Measures of 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: Mon, 04 Oct 2010 17:49:34 +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/Oct/04/t1286214612xhkiesdkkx9bkj2.htm/, Retrieved Mon, 04 Oct 2010 19:50:14 +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/2010/Oct/04/t1286214612xhkiesdkkx9bkj2.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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8.95 57.47 86.58 137.55 171.26 236.71 239.89 252.64 259.7 308.16 324.04 388.3 392.25 440.31 694.87 756.46 4813 33999 37028 39047 59609 62156 64016 70939 72844 85094 103898 109215 131812 136452 136813 140321 150034 156187 158047 169861 171328 180818 183186 183613 184641 187881 190157 190379 191835 192797 193299 197549 198296 199297 199746 200156 203077 204386 206771 206893 207533 208108 211655 213361 213511 213923 216046 216548 216886 217465 218761 220553 221588 223166 226731 229641 232444 232669 235577 236302 238502 240755 241171 242205 242344 246542 249148 250407 251422 257102 257567 260642 261596 262517 262875 263906 265777 266793 274482 275562 278741 287069 289714 293671 295281 308174 308532 313906 315955 330563 348821 350089 356725 366936 380155 380531 383703 386688 401422 401915 403064 403556 406167 421403 426113 435956 438555 441437 449594 475834 506652 556277 601162 611281 6 etc...

