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WS3_Part 2 Central Tendency X(t)/Y(t)

*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, 19 Oct 2009 13:44:46 -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/19/t12559818485r13yy2ckp657cr.htm/, Retrieved Mon, 19 Oct 2009 21:50:50 +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/19/t12559818485r13yy2ckp657cr.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:
 
IsPrivate?
No (this computation is public)
 
User-defined keywords:
 
Dataseries X:
» Textbox « » Textfile « » CSV «
4409.665392 4414.796117 4262.170253 9180.230906 5623.040685 4793.538038 4556.98594 5228.326489 7065.351724 6277.678356 5815.624297 4815.23135 5050.863095 5328.265957 4828.295238 9732.017094 6617.739726 5110.450928 5026.346667 6108.75 7446.818792 6799.576784 6224.794673 5076.160878 5662.489583 6016.390135 4974.537122 12088.18737 6574.413348 5688.14299 5906.512077 6522.140505 7461.829574 8217.872045 7109.517491 6486.437292 5691.161867 6256.405733 5230.036765 14075.62358 6718.846154 5849.459963 6343.181347 6361.153846 7805.764706 7736.740548 7024.666667 5612.768635 6488.19209 5785.606061 5410.217186 13860.55928 6657.542553 5788.197946 5607.108779 5519.434146 7172.123552 6579.530516 6011.544359 4992.234742 5691.764069 5241.415385 4660.392523 10868.1409 5726.075051 5299.970646 4611.286089 5129.235412 7093.213793 5600.455038 5106.247485 5734.831199 4481.197441 4808.985656 4561.413563 9286.098418 6154.4406 4773.686636 4895.299807 5822.111 etc...
 
Output produced by software:


Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time3 seconds
R Server'George Udny Yule' @ 72.249.76.132


