# ::Free Statistics and Forecasting Software::

v1.2.1

### :: Variability - Free Statistics Software (Calculator) ::

All rights reserved. The non-commercial (academic) use of this software is free of charge. The only thing that is asked in return is to cite this software when results are used in publications.

This free online software (calculator) computes the following statistics: absolute range, relative range, variance, standard error, coefficient of variation, mean squared error, mean absolute deviation, median absolute deviation, mean squared deviation, interquartile difference, semi interquartile difference, coefficient of quartile variation, number of pairs of observations, squared differences between observations, mean absolute differences between observations, Gini mean difference, Leik measure of dispersion, index of diversity, index of qualitative variation, and coefficient of dispersion.

Enter (or paste) your data delimited by hard returns.

 Send output to: Browser Blue - Charts White Browser Black/White CSV Data[reset data] 13 16 19 15 14 13 19 15 14 15 16 16 16 16 17 15 15 20 18 16 16 16 19 16 17 17 16 15 16 14 15 12 14 16 14 7 10 14 16 16 16 14 20 14 14 11 14 15 16 14 16 14 12 16 9 14 16 16 15 16 12 16 16 14 16 17 18 18 12 16 10 14 18 18 16 17 16 16 13 16 16 20 16 15 15 16 14 16 16 15 12 17 16 15 13 16 16 16 16 14 16 16 20 15 16 13 17 16 16 12 16 16 17 13 12 18 14 14 13 16 13 16 13 16 15 16 15 17 15 12 16 10 16 12 14 15 13 15 11 12 8 16 15 17 16 10 18 13 16 13 10 15 16 16 14 10 17 13 15 16 12 13 Sample Range:(leave blank to include all observations) From: To:

 Source code of R module num <- 50 res <- array(NA,dim=c(num,3)) x <- as.numeric(na.omit(x)) 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 <- "Interquartile Difference" mylink2 <- paste(mylink1,"(Weighted Average at Xnp)",sep=" ") res[18,] <- c("", mylink2, qarr[1,1]) mylink2 <- paste(mylink1,"(Weighted Average at X(n+1)p)",sep=" ") res[19,] <- c("", mylink2, qarr[2,1]) mylink2 <- paste(mylink1,"(Empirical Distribution Function)",sep=" ") res[20,] <- c("", mylink2, qarr[3,1]) mylink2 <- paste(mylink1,"(Empirical Distribution Function - Averaging)",sep=" ") res[21,] <- c("", mylink2, qarr[4,1]) mylink2 <- paste(mylink1,"(Empirical Distribution Function - Interpolation)",sep=" ") res[22,] <- c("", mylink2, qarr[5,1]) mylink2 <- paste(mylink1,"(Closest Observation)",sep=" ") res[23,] <- c("", mylink2, qarr[6,1]) mylink2 <- paste(mylink1,"(True Basic - Statistics Graphics Toolkit)",sep=" ") res[24,] <- c("", mylink2, qarr[7,1]) mylink2 <- paste(mylink1,"(MS Excel (old versions))",sep=" ") res[25,] <- c("", mylink2, qarr[8,1]) mylink1 <- "Semi Interquartile Difference" mylink2 <- paste(mylink1,"(Weighted Average at Xnp)",sep=" ") res[26,] <- c("", mylink2, qarr[1,2]) mylink2 <- paste(mylink1,"(Weighted Average at X(n+1)p)",sep=" ") res[27,] <- c("", mylink2, qarr[2,2]) mylink2 <- paste(mylink1,"(Empirical Distribution Function)",sep=" ") res[28,] <- c("", mylink2, qarr[3,2]) mylink2 <- paste(mylink1,"(Empirical Distribution Function - Averaging)",sep=" ") res[29,] <- c("", mylink2, qarr[4,2]) mylink2 <- paste(mylink1,"(Empirical Distribution Function - Interpolation)",sep=" ") res[30,] <- c("", mylink2, qarr[5,2]) mylink2 <- paste(mylink1,"(Closest Observation)",sep=" ") res[31,] <- c("", mylink2, qarr[6,2]) mylink2 <- paste(mylink1,"(True Basic - Statistics Graphics Toolkit)",sep=" ") res[32,] <- c("", mylink2, qarr[7,2]) mylink2 <- paste(mylink1,"(MS Excel (old versions))",sep=" ") res[33,] <- c("", mylink2, qarr[8,2]) mylink1 <- "Coefficient of Quartile Variation" mylink2 <- paste(mylink1,"(Weighted Average at Xnp)",sep=" ") res[34,] <- c("", mylink2, qarr[1,3]) mylink2 <- paste(mylink1,"(Weighted Average at X(n+1)p)",sep=" ") res[35,] <- c("", mylink2, qarr[2,3]) mylink2 <- paste(mylink1,"(Empirical Distribution Function)",sep=" ") res[36,] <- c("", mylink2, qarr[3,3]) mylink2 <- paste(mylink1,"(Empirical Distribution Function - Averaging)",sep=" ") res[37,] <- c("", mylink2, qarr[4,3]) mylink2 <- paste(mylink1,"(Empirical Distribution Function - Interpolation)",sep=" ") res[38,] <- c("", mylink2, qarr[5,3]) mylink2 <- paste(mylink1,"(Closest Observation)",sep=" ") res[39,] <- c("", mylink2, qarr[6,3]) mylink2 <- paste(mylink1,"(True Basic - Statistics Graphics Toolkit)",sep=" ") res[40,] <- c("", mylink2, qarr[7,3]) mylink2 <- paste(mylink1,"(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) print(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,res[i,1],header=TRUE) } else { a<-table.element(a,res[i,2],header=TRUE) } a<-table.element(a,signif(as.numeric(res[i,3],6))) a<-table.row.end(a) } a<-table.end(a) table.save(a,file="mytable.tab")
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 Cite this software as: Wessa P., (2021), Variability (v1.0.8) in Free Statistics Software (v1.2.1), Office for Research Development and Education, URL http://www.wessa.net/rwasp_variability.wasp/ The R code is based on : Borghers, E, and P. Wessa, Statistics - Econometrics - Forecasting, Office for Research Development and Education, http://www.xycoon.com/
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