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WS3_Part2 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 12:16:03 -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/t1255976215zx7uke3qzjtlhnc.htm/, Retrieved Mon, 19 Oct 2009 20:16:57 +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/t1255976215zx7uke3qzjtlhnc.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 «

104,6 103 106,9 56,3 93,4 109,1 113,8 97,4 72,5 82,7 88,9 105,9 100,8 94 105 58,5 87,6 113,1 112,5 89,6 74,5 82,7 90,1 109,4 96 89,2 109,1 49,1 92,9 107,7 103,5 91,1 79,8 71,9 82,9 90,1 100,7 90,7 108,8 44,1 93,6 107,4 96,5 93,6 76,5 76,7 84 103,3 88,5 99 105,9 44,7 94 107,1 104,8 102,5 77,7 85,2 91,3 106,5 92,4 97,5 107 51,1 98,6 102,2 114,3 99,4 72,5 92,3 99,4 85,9 109,4 97,6 104,7 56,9 86,7 108,5 103,4 86,2 71 75,9 87,1 102 88,5 87,8 100,8

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	91.6988505747127	1.7395339412247	52.7146084371041
Geometric Mean	89.9751089483086
Harmonic Mean	87.8502737741544
Quadratic Mean	93.1069993788509
Winsorized Mean ( 1 / 29 )	91.7	1.73649214565821	52.807610002314
Winsorized Mean ( 2 / 29 )	91.7850574712644	1.70346638490476	53.8813435267149
Winsorized Mean ( 3 / 29 )	91.8333333333333	1.68071491855132	54.6394467733341
Winsorized Mean ( 4 / 29 )	91.9298850574713	1.59665524738321	57.5765402131343
Winsorized Mean ( 5 / 29 )	91.964367816092	1.58775383737302	57.9210490010527
Winsorized Mean ( 6 / 29 )	92.0540229885057	1.55712039022154	59.118115443475
Winsorized Mean ( 7 / 29 )	93.0597701149425	1.33200427589104	69.8644680045718
Winsorized Mean ( 8 / 29 )	93.1149425287356	1.31240817067602	70.9496821257765
Winsorized Mean ( 9 / 29 )	93.1459770114943	1.29653456700833	71.8422627376782
Winsorized Mean ( 10 / 29 )	93.0540229885057	1.28410470705626	72.4660710899713
Winsorized Mean ( 11 / 29 )	93.2689655172414	1.23312919809955	75.636004451913
Winsorized Mean ( 12 / 29 )	93.4206896551724	1.19404445016659	78.2388709583959
Winsorized Mean ( 13 / 29 )	93.4954022988506	1.17687545679533	79.443752318059
Winsorized Mean ( 14 / 29 )	93.5114942528736	1.16933832251533	79.9695797634717
Winsorized Mean ( 15 / 29 )	93.6149425287356	1.13166795348799	82.7229773894351
Winsorized Mean ( 16 / 29 )	93.8908045977011	1.05545341397534	88.9577913667112
Winsorized Mean ( 17 / 29 )	94.4574712643678	0.97142360085939	97.2361297180798
Winsorized Mean ( 18 / 29 )	94.271264367816	0.946392449741666	99.6111754627259
Winsorized Mean ( 19 / 29 )	94.271264367816	0.934457128903064	100.883455700615
Winsorized Mean ( 20 / 29 )	94.5011494252874	0.896301310480094	105.434576877578
Winsorized Mean ( 21 / 29 )	94.7666666666667	0.854484321333095	110.905097145398
Winsorized Mean ( 22 / 29 )	94.6655172413793	0.795003824828618	119.075549431208
Winsorized Mean ( 23 / 29 )	94.7183908045977	0.781477515363385	121.204243170776
Winsorized Mean ( 24 / 29 )	94.8287356321839	0.76059549126049	124.676962619157
Winsorized Mean ( 25 / 29 )	94.8574712643678	0.735268095211646	129.010726675232
Winsorized Mean ( 26 / 29 )	94.8574712643678	0.697928412462555	135.912895320703
Winsorized Mean ( 27 / 29 )	94.8264367816092	0.678629883292827	139.732185564089
Winsorized Mean ( 28 / 29 )	94.987356321839	0.643715750038507	147.561025679668
Winsorized Mean ( 29 / 29 )	94.587356321839	0.594017678524494	159.233234534010
