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robustness

*Unverified author*
R Software Module: /rwasp_centraltendency.wasp (opens new window with default values)
Title produced by software: Central Tendency
Date of computation: Wed, 20 Oct 2010 07:29:54 +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/20/t1287559844skay8qk6ii4dvg1.htm/, Retrieved Wed, 20 Oct 2010 09:30:46 +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/20/t1287559844skay8qk6ii4dvg1.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:
KDGP1W52
 
Dataseries X:
» Textbox « » Textfile « » CSV «
7,24 7,52 7,57 7,59 7,58 7,55 7,52 7,55 7,62 7,64 7,68 7,69 7,7 7,6 7,51 7,66 7,69 7,66 7,7 7,72 7,74 7,76 7,72 7,73 7,75 8,1 8,22 8,32 8,07 8,18 8,33 8,34 8,25 8,36 8,36 8,34 8,41 8,39 8,43 8,44 8,49 8,47 8,53 8,52 8,51 8,53 8,54 8,53 8,47 8,63 8,67 8,73 8,57 8,55 8,63 8,65 8,44 8,62 8,37 8,59 8,84 8,72 8,8 8,69 8,68 8,57 8,85 8,85 8,85 8,93 8,75 8,78 8,77 9,03 9,01 9,07 8,99 9,02 8,99 8,98 8,94 8,94 8,75 8,86
 
Output produced by software:


Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time4 seconds
R Server'Gwilym Jenkins' @ 72.249.127.135


