66  Variability

66.1 How to Use This Chapter

This chapter lists several variability measures because different analyses need different robustness and scale properties.

• Use Variance/Standard Deviation when means and squared-error loss are central.
• Use IQR or MAD-type measures when robustness to outliers matters.
• Use Coefficient of Variation when comparing spread across variables with different units or magnitudes.

66.2 Absolute Range

66.2.1 Definition

If the observations are sorted in ascending order then

\[ Range = x_{max} - x_{min} = x_n - x_1 \]

66.3 Relative Range (biased)

66.3.1 Definition

\[ Range = \frac{x_{max} - x_{min}}{s_x} \]

where

\[ s_x = \sqrt{\frac{1}{n} \sum_{i=1}^{n} \left( x_i - \bar{x} \right)^2 } \]

Note: \(\bar{x}\) is the Arithmetic Mean as defined in Section 65.2.

66.4 Relative Range (\(n-1\) denominator)

66.4.1 Definition

\[ Range = \frac{x_{max} - x_{min}}{s_x} \]

where

\[ s_x = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} \left( x_i - \bar{x} \right)^2 } \]

Note: \(\bar{x}\) is the Arithmetic Mean as defined in Section 65.2.

66.5 Variance (biased)

66.5.1 Definition

\[ V(x) = s_x^2 = \frac{1}{n} \sum_{i=1}^{n} \left( x_i - \bar{x} \right)^2 \]

Note: \(\bar{x}\) is the Arithmetic Mean as defined in Section 65.2.

66.5.2 Alternative Formulations

\[ \begin{align*} & V(x) = \frac{1}{n} \sum_{i=1}^{n} \left( x_i - \bar{x} \right)^2 \\ & V(x) = \frac{1}{n} \sum_{i=1}^{n} \left( x_i^2 - 2 \bar{x} x_i + \bar{x}^2 \right) \\ & V(x) = \frac{1}{n} \sum_{i=1}^{n} x_i^2 - 2 \bar{x} \frac{1}{n} \sum_{i=1}^{n} x_i + \frac{1}{n} n \bar{x}^2 \\ & V(x) = \frac{1}{n} \sum_{i=1}^{n} x_i^2 - 2 \bar{x} \bar{x} + \bar{x}^2 \\ & V(x) = \frac{1}{n} \sum_{i=1}^{n} x_i^2 - \bar{x}^2 \\ & V(x) = \left( \frac{1}{n} \sum_{i=1}^{n} x_i^2 \right) - \left( \frac{1}{n} \sum_{i=1}^{n} x_i \right)^2 \\ & V(x) = \left( Mean \, of \, Squares \right) - \left( Square \, of \, Mean \right) \end{align*} \]

66.5.3 Property 1

The first property holds for any real number a

\[ \begin{align*} & V(a) = \frac{1}{n} \sum_{i=1}^{n} a^2 - \left( \frac{1}{n} \sum_{i=1}^{n} a \right)^2 \\ & V(a) = \frac{1}{n} n a^2 - \left( \frac{1}{n} n a \right)^2 \\ & V(a) = a^2 - a^2 = 0 \end{align*} \]

66.5.4 Property 2

\[ \begin{align*} & V(x+a) = \frac{1}{n} \sum_{i=1}^n \left( x_i + a \right)^2 - \left( \frac{1}{n} \sum_{i=1}^n \left( x_i + a \right) \right)^2 \\ & V(x+a) = \frac{1}{n} \sum_{i=1}^{n} \left( x_i^2 + 2ax_i + a^2 \right) - \left( \bar{x} + a \right)^2 \\ & V(x+a) = \frac{1}{n} \sum_{i=1}^{n} x_i^2 + \frac{2a}{n} \sum_{i=1}^{n} x_i + \frac{1}{n} n a^2 - \bar{x}^2 - a^2 - 2a\bar{x} \\ & V(x+a) = \frac{1}{n} \sum_{i=1}^{n} x_i^2 + 2 a \bar{x} + a^2 - \bar{x}^2 - a^2 - 2a\bar{x} \\ & V(x+a) = \frac{1}{n} \sum_{i=1}^{n} x_i^2 - \bar{x}^2 = V(x) \end{align*} \]

66.5.5 Property 3

\[ \begin{align*} & V(ax) = \frac{1}{n} \sum_{i=1}^n \left( a x_i \right)^2 - \left( \frac{1}{n} \sum_{i=1}^n \left( a x_i \right) \right)^2 \\ & V(ax) = \frac{a^2}{n} \sum_{i=1}^n x_i^2 - \left( a \bar{x} \right)^2 \\ & V(ax) = \frac{a^2}{n} \sum_{i=1}^n x_i^2 - a^2\bar{x}^2 \\ & V(ax) = a^2 \left( \frac{1}{n} \sum_{i=1}^n x_i^2 - \bar{x}^2 \right) = a^2 V(x) \end{align*} \]

66.6 Variance (unbiased)

66.6.1 Definition

\[ V(x) = s_x^2 = \frac{1}{n-1} \sum_{i=1}^{n} \left( x_i - \bar{x} \right)^2 \]

Note: \(\bar{x}\) is the Arithmetic Mean as defined in Section 65.2.