Central Tendency - Ungrouped Data
MeasureValueS.E.Value/S.E.
Arithmetic Mean6270.96022914943194.62826605304232.2201926591711
Geometric Mean6076.8010518956
Harmonic Mean5926.25080181031
Quadratic Mean6525.53722868852
Winsorized Mean ( 1 / 29 )6270.18357212644193.29088574487832.4391062101209
Winsorized Mean ( 2 / 29 )6229.55733798851176.02442001100935.3903017410817
Winsorized Mean ( 3 / 29 )6189.77647088506160.42496461992638.5836237243354
Winsorized Mean ( 4 / 29 )6141.02542228736143.45742193518842.8073036546118
Winsorized Mean ( 5 / 29 )6115.65237326437136.21995554906944.8954218830396
Winsorized Mean ( 6 / 29 )6111.79065009195133.80596909541545.6765172092862
Winsorized Mean ( 7 / 29 )6038.31056975862114.22210978461852.8646387388985
Winsorized Mean ( 8 / 29 )6010.83349150575104.77877611884457.366899234326
Winsorized Mean ( 9 / 29 )6005.74665467816103.08905066881258.2578519805409
Winsorized Mean ( 10 / 29 )5975.9232804252996.93423873669961.6492516814166
Winsorized Mean ( 11 / 29 )5974.8150509080596.48630034318361.9239729335335
Winsorized Mean ( 12 / 29 )5952.1718295287492.026518422615164.6788766059199
Winsorized Mean ( 13 / 29 )5947.6198230459888.232922649773267.408169699355
Winsorized Mean ( 14 / 29 )5950.296116793184.882926021466870.1000353744672
Winsorized Mean ( 15 / 29 )5950.5364481724184.03124656008370.8133782582618
Winsorized Mean ( 16 / 29 )5951.6858469770182.397854893806972.2310775522913
Winsorized Mean ( 17 / 29 )5948.5264597011580.53216074342773.8652285594692
Winsorized Mean ( 18 / 29 )5940.2666850804677.778151704138276.3744901997254
Winsorized Mean ( 19 / 29 )5911.9232169080571.747700450439882.398783233363
Winsorized Mean ( 20 / 29 )5894.3307601264468.995555385489385.4305864659522
Winsorized Mean ( 21 / 29 )5884.0675249885166.364945062980688.6622827669714
Winsorized Mean ( 22 / 29 )5899.0599560229961.760087185042895.5157323264212
Winsorized Mean ( 23 / 29 )5889.4108125517260.347678401536397.5913401898455
Winsorized Mean ( 24 / 29 )5891.1381096551759.762178395079198.5763616364469
Winsorized Mean ( 25 / 29 )5880.8997751724157.1787203488102.851196027087
Winsorized Mean ( 26 / 29 )5883.2796439310354.2837713029972108.380083083988
Winsorized Mean ( 27 / 29 )5888.8217474137953.4559696801784110.16209756639
Winsorized Mean ( 28 / 29 )5851.2950460804647.9876031897076121.933471504062
Winsorized Mean ( 29 / 29 )5872.6212894137943.8349157005312133.971314774141
Trimmed Mean ( 1 / 29 )6202.77348356471174.76521375146535.4920372905887
Trimmed Mean ( 2 / 29 )6132.11471603614151.30194696115540.5289874928741
Trimmed Mean ( 3 / 29 )6079.78441906173134.46226467949445.2155438074288
Trimmed Mean ( 4 / 29 )6039.40758991139121.88768776798749.5489552760029
Trimmed Mean ( 5 / 29 )6010.70385154545113.69789567804052.8655681417893
Trimmed Mean ( 6 / 29 )5986.35579450667106.35392858206656.2871148656008
Trimmed Mean ( 7 / 29 )5961.440651958998.15648216630360.734049554248
Trimmed Mean ( 8 / 29 )5947.9845496478993.922880597603463.3283872023796
Trimmed Mean ( 9 / 29 )5938.0790098985591.177061044787865.1268964129219
Trimmed Mean ( 10 / 29 )5928.3160163731388.269784759083267.1613285628081
Trimmed Mean ( 11 / 29 )5921.9439671846286.111304721195568.7708075769869
Trimmed Mean ( 12 / 29 )5915.306471825483.592868802217870.763290656062
Trimmed Mean ( 13 / 29 )5910.9249334098481.44421827982672.5763603390616
Trimmed Mean ( 14 / 29 )5906.7626708305179.548384894472374.2537095965724
Trimmed Mean ( 15 / 29 )5902.0165432631677.833448794111275.8287938502565
Trimmed Mean ( 16 / 29 )5896.8998987454575.854699667925877.7394139659205
Trimmed Mean ( 17 / 29 )5891.2791705660473.686788898248279.9502768223639
Trimmed Mean ( 18 / 29 )5885.5346329019671.323130847767382.5192972173936
Trimmed Mean ( 19 / 29 )5880.1358930612268.889456231939485.3561083899949
Trimmed Mean ( 20 / 29 )5877.039031723467.041674915168487.6624732177404
Trimmed Mean ( 21 / 29 )5875.3674979777865.214769614659690.0925899561417
Trimmed Mean ( 22 / 29 )5874.5292893953563.369453666441792.702855232383
Trimmed Mean ( 23 / 29 )5872.163249487861.891735598574794.8779864176734
Trimmed Mean ( 24 / 29 )5870.4904089230860.191768242156697.529788214654
Trimmed Mean ( 25 / 29 )5868.467492297357.981333486896101.213048051466
Trimmed Mean ( 26 / 29 )5867.2313681714355.5947494186626105.535710287811
Trimmed Mean ( 27 / 29 )5865.6040954545553.0380018881078110.592478725518
Trimmed Mean ( 28 / 29 )5863.1907911290349.5876136570597118.239018954974
Trimmed Mean ( 29 / 29 )5864.4653352413846.4589354222936126.229008089309
Median5815.624297
Midrange9168.8969165
Midmean - Weighted Average at Xnp5857.59079218182
Midmean - Weighted Average at X(n+1)p5875.36749797778
Midmean - Empirical Distribution Function5875.36749797778
Midmean - Empirical Distribution Function - Averaging5875.36749797778
Midmean - Empirical Distribution Function - Interpolation5874.52928939535
Midmean - Closest Observation5857.59079218182
Midmean - True Basic - Statistics Graphics Toolkit5875.36749797778
Midmean - MS Excel (old versions)5875.36749797778
Number of observations87
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Oct/19/t12559818485r13yy2ckp657cr/1w6we1255981483.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Oct/19/t12559818485r13yy2ckp657cr/1w6we1255981483.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Oct/19/t12559818485r13yy2ckp657cr/2291o1255981483.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Oct/19/t12559818485r13yy2ckp657cr/2291o1255981483.ps (open in new window)


 
Parameters (Session):
par1 = 0.000 ; par2 = 0.99 ; par3 = 0.005 ;
 
Parameters (R input):
par1 = 0.000 ; par2 = 0.99 ; par3 = 0.005 ;
 
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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