Central Tendency - Ungrouped Data
MeasureValueS.E.Value/S.E.
Arithmetic Mean8.320595238095240.0550257727104073151.212692312116
Geometric Mean8.3052108908951
Harmonic Mean8.28956303940615
Quadratic Mean8.33568326390767
Winsorized Mean ( 1 / 28 )8.323333333333330.0542783283518114153.345425072502
Winsorized Mean ( 2 / 28 )8.323333333333330.0541984658854951153.571382461600
Winsorized Mean ( 3 / 28 )8.322976190476190.0541435110297579153.720658896720
Winsorized Mean ( 4 / 28 )8.323452380952380.0537480720671515154.860482633745
Winsorized Mean ( 5 / 28 )8.323452380952380.0537480720671515154.860482633745
Winsorized Mean ( 6 / 28 )8.324166666666660.0533964615811754155.893600814952
Winsorized Mean ( 7 / 28 )8.321666666666670.052773026731028157.687879247107
Winsorized Mean ( 8 / 28 )8.322619047619050.0526125052071963158.187089074038
Winsorized Mean ( 9 / 28 )8.322619047619050.0522826948351009159.184966916271
Winsorized Mean ( 10 / 28 )8.316666666666670.0507664529043122163.822094924427
Winsorized Mean ( 11 / 28 )8.317976190476190.0501686140755814165.80039819208
Winsorized Mean ( 12 / 28 )8.320833333333330.0497071770663236167.397020398703
Winsorized Mean ( 13 / 28 )8.320833333333330.0497071770663236167.397020398703
Winsorized Mean ( 14 / 28 )8.32250.0489636941499291169.972877751342
Winsorized Mean ( 15 / 28 )8.317142857142860.0477916121677297174.029342805030
Winsorized Mean ( 16 / 28 )8.313333333333330.0473334666632466175.633308087879
Winsorized Mean ( 17 / 28 )8.313333333333330.046773809469051177.734792776159
Winsorized Mean ( 18 / 28 )8.309047619047620.0462757019644346179.555301515114
Winsorized Mean ( 19 / 28 )8.313571428571430.0455619728435452182.467327679583
Winsorized Mean ( 20 / 28 )8.308809523809520.0450187711094617184.563223718544
Winsorized Mean ( 21 / 28 )8.308809523809520.0443446590735647187.368889453537
Winsorized Mean ( 22 / 28 )8.303571428571430.0430684930148054192.799209986682
Winsorized Mean ( 23 / 28 )8.303571428571430.0423423615776037196.105534013565
Winsorized Mean ( 24 / 28 )8.303571428571430.0415876966589945199.664133761915
Winsorized Mean ( 25 / 28 )8.389880952380950.0269220917841857311.635552676827
Winsorized Mean ( 26 / 28 )8.392976190476190.0248828464198234337.299682233691
Winsorized Mean ( 27 / 28 )8.418690476190480.0213175992801215394.917380966103
Winsorized Mean ( 28 / 28 )8.428690476190480.0191416345519165440.33284897013
Trimmed Mean ( 1 / 28 )8.324634146341460.054015843840358154.114673667686
Trimmed Mean ( 2 / 28 )8.3260.0536899797359864155.075491571091
Trimmed Mean ( 3 / 28 )8.32743589743590.0533379439078868156.125926260247
Trimmed Mean ( 4 / 28 )8.329078947368420.0529293397334296157.36223027373
Trimmed Mean ( 5 / 28 )8.330675675675670.0525623802391438158.491218201563
Trimmed Mean ( 6 / 28 )8.332361111111110.052107949321147159.905757560288
Trimmed Mean ( 7 / 28 )8.3340.0516387355210328161.390473951584
Trimmed Mean ( 8 / 28 )8.336176470588240.0512002684518539162.815093019036
Trimmed Mean ( 9 / 28 )8.338333333333330.0506869285335452164.506581372650
Trimmed Mean ( 10 / 28 )8.3406250.0501143904775515166.431735885048
Trimmed Mean ( 11 / 28 )8.343870967741940.0496856085333673167.933355634327
Trimmed Mean ( 12 / 28 )8.347166666666670.0492434811233427169.508054188108
Trimmed Mean ( 13 / 28 )8.35034482758620.0487552038684008171.27084218795
Trimmed Mean ( 14 / 28 )8.353750.0481195497841425173.604076461100
Trimmed Mean ( 15 / 28 )8.357222222222220.0474385229932198176.169528368678
Trimmed Mean ( 16 / 28 )8.361538461538460.0467595374768788178.819956584750
Trimmed Mean ( 17 / 28 )8.36660.0459419420662245182.112458109405
Trimmed Mean ( 18 / 28 )8.372083333333330.0449713840587629186.164680241857
Trimmed Mean ( 19 / 28 )8.378478260869570.0437725427265127191.409448457624
Trimmed Mean ( 20 / 28 )8.3850.0423459775915236198.011723353824
Trimmed Mean ( 21 / 28 )8.392619047619050.0405322594508569207.060232055275
Trimmed Mean ( 22 / 28 )8.4010.0382297373848268219.750397849563
Trimmed Mean ( 23 / 28 )8.410789473684210.0353390035911414238.003017034487
Trimmed Mean ( 24 / 28 )8.421666666666670.0314024061356946268.185394147039
Trimmed Mean ( 25 / 28 )8.433823529411760.0256467665031011328.845491239294
Trimmed Mean ( 26 / 28 )8.43843750.0237217972403972355.725049602469
Trimmed Mean ( 27 / 28 )8.443333333333330.0216095622370428390.722090559515
Trimmed Mean ( 28 / 28 )8.446071428571430.0199823949348375422.675633031677
Median8.455
Midrange8.155
Midmean - Weighted Average at Xnp8.36204545454545
Midmean - Weighted Average at X(n+1)p8.39261904761905
Midmean - Empirical Distribution Function8.36204545454545
Midmean - Empirical Distribution Function - Averaging8.39261904761905
Midmean - Empirical Distribution Function - Interpolation8.39261904761905
Midmean - Closest Observation8.36204545454545
Midmean - True Basic - Statistics Graphics Toolkit8.39261904761905
Midmean - MS Excel (old versions)8.37022222222222
Number of observations84
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2010/Oct/20/t1287559844skay8qk6ii4dvg1/10tu91287559789.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Oct/20/t1287559844skay8qk6ii4dvg1/10tu91287559789.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Oct/20/t1287559844skay8qk6ii4dvg1/2s2bu1287559789.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Oct/20/t1287559844skay8qk6ii4dvg1/2s2bu1287559789.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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