66.7 Standard Deviation (biased)

66.7.1 Definition

\[ s_x = \sqrt{V(x)} = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} \left( x_i - \bar{x} \right)^2 } \]

Note: \(\bar{x}\) is the Arithmetic Mean as defined in Section 65.2.

66.8 Standard Deviation (\(n-1\) denominator)

66.8.1 Definition

\[ s_x = \sqrt{V(x)} = \sqrt{ \frac{1}{n-1} \sum_{i=1}^{n} \left( x_i - \bar{x} \right)^2 } \]

Note: \(\bar{x}\) is the Arithmetic Mean as defined in Section 65.2. Also note that it can be shown that the square root of the sample variance underestimates the true standard deviation (this is because the square root is a nonlinear function). In other words, the standard deviation computed from the sample variance with denominator \(n-1\) is, in fact, still biased. For practical purposes, this (rather small) bias is ignored.

66.9 Coefficient of Relative Variation (biased)

66.9.1 Definition

The Coefficient of Variation is defined for any Arithmetic mean \(\bar{x} \neq 0\)

\[ CV =\frac{s_x}{\bar{x}} \]

\[ CV =\frac{ \sqrt{\frac{1}{n} \sum_{i=1}^{n} \left( x_i - \bar{x} \right)^2 } }{ \frac{1}{n} \sum_{i=1}^n x_i } \]

Note: \(\bar{x}\) is the Arithmetic Mean as defined in Section 65.2.

66.10 Coefficient of Relative Variation (\(n-1\) denominator)

66.10.1 Definition

The Coefficient of Variation is defined for any Arithmetic mean \(\bar{x} \neq 0\)

\[ CV =\frac{s_x}{\bar{x}} \]

\[ CV =\frac{ \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} \left( x_i - \bar{x} \right)^2 } }{ \frac{1}{n} \sum_{i=1}^n x_i } \]

Note: \(\bar{x}\) is the Arithmetic Mean as defined in Section 65.2.

66.11 Squared Coefficient of Relative Variation

66.11.1 Definition

The Squared Coefficient of Variation is defined for any Arithmetic mean \(\bar{x} \neq 0\)

\[ CV^2 =\frac{s_x^2}{\bar{x}^2} = \frac{V(x)}{\bar{x}^2} = \frac{1}{n} \sum_{i=1}^n \left( \frac{x_i -\bar{x} }{\bar{x}} \right)^2 \]

or

\[ CV^2 =\frac{\frac{1}{n} \sum_{i=1}^n \left( x_i - \bar{x} \right)^2 }{\bar{x}^2} = \frac{\sum_{i=1}^n \left( \frac{x_i - \bar{x} }{\bar{x} } \right)^2 }{n} = \frac{1}{n} \sum_{i=1}^n \left( \frac{x_i -\bar{x} }{\bar{x}} \right)^2 \]

Note: \(\bar{x}\) is the Arithmetic Mean as defined in Section 65.2.

66.12 Mean Squared Error (MSE)

66.12.1 Definition

For any real number a, the MSE is defined as follows

\[ MSE = \frac{1}{n} \sum_{i=1}^{n} \left( x_i - a \right)^2 \]

66.12.2 Property

There is a relationship between MSE and the Variance. If \(a = \bar{x}\) then the MSE is equal to the (biased) variance.

\[ \begin{align*} & MSE = \frac{1}{n} \sum_{i=1}^{n} \left( x_i -\bar{x} + \bar{x} - a \right)^2 \\ & MSE = \frac{1}{n} \sum_{i=1}^{n} \left( \left( x_i -\bar{x} \right)^2 + 2 \left( \left( x_i - \bar{x} \right) \left( \bar{x} - a \right) \right) + \left( \bar{x} - a \right)^2 \right) \\ & MSE = \frac{1}{n} \sum_{i=1}^n \left( x_i - \bar{x} \right) ^2 + \left( \bar{x} - a \right)^2 \\ & MSE = V(x) + b \end{align*} \]

where \(b = \left( \bar{x} - a \right)^2\). The MSE is minimal and equal to the (biased) variance if \(b=0\). This minimum can only be reached if \(a\) is equal to the arithmetic mean.

