) 16714.3809523810 183.941369454132 90.8679814768308 Winsorized Mean ( 13 / 28 ) 16708.8095238095 182.980121186258 91.3148893742476 Winsorized Mean ( 14 / 28 ) 16713.8095238095 181.680587988067 91.9955715076577 Winsorized Mean ( 15 / 28 ) 16736.6666666667 170.199875187572 98.33536392563 Winsorized Mean ( 16 / 28 ) 16758 166.879663667344 100.419665474669 Winsorized Mean ( 17 / 28 ) 16757.3928571429 165.851009576273 101.038835397841 Winsorized Mean ( 18 / 28 ) 16765.1071428571 164.691548315605 101.797009708898 Winsorized Mean ( 19 / 28 ) 16742.9404761905 153.658430232897 108.962068991682 Winsorized Mean ( 20 / 28 ) 16716.0357142857 148.536903939802 112.537930109680 Winsorized Mean ( 21 / 28 ) 16756.2857142857 138.328354538653 121.134136021285 Winsorized Mean ( 22 / 28 ) 16699.7142857143 122.598179855146 136.215026238118 Winsorized Mean ( 23 / 28 ) 16701.0833333333 117.560599732180 142.063611204611 Winsorized Mean ( 24 / 28 ) 16670.2261904762 110.171874995066 151.311087255461 Winsorized Mean ( 25 / 28 ) 16690.7619047619 104.635569001931 159.513271289745 Winsorized Mean ( 26 / 28 ) 16685.5 99.9560380992349 166.928384890915 Winsorized Mean ( 27 / 28 ) 16583.2857142857 86.131987822671 192.533414512939 Winsorized Mean ( 28 / 28 ) 16591.2857142857 84.5556635168089 196.217320333457 Trimmed Mean ( 1 / 28 ) 16733.9024390244 218.643550765635 76.5350836117801 Trimmed Mean ( 2 / 28 ) 16724.275 211.561156863339 79.0517278689456 Trimmed Mean ( 3 / 28 ) 16722.5641025641 207.612960550458 80.5468216349618 Trimmed Mean ( 4 / 28 ) 16719.3026315789 203.538382965618 82.1432419181752 Trimmed Mean ( 5 / 28 ) 16715.7972972973 199.062745554764 83.9725045020974 Trimmed Mean ( 6 / 28 ) 16712.1805555556 194.757940655854 85.8100085638447 Trimmed Mean ( 7 / 28 ) 16704.2714285714 190.822363193069 87.5383322429066 Trimmed Mean ( 8 / 28 ) 16702.1176470588 187.706182252359 88.9801148083865 Trimmed Mean ( 9 / 28 ) 16700.3636363636 184.102070397497 90.7125248526845 Trimmed Mean ( 10 / 28 ) 16699.15625 180.267124622373 92.6356166438098 Trimmed Mean ( 11 / 28 ) 16695.0645161290 176.326104775386 94.6828862203704 Trimmed Mean ( 12 / 28 ) 16692.1333333333 172.630108906128 96.6930591604394 Trimmed Mean ( 13 / 28 ) 16689.4482758621 169.181637897206 98.6481067525929 Trimmed Mean ( 14 / 28 ) 16687.2142857143 165.090920865242 101.078933948981 Trimmed Mean ( 15 / 28 ) 16684.2592592593 160.252816476993 104.112112511012 Trimmed Mean ( 16 / 28 ) 16678.6153846154 156.456674243764 106.602134202531 Trimmed Mean ( 17 / 28 ) 16670.28 152.266077640425 109.481246633060 Trimmed Mean ( 18 / 28 ) 16661.3125 147.109002610846 113.258279264356 Trimmed Mean ( 19 / 28 ) 16650.7826086957 140.690907015877 118.350097826980 Trimmed Mean ( 20 / 28 ) 16641.5227272727 134.888276879695 123.37263928514 Trimmed Mean ( 21 / 28 ) 16634.0714285714 128.488480631091 129.459632076514 Trimmed Mean ( 22 / 28 ) 16621.85 122.381044021929 135.820462497620 Trimmed Mean ( 23 / 28 ) 16614.0263157895 118.182065866699 140.579928045334 Trimmed Mean ( 24 / 28 ) 16605.1944444444 113.639333534223 146.121892200677 Trimmed Mean ( 25 / 28 ) 16598.5 109.312885978727 151.843946405643 Trimmed Mean ( 26 / 28 ) 16588.8125 104.623253034901 158.557605683186 Trimmed Mean ( 27 / 28 ) 16578.4 99.2121446959615 167.100510232946 Trimmed Mean ( 28 / 28 ) 16577.8571428571 95.9831984068943 172.716240112982 Median 16441.5 Midrange 17422 Midmean - Weighted Average at Xnp 16597.7906976744 Midmean - Weighted Average at X(n+1)p 16634.0714285714 Midmean - Empirical Distribution Function 16597.7906976744 Midmean - Empirical Distribution Function - Averaging 16634.0714285714 Midmean - Empirical Distribution Function - Interpolation 16634.0714285714 Midmean - Closest Observation 16597.7906976744 Midmean - True Basic - Statistics Graphics Toolkit 16634.0714285714 Midmean - MS Excel (old versions) 16641.5227272727 Number of observations 84

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Nov/29/t12910387111c8qnjs6v4eempk/19ti01291038774.png (open in new window) http://www.freestatistics.org/blog/date/2010/Nov/29/t12910387111c8qnjs6v4eempk/19ti01291038774.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Nov/29/t12910387111c8qnjs6v4eempk/2kkil1291038774.png (open in new window) http://www.freestatistics.org/blog/date/2010/Nov/29/t12910387111c8qnjs6v4eempk/2kkil1291038774.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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