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 234254.307692308 7145.66825097824 32.7827012764325 Geometric Mean 223833.163772138 Harmonic Mean 212038.924447132 Quadratic Mean 243865.808181827 Winsorized Mean ( 1 / 30 ) 234227.010989011 7072.55735009978 33.1177252292916 Winsorized Mean ( 2 / 30 ) 234574.681318681 6965.36847104697 33.677282442952 Winsorized Mean ( 3 / 30 ) 234645.395604396 6904.69266761081 33.9834670274453 Winsorized Mean ( 4 / 30 ) 235622.142857143 6715.01762971749 35.0888345868804 Winsorized Mean ( 5 / 30 ) 235150.769230769 6502.91210331664 36.1608407886732 Winsorized Mean ( 6 / 30 ) 234501.318681319 6351.04652960564 36.9232562835403 Winsorized Mean ( 7 / 30 ) 234260.703296703 6197.6651681407 37.7982186744983 Winsorized Mean ( 8 / 30 ) 235003.120879121 6035.65986735435 38.9357793586422 Winsorized Mean ( 9 / 30 ) 233805.923076923 5577.17888721378 41.9218977560404 Winsorized Mean ( 10 / 30 ) 232405.043956044 5248.21392238065 44.2826926251936 Winsorized Mean ( 11 / 30 ) 233585.428571429 4990.16688688958 46.8091416311378 Winsorized Mean ( 12 / 30 ) 233070.219780220 4837.91815837718 48.1757260355145 Winsorized Mean ( 13 / 30 ) 234374.791208791 4645.83988325934 50.448314427135 Winsorized Mean ( 14 / 30 ) 232755.560439560 4261.6762917805 54.6159643538568 Winsorized Mean ( 15 / 30 ) 232560.560439560 4209.5453482968 55.2460043062977 Winsorized Mean ( 16 / 30 ) 232045.571428571 4075.36001747992 56.9386681012936 Winsorized Mean ( 17 / 30 ) 232156.725274725 3920.95522942142 59.209226244846 Winsorized Mean ( 18 / 30 ) 230959.626373626 3615.0627729229 63.8881371863117 Winsorized Mean ( 19 / 30 ) 230342.230769231 3513.00474904426 65.568437057165 Winsorized Mean ( 20 / 30 ) 230424.868131868 3438.9601475251 67.0042275126964 Winsorized Mean ( 21 / 30 ) 228872.483516484 3166.39408398299 72.2817430319938 Winsorized Mean ( 22 / 30 ) 228748.219780220 3118.50157033020 73.3519655582533 Winsorized Mean ( 23 / 30 ) 229349.505494505 2923.40115175942 78.4529709022224 Winsorized Mean ( 24 / 30 ) 229274.604395604 2864.07601863284 80.0518571797715 Winsorized Mean ( 25 / 30 ) 229451.252747253 2818.38324200273 81.4123676750951 Winsorized Mean ( 26 / 30 ) 229316.395604396 2768.95204240834 82.8170340591903 Winsorized Mean ( 27 / 30 ) 229154.989010989 2717.97363709438 84.3109682461691 Winsorized Mean ( 28 / 30 ) 229107.604395604 2491.5118876733 91.9552523626756 Winsorized Mean ( 29 / 30 ) 229376.571428571 2424.59000887696 94.6042714804452 Winsorized Mean ( 30 / 30 ) 228290.307692308 2100.99689738144 108.658088918092 Trimmed Mean ( 1 / 30 ) 234189.617977528 6819.08532491074 34.3432596629950 Trimmed Mean ( 2 / 30 ) 234150.505747126 6526.5482666284 35.87662209508 Trimmed Mean ( 3 / 30 ) 233923.447058824 6255.14881556815 37.3969435349999 Trimmed Mean ( 4 / 30 ) 233659.602409639 5966.82746012148 39.1597719175344 Trimmed Mean ( 5 / 30 ) 233108.395061728 5698.9247407312 40.9039258573924 Trimmed Mean ( 6 / 30 ) 232637.873417722 5452.36965731674 42.667296613966 Trimmed Mean ( 7 / 30 ) 232270.831168831 5205.87115220075 44.6170918138533 Trimmed Mean ( 8 / 30 ) 231925.92 4954.76456661102 46.8086660591087 Trimmed Mean ( 9 / 30 ) 231446.424657534 4695.10375089879 49.2952737441059 Trimmed Mean ( 10 / 30 ) 231110.408450704 4493.84034197321 51.4282642158189 Trimmed Mean ( 11 / 30 ) 230939.666666667 4326.51496056991 53.3777575650047 Trimmed Mean ( 12 / 30 ) 230612.985074627 4178.48950061872 55.1905144288335 Trimmed Mean ( 13 / 30 ) 230326.307692308 4031.59691625481 57.1302916627566 Trimmed Mean ( 14 / 30 ) 229876.476190476 3889.52717335762 59.1013935485727 Trimmed Mean ( 15 / 30 ) 229569.68852459 3792.0005445416 60.5405209804214 Trimmed Mean ( 16 / 30 ) 229262.152542373 3682.35981141454 62.2595738286379 Trimmed Mean ( 17 / 30 ) 228984.421052632 3573.06812340646 64.086217543013 Trimmed Mean ( 18 / 30 ) 228675.672727273 3466.19672621418 65.9730796575517 Trimmed Mean ( 19 / 30 ) 228457.811320755 3390.97147800264 67.3723777397625 Trimmed Mean ( 20 / 30 ) 228280.843137255 3314.84984788417 68.8661187121233 Trimmed Mean ( 21 / 30 ) 228081.755102041 3231.11972211097 70.5890758368525 Trimmed Mean ( 22 / 30 ) 228008.851063830 3176.24249930817 71.7857188528563 Trimmed Mean ( 23 / 30 ) 227940.888888889 3112.37005768627 73.2370780672333 Trimmed Mean ( 24 / 30 ) 227811.279069767 3064.64851252863 74.3352061870877 Trimmed Mean ( 25 / 30 ) 227675.951219512 3010.21758490596 75.6343834947813 Trimmed Mean ( 26 / 30 ) 227510.256410256 2943.50516324383 77.292290583086 Trimmed Mean ( 27 / 30 ) 227339.405405405 2861.49308961093 79.4478261124568 Trimmed Mean ( 28 / 30 ) 227164.571428571 2758.87199890871 82.3396560327655 Trimmed Mean ( 29 / 30 ) 227164.571428571 2674.19782887317 84.9468087124625 Trimmed Mean ( 30 / 30 ) 226729.935483871 2567.80062240439 88.2973286576936 Median 223166 Midrange 237133 Midmean - Weighted Average at Xnp 227187.804347826 Midmean - Weighted Average at X(n+1)p 228008.851063830 Midmean - Empirical Distribution Function 228008.851063830 Midmean - Empirical Distribution Function - Averaging 228008.851063830 Midmean - Empirical Distribution Function - Interpolation 227940.888888889 Midmean - Closest Observation 227187.804347826 Midmean - True Basic - Statistics Graphics Toolkit 228008.851063830 Midmean - MS Excel (old versions) 228008.851063830 Number of observations 91

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Oct/04/t1286214612xhkiesdkkx9bkj2/1q5d11286214573.png (open in new window) http://www.freestatistics.org/blog/date/2010/Oct/04/t1286214612xhkiesdkkx9bkj2/1q5d11286214573.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Oct/04/t1286214612xhkiesdkkx9bkj2/20wcm1286214573.png (open in new window) http://www.freestatistics.org/blog/date/2010/Oct/04/t1286214612xhkiesdkkx9bkj2/20wcm1286214573.ps (open in new window)

Parameters (Session):

par1 = 15 ; par2 = grey ; par3 = FALSE ; par4 = Unknown ;

Parameters (R input):

par1 = 15 ; par2 = grey ; par3 = FALSE ; par4 = Unknown ;

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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As a general rule, we do not log on-line behavior of individuals (other than normal logging of webserver 'hits'). However, in cases of abuse, hacking, unauthorized access, Denial of Service attacks, illegal copying, hotlinking, non-compliance with international webstandards (such as robots.txt), or any other harmful behavior, our system engineers are empowered to log, track, identify, publish, and ban misbehaving individuals - even if this leads to ban entire blocks of IP addresses, or disclosing user's identity.

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