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*The author of this computation has been verified*

R Software Module: /rwasp_variability.wasp (opens new window with default values)

Title produced by software: Variability

Date of computation: Sun, 03 Oct 2010 13:06:35 +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/03/t1286111394pbtjvbpqtkcfbbq.htm/, Retrieved Sun, 19 May 2013 10:42:26 +0000

Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)

F     [Variability] [Variability of th...] [2010-09-25 09:46:38] [b98453cac15ba1066b407e146608df68]
-    D    [Variability] [] [2010-10-03 13:06:35] [5f45e5b827d1a020c3ecc9d930121b4e] [Current]

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Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

System-generated keywords (parent):

t1285407961ynt1v7hvkzmrn5a (pk = 79535)

Estimated Impact

Dataseries X:

» Textfile « » CSV « » Stem and Leaf « » Histogram « » Kernel Density « » Harrell-Davis Quantiles « » Central Tendency « » Variability «

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	'Sir Ronald Aylmer Fisher' @ 193.190.124.24

Variability - Ungrouped Data
Absolute range	4197.548
Relative range (unbiased)	9.65231168825579
Relative range (biased)	9.68722070523478
Variance (unbiased)	189116.171423438
Variance (biased)	187755.623427586
Standard Deviation (unbiased)	434.874891691206
Standard Deviation (biased)	433.307769867546
Coefficient of Variation (unbiased)	1.3045953732984
Coefficient of Variation (biased)	1.29989411341975
Mean Squared Error (MSE versus 0)	298871.725801784
Mean Squared Error (MSE versus Mean)	187755.623427586
Mean Absolute Deviation from Mean (MAD Mean)	190.658904093991
Mean Absolute Deviation from Median (MAD Median)	164.434194244604
Median Absolute Deviation from Mean	116.454820143885
Median Absolute Deviation from Median	67.003
Mean Squared Deviation from Mean	187755.623427586
Mean Squared Deviation from Median	196250.899172942
Interquartile Difference (Weighted Average at Xnp)	166.72125
Interquartile Difference (Weighted Average at X(n+1)p)	174.139
Interquartile Difference (Empirical Distribution Function)	174.139
Interquartile Difference (Empirical Distribution Function - Averaging)	174.139
Interquartile Difference (Empirical Distribution Function - Interpolation)	168.7825
Interquartile Difference (Closest Observation)	163.928
Interquartile Difference (True Basic - Statistics Graphics Toolkit)	174.139
Interquartile Difference (MS Excel (old versions))	174.139
Semi Interquartile Difference (Weighted Average at Xnp)	83.360625
Semi Interquartile Difference (Weighted Average at X(n+1)p)	87.0695
Semi Interquartile Difference (Empirical Distribution Function)	87.0695
Semi Interquartile Difference (Empirical Distribution Function - Averaging)	87.0695
Semi Interquartile Difference (Empirical Distribution Function - Interpolation)	84.39125
Semi Interquartile Difference (Closest Observation)	81.964
Semi Interquartile Difference (True Basic - Statistics Graphics Toolkit)	87.0695
Semi Interquartile Difference (MS Excel (old versions))	87.0695
Coefficient of Quartile Variation (Weighted Average at Xnp)	0.302121968688968
Coefficient of Quartile Variation (Weighted Average at X(n+1)p)	0.311110833200830
Coefficient of Quartile Variation (Empirical Distribution Function)	0.311110833200830
Coefficient of Quartile Variation (Empirical Distribution Function - Averaging)	0.311110833200830
Coefficient of Quartile Variation (Empirical Distribution Function - Interpolation)	0.304179203194934
Coefficient of Quartile Variation (Closest Observation)	0.298310167745786
Coefficient of Quartile Variation (True Basic - Statistics Graphics Toolkit)	0.311110833200830
Coefficient of Quartile Variation (MS Excel (old versions))	0.311110833200830
Number of all Pairs of Observations	9591
Squared Differences between all Pairs of Observations	378232.342846877
Mean Absolute Differences between all Pairs of Observations	271.104164112189
Gini Mean Difference	271.10416411219
Leik Measure of Dispersion	0.546817227789221
Index of Diversity	0.98064946254602
Index of Qualitative Variation	0.987755618071715
Coefficient of Dispersion	0.79055485151196
Observations	139

Charts produced by software:

Parameters (Session):

Parameters (R input):

