WS 3: Part 2: 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: Wed, 21 Oct 2009 12:26:29 -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/21/t125614967596d61zinghby87f.htm/, Retrieved Wed, 21 Oct 2009 20:27:55 +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/21/t125614967596d61zinghby87f.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:

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Dataseries X:

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3.7105 3.6810 2.8824 2.7716 2.8075 3.1007 3.2734 3.4148 3.5462 3.6457 3.7869 3.8974 4.0625 4.0354 3.1678 3.0064 3.0000 3.1633 3.3504 3.5227 3.7440 3.7967 3.9573 4.0351 4.1622 4.1161 3.3056 3.1733 3.1611 3.3806 3.6016 3.8103 3.7949 3.9459 4.0667 4.1569 4.3789 4.3656 3.4758 3.3385 3.3710 3.5826 3.8113 3.8952 3.9238 4.0198 4.1895 4.2688 4.5833 4.4828 3.5603 3.4417 3.4274 3.6422 3.7810 3.8224 3.8440 3.8899 3.9630 4.0187

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 3.68520666666667 0.0546306734822335 67.4567313885511 Geometric Mean 3.66066558034198 Harmonic Mean 3.63546562685972 Quadratic Mean 3.70902066517115 Winsorized Mean ( 1 / 20 ) 3.68413 0.0540202224793934 68.1990897280245 Winsorized Mean ( 2 / 20 ) 3.68316333333333 0.0525307595640559 70.1144122776693 Winsorized Mean ( 3 / 20 ) 3.68837833333333 0.0509469965566667 72.396384136813 Winsorized Mean ( 4 / 20 ) 3.68235166666667 0.0494723331624908 74.432545046381 Winsorized Mean ( 5 / 20 ) 3.68360166666667 0.0464488260718798 79.3045159196547 Winsorized Mean ( 6 / 20 ) 3.68691166666667 0.0447055991318563 82.4709150142996 Winsorized Mean ( 7 / 20 ) 3.68655 0.0445434212393504 82.7630634878858 Winsorized Mean ( 8 / 20 ) 3.68171 0.0434758426135762 84.6840401168052 Winsorized Mean ( 9 / 20 ) 3.675125 0.0420976271608106 87.3000510447115 Winsorized Mean ( 10 / 20 ) 3.69110833333333 0.0387703898818588 95.2043129971323 Winsorized Mean ( 11 / 20 ) 3.69204333333333 0.0369119799246609 100.022901531399 Winsorized Mean ( 12 / 20 ) 3.69856333333333 0.0357567947765601 103.436657464552 Winsorized Mean ( 13 / 20 ) 3.69782666666667 0.0347941277650456 106.277320461573 Winsorized Mean ( 14 / 20 ) 3.70237666666667 0.0339492735888827 109.056138034102 Winsorized Mean ( 15 / 20 ) 3.69085166666667 0.0314255271107774 117.447565912137 Winsorized Mean ( 16 / 20 ) 3.69845166666667 0.0296998396856624 124.527664317733 Winsorized Mean ( 17 / 20 ) 3.69879166666667 0.0286495491816974 129.104707484529 Winsorized Mean ( 18 / 20 ) 3.69645166666667 0.0270041430689954 136.884612750801 Winsorized Mean ( 19 / 20 ) 3.69889 0.0241127758409741 153.399592995618 Winsorized Mean ( 20 / 20 ) 3.71379 0.0216092061229823 171.861473247285 Trimmed Mean ( 1 / 20 ) 3.68547413793103 0.0519548706839107 70.9360660399516 Trimmed Mean ( 2 / 20 ) 3.68691428571429 0.0494030301944458 74.6293146635526 Trimmed Mean ( 3 / 20 ) 3.68899814814815 0.0472548622051925 78.0660015921661 Trimmed Mean ( 4 / 20 ) 3.68923653846154 0.0453676015312047 81.3187476072328 Trimmed Mean ( 5 / 20 ) 3.691302 0.0435937865344152 84.6749569938342 Trimmed Mean ( 6 / 20 ) 3.69322708333333 0.0424161395980417 87.0712685862587 Trimmed Mean ( 7 / 20 ) 3.6946 0.0414452275767535 89.1441600400884 Trimmed Mean ( 8 / 20 ) 3.69616818181818 0.0402131219541776 91.9144797071445 Trimmed Mean ( 9 / 20 ) 3.69875 0.0388850747576528 95.1200434370276 Trimmed Mean ( 10 / 20 ) 3.7026875 0.0374819904083007 98.7857757729964 Trimmed Mean ( 11 / 20 ) 3.70451578947368 0.0365449855336872 101.368648403455 Trimmed Mean ( 12 / 20 ) 3.70640555555556 0.0357351016674314 103.718903336255 Trimmed Mean ( 13 / 20 ) 3.70755882352941 0.0348868438491482 106.273838916498 Trimmed Mean ( 14 / 20 ) 3.7089625 0.0339065523065162 109.387780464108 Trimmed Mean ( 15 / 20 ) 3.70990333333333 0.0326998634336961 113.453175144163 Trimmed Mean ( 16 / 20 ) 3.712625 0.0316555643131524 117.281908585577 Trimmed Mean ( 17 / 20 ) 3.71466923076923 0.0306065421710562 121.368471159153 Trimmed Mean ( 18 / 20 ) 3.71700416666667 0.0292655306713727 127.009628098171 Trimmed Mean ( 19 / 20 ) 3.72011818181818 0.0276810026251957 134.392465193153 Trimmed Mean ( 20 / 20 ) 3.72347 0.0262907329080440 141.626709800120 Median 3.7625 Midrange 3.67745 Midmean - Weighted Average at Xnp 3.69897096774193 Midmean - Weighted Average at X(n+1)p 3.70990333333333 Midmean - Empirical Distribution Function 3.69897096774193 Midmean - Empirical Distribution Function - Averaging 3.70990333333333 Midmean - Empirical Distribution Function - Interpolation 3.70990333333333 Midmean - Closest Observation 3.69897096774193 Midmean - True Basic - Statistics Graphics Toolkit 3.70990333333333 Midmean - MS Excel (old versions) 3.7089625 Number of observations 60

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Oct/21/t125614967596d61zinghby87f/1tppo1256149584.png (open in new window) http://www.freestatistics.org/blog/date/2009/Oct/21/t125614967596d61zinghby87f/1tppo1256149584.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Oct/21/t125614967596d61zinghby87f/2aws71256149584.png (open in new window) http://www.freestatistics.org/blog/date/2009/Oct/21/t125614967596d61zinghby87f/2aws71256149584.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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