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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: Sun, 14 Dec 2008 05:46:40 -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/14/t1229259228hokx1oxf655n7f9.htm/, Retrieved Sun, 14 Dec 2008 13:53:48 +0100

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/14/t1229259228hokx1oxf655n7f9.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},
}

Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

» Textbox « » Textfile « » CSV «

493 481 462 457 442 439 488 521 501 485 464 460 467 460 448 443 436 431 484 510 513 503 471 471 476 475 470 461 455 456 517 525 523 519 509 512 519 517 510 509 501 507 569 580 578 565 547 555 562 561 555 544 537 543 594 611 613 611 594 595 591 589 584 573 567 569 621 629 628 612 595 597 593 590 580 574 573 573 620 626 620 588 566 557 561 549 532 526 511 499 555 565 542 527 510 514 517 508 493 490 469 478

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	3 seconds
R Server	'Herman Ole Andreas Wold' @ 193.190.124.10:1001

Central Tendency - Ungrouped Data
Measure	Value	S.E.	Value/S.E.
Arithmetic Mean	531.039215686274	5.29381205212465	100.313197835035
Geometric Mean	528.36044797212
Harmonic Mean	525.67535219736
Quadratic Mean	533.697590070499
Winsorized Mean ( 1 / 34 )	531.078431372549	5.28307014749609	100.524584483182
Winsorized Mean ( 2 / 34 )	531.098039215686	5.26567530345921	100.860385156447
Winsorized Mean ( 3 / 34 )	531.039215686274	5.22493316057207	101.635599799354
Winsorized Mean ( 4 / 34 )	531.039215686274	5.21168887647575	101.893882822371
Winsorized Mean ( 5 / 34 )	531.28431372549	5.17164846685104	102.730167591802
Winsorized Mean ( 6 / 34 )	531.28431372549	5.03989861342288	105.415674892846
Winsorized Mean ( 7 / 34 )	531.28431372549	5.01868721916412	105.861212409642
Winsorized Mean ( 8 / 34 )	531.28431372549	4.99464627861201	106.370758626201
Winsorized Mean ( 9 / 34 )	531.549019607843	4.95623689002775	107.248509585438
Winsorized Mean ( 10 / 34 )	530.176470588235	4.75147060100352	111.581553398702
Winsorized Mean ( 11 / 34 )	530.06862745098	4.70599933928878	112.636783228085
Winsorized Mean ( 12 / 34 )	530.186274509804	4.6889820005981	113.070656795478
Winsorized Mean ( 13 / 34 )	530.313725490196	4.6351861420208	114.410448521706
Winsorized Mean ( 14 / 34 )	530.725490196078	4.57764093368124	115.938645665963
Winsorized Mean ( 15 / 34 )	530.872549019608	4.5173509100512	117.518554478114
Winsorized Mean ( 16 / 34 )	530.71568627451	4.45374374249539	119.161702369781
Winsorized Mean ( 17 / 34 )	530.71568627451	4.40905814430434	120.369400653084
Winsorized Mean ( 18 / 34 )	530.539215686274	4.3856659168912	120.971187897128
Winsorized Mean ( 19 / 34 )	531.098039215686	4.26241969618869	124.600127878204
Winsorized Mean ( 20 / 34 )	530.509803921569	4.13476027070759	128.304851838673
Winsorized Mean ( 21 / 34 )	530.098039215686	3.9772579524114	133.282287837099
Winsorized Mean ( 22 / 34 )	530.745098039216	3.89437034939799	136.285214404753
Winsorized Mean ( 23 / 34 )	530.970588235294	3.75345452434262	141.461841296262
Winsorized Mean ( 24 / 34 )	530.264705882353	3.60939708606652	146.912266297702
Winsorized Mean ( 25 / 34 )	530.754901960784	3.48974247170664	152.089991242598
Winsorized Mean ( 26 / 34 )	531.264705882353	3.42844107522468	154.958097346776
Winsorized Mean ( 27 / 34 )	532.058823529412	3.33506527116369	159.534755775189
Winsorized Mean ( 28 / 34 )	530.960784313726	3.20374954954412	165.731052350606
Winsorized Mean ( 29 / 34 )	532.666666666667	3.00902458486251	177.023035752633
Winsorized Mean ( 30 / 34 )	532.666666666667	2.87439554417728	185.314323822168
Winsorized Mean ( 31 / 34 )	532.362745098039	2.83859433276403	187.544496567659
Winsorized Mean ( 32 / 34 )	532.676470588235	2.73377608734332	194.850073147684
