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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, 06 Dec 2008 12:34:56 -0700
 
Cite this page as follows:
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/Dec/06/t1228592968jx7nbb4lrxlhfpw.htm/, Retrieved Sat, 06 Dec 2008 19:49:30 +0000
 
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/2008/Dec/06/t1228592968jx7nbb4lrxlhfpw.htm/},
    year = {2008},
}
@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 = {2008},
    note = {{ISBN} 3-900051-07-0},
    url = {http://www.R-project.org},
}
 
Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
 
Feedback Forum:
2008-11-27 13:41:43 [a2386b643d711541400692649981f2dc] [reply
test

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Original text written by user:
 
IsPrivate?
No (this computation is public)
 
User-defined keywords:
 
Dataseries X:
» Textbox « » Textfile « » CSV «
3595 3914 4159 3676 3794 3446 3504 3958 3353 3480 3098 2944 3389 3497 4404 3849 3734 3060 3507 3287 3215 3764 2734 2837 2766 3851 3289 3848 3348 3682 4058 3655 3811 3341 3032 3475 3353 3186 3902 4164 3499 4145 3796 3711 3949 3740 3243 4407 4814 3908 5250 3937 4004 5560 3922 3759 4138 4634 3996 4308 4142 4429 5219 4929 5754 5592 4163 4962 5208 4755 4491 5732 5730 5024 6056 4901 5353 5578 4618 4724 5011 5298 4143 4617 4736 4214 5112 4197 4119 5104 4194 4583 3790 5557 4304 3838 4277 4951 4479 4677 4274 4782
 
Output produced by software:


Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time1 seconds
R Server'Gwilym Jenkins' @ 72.249.127.135