R code (references can be found in the software module):

num <- 50
res <- array(NA,dim=c(num,3))
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]
}
}
}
}
iqd <- 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)
}
iqdiff <- qvalue3 - qvalue1
return(c(iqdiff,iqdiff/2,iqdiff/(qvalue3 + qvalue1)))
}
range <- max(x) - min(x)
lx <- length(x)
biasf <- (lx-1)/lx
varx <- var(x)
bvarx <- varx*biasf
sdx <- sqrt(varx)
mx <- mean(x)
bsdx <- sqrt(bvarx)
x2 <- x*x
mse0 <- sum(x2)/lx
xmm <- x-mx
xmm2 <- xmm*xmm
msem <- sum(xmm2)/lx
axmm <- abs(x - mx)
medx <- median(x)
axmmed <- abs(x - medx)
xmmed <- x - medx
xmmed2 <- xmmed*xmmed
msemed <- sum(xmmed2)/lx
qarr <- array(NA,dim=c(8,3))
for (j in 1:8) {
qarr[j,] <- iqd(x,j)
}
sdpo <- 0
adpo <- 0
for (i in 1:(lx-1)) {
for (j in (i+1):lx) {
ldi <- x[i]-x[j]
aldi <- abs(ldi)
sdpo = sdpo + ldi * ldi
adpo = adpo + aldi
}
}
denom <- (lx*(lx-1)/2)
sdpo = sdpo / denom
adpo = adpo / denom
gmd <- 0
for (i in 1:lx) {
for (j in 1:lx) {
ldi <- abs(x[i]-x[j])
gmd = gmd + ldi
}
}
gmd <- gmd / (lx*(lx-1))
sumx <- sum(x)
pk <- x / sumx
ck <- cumsum(pk)
dk <- array(NA,dim=lx)
for (i in 1:lx) {
if (ck[i] <= 0.5) dk[i] <- ck[i] else dk[i] <- 1 - ck[i]
}
bigd <- sum(dk) * 2 / (lx-1)
iod <- 1 - sum(pk*pk)
res[1,] <- c('Absolute range','http://www.xycoon.com/absolute.htm', range)
res[2,] <- c('Relative range (unbiased)','http://www.xycoon.com/relative.htm', range/sd(x))
res[3,] <- c('Relative range (biased)','http://www.xycoon.com/relative.htm', range/sqrt(varx*biasf))
res[4,] <- c('Variance (unbiased)','http://www.xycoon.com/unbiased.htm', varx)
res[5,] <- c('Variance (biased)','http://www.xycoon.com/biased.htm', bvarx)
res[6,] <- c('Standard Deviation (unbiased)','http://www.xycoon.com/unbiased1.htm', sdx)
res[7,] <- c('Standard Deviation (biased)','http://www.xycoon.com/biased1.htm', bsdx)
res[8,] <- c('Coefficient of Variation (unbiased)','http://www.xycoon.com/variation.htm', sdx/mx)
res[9,] <- c('Coefficient of Variation (biased)','http://www.xycoon.com/variation.htm', bsdx/mx)
res[10,] <- c('Mean Squared Error (MSE versus 0)','http://www.xycoon.com/mse.htm', mse0)
res[11,] <- c('Mean Squared Error (MSE versus Mean)','http://www.xycoon.com/mse.htm', msem)
res[12,] <- c('Mean Absolute Deviation from Mean (MAD Mean)', 'http://www.xycoon.com/mean2.htm', sum(axmm)/lx)
res[13,] <- c('Mean Absolute Deviation from Median (MAD Median)', 'http://www.xycoon.com/median1.htm', sum(axmmed)/lx)
res[14,] <- c('Median Absolute Deviation from Mean', 'http://www.xycoon.com/mean3.htm', median(axmm))
res[15,] <- c('Median Absolute Deviation from Median', 'http://www.xycoon.com/median2.htm', median(axmmed))
res[16,] <- c('Mean Squared Deviation from Mean', 'http://www.xycoon.com/mean1.htm', msem)
res[17,] <- c('Mean Squared Deviation from Median', 'http://www.xycoon.com/median.htm', msemed)
load(file='createtable')
mylink1 <- hyperlink('http://www.xycoon.com/difference.htm','Interquartile Difference','')
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_1.htm','(Weighted Average at Xnp)',''),sep=' ')
res[18,] <- c('', mylink2, qarr[1,1])
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_2.htm','(Weighted Average at X(n+1)p)',''),sep=' ')
res[19,] <- c('', mylink2, qarr[2,1])
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_3.htm','(Empirical Distribution Function)',''),sep=' ')
res[20,] <- c('', mylink2, qarr[3,1])
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_4.htm','(Empirical Distribution Function - Averaging)',''),sep=' ')
res[21,] <- c('', mylink2, qarr[4,1])
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_5.htm','(Empirical Distribution Function - Interpolation)',''),sep=' ')
res[22,] <- c('', mylink2, qarr[5,1])
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_6.htm','(Closest Observation)',''),sep=' ')
res[23,] <- c('', mylink2, qarr[6,1])
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_7.htm','(True Basic - Statistics Graphics Toolkit)',''),sep=' ')
res[24,] <- c('', mylink2, qarr[7,1])