Trimmed Mean ( 1 / 29 )	91.9929411764706	1.66764211362825	55.1634792769319
Trimmed Mean ( 2 / 29 )	92.3	1.58677683091770	58.1682302145909
Trimmed Mean ( 3 / 29 )	92.5765432098765	1.51288118511077	61.1922100168749
Trimmed Mean ( 4 / 29 )	92.8493670886076	1.43609729515380	64.6539530447786
Trimmed Mean ( 5 / 29 )	93.109090909091	1.37682853228262	67.6257709118845
Trimmed Mean ( 6 / 29 )	93.3746666666667	1.30912265969929	71.326140430657
Trimmed Mean ( 7 / 29 )	93.6369863013699	1.23703705706537	75.6945685390424
Trimmed Mean ( 8 / 29 )	93.7380281690141	1.21118091964388	77.3939108920043
Trimmed Mean ( 9 / 29 )	93.836231884058	1.18479636095895	79.2003039308028
Trimmed Mean ( 10 / 29 )	93.9358208955224	1.15660199898057	81.2170660074231
Trimmed Mean ( 11 / 29 )	94.0538461538462	1.12506212846198	83.5988020345352
Trimmed Mean ( 12 / 29 )	94.152380952381	1.09746451058153	85.790820609307
Trimmed Mean ( 13 / 29 )	94.239344262295	1.07164721452366	87.9387759190736
Trimmed Mean ( 14 / 29 )	94.3237288135593	1.04337642292910	90.4023962404306
Trimmed Mean ( 15 / 29 )	94.4122807017544	1.00999103626211	93.4783352644058
Trimmed Mean ( 16 / 29 )	94.4963636363636	0.976198093672586	96.8003976332876
Trimmed Mean ( 17 / 29 )	94.5584905660377	0.9493846221786	99.5997705851288
Trimmed Mean ( 18 / 29 )	94.5686274509804	0.93196183628031	101.472639511106
Trimmed Mean ( 19 / 29 )	94.5979591836735	0.914055448104867	103.492582840358
Trimmed Mean ( 20 / 29 )	94.6297872340426	0.893008691225989	105.967375417285
Trimmed Mean ( 21 / 29 )	94.6422222222222	0.873200827843373	108.385401392677
Trimmed Mean ( 22 / 29 )	94.6302325581395	0.85537538716979	110.630062517050
Trimmed Mean ( 23 / 29 )	94.6268292682927	0.84337464109782	112.200230665125
Trimmed Mean ( 24 / 29 )	94.6179487179487	0.829010692739082	114.133568537370
Trimmed Mean ( 25 / 29 )	94.5972972972973	0.813008782595782	116.354582290324
Trimmed Mean ( 26 / 29 )	94.5714285714286	0.795782701522983	118.840769459346
Trimmed Mean ( 27 / 29 )	94.5424242424242	0.779775490987804	121.243133870058
Trimmed Mean ( 28 / 29 )	94.5129032258065	0.760570554587166	124.265793167746
Trimmed Mean ( 29 / 29 )	94.4620689655172	0.740862461609611	127.502841431981
Median	93.6
Midrange	79.2
Midmean - Weighted Average at Xnp	94.415909090909
Midmean - Weighted Average at X(n+1)p	94.6422222222222
Midmean - Empirical Distribution Function	94.6422222222222
Midmean - Empirical Distribution Function - Averaging	94.6422222222222
Midmean - Empirical Distribution Function - Interpolation	94.6302325581395
Midmean - Closest Observation	94.415909090909
Midmean - True Basic - Statistics Graphics Toolkit	94.6422222222222
Midmean - MS Excel (old versions)	94.6422222222222
Number of observations	87

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Oct/19/t1255976215zx7uke3qzjtlhnc/15k7j1255976161.png (open in new window)

http://www.freestatistics.org/blog/date/2009/Oct/19/t1255976215zx7uke3qzjtlhnc/15k7j1255976161.ps (open in new window)

http://www.freestatistics.org/blog/date/2009/Oct/19/t1255976215zx7uke3qzjtlhnc/2zu3h1255976161.png (open in new window)

http://www.freestatistics.org/blog/date/2009/Oct/19/t1255976215zx7uke3qzjtlhnc/2zu3h1255976161.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')

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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