Central Tendency - Ungrouped Data
MeasureValueS.E.Value/S.E.
Arithmetic Mean4169.5686274509875.630009725981155.1311396436144
Geometric Mean4101.54042520798
Harmonic Mean4034.86803377417
Quadratic Mean4238.2794002413
Winsorized Mean ( 1 / 34 )4166.9215686274574.895011854425555.6368370262997
Winsorized Mean ( 2 / 34 )4167.8823529411874.552998213592555.9049595966657
Winsorized Mean ( 3 / 34 )4170.9705882352974.004259685101756.3612230699064
Winsorized Mean ( 4 / 34 )4169.0098039215772.360732551802757.6142564744904
Winsorized Mean ( 5 / 34 )4169.6960784313772.016010198772257.899570761037
Winsorized Mean ( 6 / 34 )4170.8725490196171.47499799981658.3542870337729
Winsorized Mean ( 7 / 34 )4176.7058823529470.566564249299859.1881711513876
Winsorized Mean ( 8 / 34 )4162.9803921568667.29637849693461.8603925669868
Winsorized Mean ( 9 / 34 )4160.5980392156966.116095189147162.9286715634508
Winsorized Mean ( 10 / 34 )4160.2058823529464.738558995534864.261638610768
Winsorized Mean ( 11 / 34 )4157.0784313725564.157150123077264.7952476598124
Winsorized Mean ( 12 / 34 )4161.9019607843163.1413250471865.9140738284236
Winsorized Mean ( 13 / 34 )4150.5588235294161.067914260371667.9662777712192
Winsorized Mean ( 14 / 34 )4150.1470588235360.807877937245868.2501544143098
Winsorized Mean ( 15 / 34 )4138.3823529411859.019762547920370.118586966206
Winsorized Mean ( 16 / 34 )4141.9901960784357.981836127648371.4359956963031
Winsorized Mean ( 17 / 34 )4143.3235294117655.578904196503274.5484926216384
Winsorized Mean ( 18 / 34 )4146.554.666807682200475.8504140959763
Winsorized Mean ( 19 / 34 )4143.3333333333353.95952403062576.7859503538571
Winsorized Mean ( 20 / 34 )4141.1764705882452.768056209397978.4788519432084
Winsorized Mean ( 21 / 34 )4123.6764705882450.220709073388882.1110762208117
Winsorized Mean ( 22 / 34 )4117.8529411764749.156238058479483.7707095542505
Winsorized Mean ( 23 / 34 )4112.4411764705948.263874552142885.2074396146467
Winsorized Mean ( 24 / 34 )4128.6764705882445.172432552810991.3981434531209
Winsorized Mean ( 25 / 34 )4140.4411764705943.098528099582496.0692014099364
Winsorized Mean ( 26 / 34 )4133.8137254902040.9125799989785101.040162355770
Winsorized Mean ( 27 / 34 )4124.0196078431439.2555851679566105.055614129769
Winsorized Mean ( 28 / 34 )4127.5882352941237.8129877797859109.157950155440
Winsorized Mean ( 29 / 34 )4133.843137254937.0704983669567111.513017611321
Winsorized Mean ( 30 / 34 )4125.6078431372635.6002039438412115.887196872393
Winsorized Mean ( 31 / 34 )4103.4215686274531.4915082441576130.302478268526
Winsorized Mean ( 32 / 34 )4101.2254901960830.8642260205493132.879583225754
Winsorized Mean ( 33 / 34 )4093.4607843137328.0177533075298146.102392271889
Winsorized Mean ( 34 / 34 )4087.4607843137327.0101998737097151.330267951561
Trimmed Mean ( 1 / 34 )4165.0673.378002676331256.7616976217244
Trimmed Mean ( 2 / 34 )4163.1224489795971.665595346335158.0909490650382
Trimmed Mean ( 3 / 34 )4160.5937569.93247146625359.4944474686235
Trimmed Mean ( 4 / 34 )4156.8404255319168.196830314027360.953572275879
Trimmed Mean ( 5 / 34 )4153.4673913043566.776295841834362.1997272975759
Trimmed Mean ( 6 / 34 )4149.7888888888965.25201599660863.5963322436599
Trimmed Mean ( 7 / 34 )4145.7159090909163.641629771188565.1415735894264
Trimmed Mean ( 8 / 34 )4140.4651162790761.993357610897566.7888508679717
Trimmed Mean ( 9 / 34 )4137.0476190476260.791669268308468.0528708759166
Trimmed Mean ( 10 / 34 )4133.7926829268359.630892511396969.3230053891405
Trimmed Mean ( 11 / 34 )4130.42558.534350444521270.564121214855
Trimmed Mean ( 12 / 34 )4127.2564102564157.359265975422371.954484424973
Trimmed Mean ( 13 / 34 )4123.3815789473756.15283957171573.4313992025504
Trimmed Mean ( 14 / 34 )4120.555.088546574738674.7977620794501
Trimmed Mean ( 15 / 34 )4117.553.873315433579476.4293039487514
Trimmed Mean ( 16 / 34 )4115.4714285714352.731283792892978.0461072166445
Trimmed Mean ( 17 / 34 )4112.9852941176551.535017087768979.8095261540876
Trimmed Mean ( 18 / 34 )4110.2272727272750.484578620937281.4155012283823
Trimmed Mean ( 19 / 34 )4107.01562549.358028991103983.2086634930303
Trimmed Mean ( 20 / 34 )4103.8709677419448.109140660128685.3033521578385
Trimmed Mean ( 21 / 34 )4100.746.786836600651387.646447119336
Trimmed Mean ( 22 / 34 )4098.7758620689745.612170311467989.8614521975168
Trimmed Mean ( 23 / 34 )4097.1964285714344.347143128551192.3891854024216
Trimmed Mean ( 24 / 34 )4095.9444444444442.936494608885695.3954085389356
Trimmed Mean ( 25 / 34 )4093.2692307692341.716698005078598.1206429682169
Trimmed Mean ( 26 / 34 )4089.4240.5301005804479100.898343242029
Trimmed Mean ( 27 / 34 )4085.7916666666739.4318578938677103.616514283038
Trimmed Mean ( 28 / 34 )4082.6521739130438.3459741903887106.468860424371
Trimmed Mean ( 29 / 34 )4078.9318181818237.2014173405126109.644527272885
Trimmed Mean ( 30 / 34 )4074.3333333333335.8166817048548113.755187231124
Trimmed Mean ( 31 / 34 )4074.3333333333334.3227607725731118.706457220338
Trimmed Mean ( 32 / 34 )4067.0789473684233.3466934681935121.963484962839
Trimmed Mean ( 33 / 34 )4064.0555555555632.140686385645126.445823427426
Trimmed Mean ( 34 / 34 )4061.3823529411831.2125144786796130.120319390333
Median4088.5
Midrange4395
Midmean - Weighted Average at Xnp4080.90196078431
Midmean - Weighted Average at X(n+1)p4093.26923076923
Midmean - Empirical Distribution Function4093.26923076923
Midmean - Empirical Distribution Function - Averaging4093.26923076923
Midmean - Empirical Distribution Function - Interpolation4089.42
Midmean - Closest Observation4093.26923076923
Midmean - True Basic - Statistics Graphics Toolkit4093.26923076923
Midmean - MS Excel (old versions)4093.26923076923
Number of observations102
 
Charts produced by software:
http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Dec/06/t1228592968jx7nbb4lrxlhfpw/1w0ef1228592093.png (open in new window)
http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Dec/06/t1228592968jx7nbb4lrxlhfpw/1w0ef1228592093.ps (open in new window)


http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Dec/06/t1228592968jx7nbb4lrxlhfpw/2mwqv1228592093.png (open in new window)
http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Dec/06/t1228592968jx7nbb4lrxlhfpw/2mwqv1228592093.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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