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_8.htm','(MS Excel (old versions))',''),sep=' ')
res[25,] <- c('', mylink2, qarr[8,1])
mylink1 <- hyperlink('http://www.xycoon.com/deviation.htm','Semi Interquartile Difference','')
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_1.htm','(Weighted Average at Xnp)',''),sep=' ')
res[26,] <- c('', mylink2, qarr[1,2])
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_2.htm','(Weighted Average at X(n+1)p)',''),sep=' ')
res[27,] <- c('', mylink2, qarr[2,2])
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_3.htm','(Empirical Distribution Function)',''),sep=' ')
res[28,] <- c('', mylink2, qarr[3,2])
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_4.htm','(Empirical Distribution Function - Averaging)',''),sep=' ')
res[29,] <- c('', mylink2, qarr[4,2])
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_5.htm','(Empirical Distribution Function - Interpolation)',''),sep=' ')
res[30,] <- c('', mylink2, qarr[5,2])
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_6.htm','(Closest Observation)',''),sep=' ')
res[31,] <- c('', mylink2, qarr[6,2])
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_7.htm','(True Basic - Statistics Graphics Toolkit)',''),sep=' ')
res[32,] <- c('', mylink2, qarr[7,2])
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_8.htm','(MS Excel (old versions))',''),sep=' ')
res[33,] <- c('', mylink2, qarr[8,2])
mylink1 <- hyperlink('http://www.xycoon.com/variation1.htm','Coefficient of Quartile Variation','')
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_1.htm','(Weighted Average at Xnp)',''),sep=' ')
res[34,] <- c('', mylink2, qarr[1,3])
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_2.htm','(Weighted Average at X(n+1)p)',''),sep=' ')
res[35,] <- c('', mylink2, qarr[2,3])
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_3.htm','(Empirical Distribution Function)',''),sep=' ')
res[36,] <- c('', mylink2, qarr[3,3])
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_4.htm','(Empirical Distribution Function - Averaging)',''),sep=' ')
res[37,] <- c('', mylink2, qarr[4,3])
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_5.htm','(Empirical Distribution Function - Interpolation)',''),sep=' ')
res[38,] <- c('', mylink2, qarr[5,3])
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_6.htm','(Closest Observation)',''),sep=' ')
res[39,] <- c('', mylink2, qarr[6,3])
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_7.htm','(True Basic - Statistics Graphics Toolkit)',''),sep=' ')
res[40,] <- c('', mylink2, qarr[7,3])
mylink2 <- paste(mylink1,hyperlink('http://www.xycoon.com/method_8.htm','(MS Excel (old versions))',''),sep=' ')
res[41,] <- c('', mylink2, qarr[8,3])
res[42,] <- c('Number of all Pairs of Observations', 'http://www.xycoon.com/pair_numbers.htm', lx*(lx-1)/2)
res[43,] <- c('Squared Differences between all Pairs of Observations', 'http://www.xycoon.com/squared_differences.htm', sdpo)
res[44,] <- c('Mean Absolute Differences between all Pairs of Observations', 'http://www.xycoon.com/mean_abs_differences.htm', adpo)
res[45,] <- c('Gini Mean Difference', 'http://www.xycoon.com/gini_mean_difference.htm', gmd)
res[46,] <- c('Leik Measure of Dispersion', 'http://www.xycoon.com/leiks_d.htm', bigd)
res[47,] <- c('Index of Diversity', 'http://www.xycoon.com/diversity.htm', iod)
res[48,] <- c('Index of Qualitative Variation', 'http://www.xycoon.com/qualitative_variation.htm', iod*lx/(lx-1))
res[49,] <- c('Coefficient of Dispersion', 'http://www.xycoon.com/dispersion.htm', sum(axmm)/lx/medx)
res[50,] <- c('Observations', '', lx)
res
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Variability - Ungrouped Data',2,TRUE)
a<-table.row.end(a)
for (i in 1:num) {
a<-table.row.start(a)
if (res[i,1] != '') {
a<-table.element(a,hyperlink(res[i,2],res[i,1],''),header=TRUE)
} else {
a<-table.element(a,res[i,2],header=TRUE)
}
a<-table.element(a,res[i,3])
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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