Winsorized Mean ( 33 / 34 )	533.970588235294	2.59763552575524	205.560242359269
Winsorized Mean ( 34 / 34 )	533.303921568627	2.44619003818375	218.014100803303
Trimmed Mean ( 1 / 34 )	531.06	5.2136537915924	101.859467703128
Trimmed Mean ( 2 / 34 )	531.040816326531	5.13513817648504	103.413150352660
Trimmed Mean ( 3 / 34 )	531.010416666667	5.05648998447389	105.015617216122
Trimmed Mean ( 4 / 34 )	531	4.98407438582265	106.539340887537
Trimmed Mean ( 5 / 34 )	530.989130434783	4.90588671436518	108.235098230044
Trimmed Mean ( 6 / 34 )	530.922222222222	4.82767272239443	109.974775166386
Trimmed Mean ( 7 / 34 )	530.852272727273	4.76982032560337	111.293976814551
Trimmed Mean ( 8 / 34 )	530.779069767442	4.70751650793135	112.751398507721
Trimmed Mean ( 9 / 34 )	530.702380952381	4.64017292252958	114.371250772928
Trimmed Mean ( 10 / 34 )	530.585365853659	4.56918284420196	116.122594333677
Trimmed Mean ( 11 / 34 )	530.6375	4.5227009597716	117.327567026850
Trimmed Mean ( 12 / 34 )	530.705128205128	4.47515466944673	118.589225938584
Trimmed Mean ( 13 / 34 )	530.763157894737	4.42139409708893	120.044299657475
Trimmed Mean ( 14 / 34 )	530.810810810811	4.36597169901412	121.579077329034
Trimmed Mean ( 15 / 34 )	530.819444444444	4.30876596700346	123.195236991162
Trimmed Mean ( 16 / 34 )	530.814285714286	4.24943269175021	124.914153069138
Trimmed Mean ( 17 / 34 )	530.823529411765	4.18770207020845	126.757711153349
Trimmed Mean ( 18 / 34 )	530.833333333333	4.11991486850135	128.845704408069
Trimmed Mean ( 19 / 34 )	530.859375	4.04138236527689	131.355889400885
Trimmed Mean ( 20 / 34 )	530.838709677419	3.96535722621027	133.869076452602
Trimmed Mean ( 21 / 34 )	530.866666666667	3.892442820532	136.383934496464
Trimmed Mean ( 22 / 34 )	530.931034482759	3.82724828601573	138.723959044599
Trimmed Mean ( 23 / 34 )	530.946428571429	3.75918093825379	141.239923614335
Trimmed Mean ( 24 / 34 )	530.944444444444	3.69650680917312	143.634104264835
Trimmed Mean ( 25 / 34 )	531	3.64021032657722	145.870692174889
Trimmed Mean ( 26 / 34 )	531.02	3.58733486429291	148.026325974079
Trimmed Mean ( 27 / 34 )	531	3.52825474707874	150.499336942619
Trimmed Mean ( 28 / 34 )	530.913043478261	3.46658668901006	153.151526589941
Trimmed Mean ( 29 / 34 )	530.909090909091	3.40901338882969	155.736874677207
Trimmed Mean ( 30 / 34 )	530.761904761905	3.36664128573525	157.65323944989
Trimmed Mean ( 31 / 34 )	530.761904761905	3.33182016937074	159.300885936514
Trimmed Mean ( 32 / 34 )	530.447368421053	3.2866161130079	161.396205149006
Trimmed Mean ( 33 / 34 )	530.25	3.24144172312945	163.584616134351
Trimmed Mean ( 34 / 34 )	529.911764705882	3.19964560399932	165.615768209933
Median	524
Midrange	530
Midmean - Weighted Average at Xnp	531
Midmean - Weighted Average at X(n+1)p	531
Midmean - Empirical Distribution Function	531
Midmean - Empirical Distribution Function - Averaging	531
Midmean - Empirical Distribution Function - Interpolation	531.843137254902
Midmean - Closest Observation	531
Midmean - True Basic - Statistics Graphics Toolkit	531
Midmean - MS Excel (old versions)	531
Number of observations	102

Charts produced by software:

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Dec/14/t1229259228hokx1oxf655n7f9/1mmoh1229258794.png (open in new window)

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Dec/14/t1229259228hokx1oxf655n7f9/1mmoh1229258794.ps (open in new window)

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Dec/14/t1229259228hokx1oxf655n7f9/2wf311229258794.png (open in new window)

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Dec/14/t1229259228hokx1oxf655n7f9/2wf311229258794.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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