66.13 Mean Absolute Deviation from the Mean (MAD)

66.13.1 Definition

\[ MAD = \frac{1}{n} \sum_{i=1}^n \left\| x_i - \bar{x} \right\| \]

Note: by default the MAD is computed against the mean. In general, however, the MAD could be computed against any measure of central tendency (such as the mean, median, geometric mean, winsorized mean, trimmed mean, etc.). To avoid confusion it is best to explicitly state the measure of central tendency that is used.

Alternative notation:

\[ MAD = Aver \left\| x - Aver(x) \right\| = \frac{1}{n} \sum_{i=1}^n \left\| x_i - \bar{x} \right\| \]

where Aver() is the mathematical function that computes the average (arithmetic mean) of the variable (in the R language we would use the function mean).

66.13.2 Property

The Mean Absolute Deviation is bound between a minimum and a maximum:

\[ \frac{x_{max} - x_{min}}{n} \leq MAD \leq \frac{x_{max} - x_{min}}{2} \]

66.14 Mean Absolute Deviation from the Median

66.14.1 Definition

\[ MADMed = Aver \left\| x - Med(x) \right\| = \frac{1}{n} \sum_{i=1}^n \left\| x_i - Med(x) \right\| \]

where Aver() and Med() are the mathematical functions that compute the arithmetic mean and median of the variable (in the R language we would use mean and median).

66.15 Median Absolute Deviation from the Mean

66.15.1 Definition

\[ MedAD = Med \left\| x - Aver(x) \right\| = Med \left\| x - \bar{x} \right\| \]

where Aver() and Med() are the functions that compute the average (arithmetic mean) and median of the variable.

66.16 Median Absolute Deviation from the Median

66.16.1 Definition

\[ MedADMed = Med \left\| x - Med(x) \right\| \]

where Med() is the function that computes the median of the variable.

66.17 Interquartile Difference (Interquartile Range)

66.17.1 Definition

\[ IQD = IQR = Q_3 - Q_1 \]

where \(Q_3 = Quantile(0.75)\) and \(Q_1 = Quantile(0.25)\) as defined in Chapter 64.

66.18 Quartile Deviation - Semi Interquartile Range - Quartile Range

66.18.1 Definition

\[ QD = \frac{Q_3 - Q_1}{2} \]

where \(Q_3 = Quantile(0.75)\) and \(Q_1 = Quantile(0.25)\) as defined in Chapter 64.

66.19 Coefficient of Quartile Variation

66.19.1 Definition

\[ CQV = \frac{Q_3 - Q_1}{Q_3 + Q_1} \]

where \(Q_3 = Quantile(0.75)\) and \(Q_1 = Quantile(0.25)\) as defined in Chapter 64.

66.20 Number of all Pairs of Observations

66.20.1 Definition

\[ \binom{n}{2} = \frac{n!}{2!(n-2)!} = \frac{1 \cdot 2 \cdot \cdots \cdot (n-2)(n-1)n}{(1 \cdot 2)(1 \cdot 2 \cdot \cdots \cdot (n-2))} = \frac{n(n-1)}{2} \]

66.21 Squared Differences between all Pairs of Observations

66.21.1 Definition

For all \(i,j = 1,2,…,n\) and \(i < j\):

\[ \frac{\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \left( x_i - x_j \right)^2 }{\binom{n}{2}} = \frac{\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \left( x_i - x_j \right)^2 }{\frac{n(n-1)}{2}}\]

66.21.2 Property

\[ \frac{\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \left( x_i - x_j \right)^2 }{\binom{n}{2}} = \frac{\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \left( x_i - x_j \right)^2 }{\frac{n(n-1)}{2}} = \frac{2n s_{x}^2}{n-1} \]

\[ \frac{\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \left( x_i - x_j \right)^2 }{n^2} = V(x) \]

Note: the proof is beyond the scope of this book.

66.22 Mean Absolute Differences between all Pairs of Observations

66.22.1 Definition

For all \(i,j = 1,2,…,n\) and \(i \neq j\):

\[ \frac{\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \left\| x_i - x_j \right\| }{\binom{n}{2}} = \frac{\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \left\| x_i - x_j \right\| }{\frac{n(n-1)}{2}}\]

66.23 Gini’s Mean Difference (Gini 1912)

66.23.1 Definition

For all \(i,j = 1,2,…,n\) and \(i \neq j\):

\[ GMD = \frac{\sum_{i=1}^{n} \sum_{j=1}^{n} \left\| x_i - x_j \right\| }{n(n-1)} \]

66.24 Leik’s Measure of Dispersion (Leik 1966)

66.24.1 Definition

\[ D = 2 \sum_{k=1}^{K} \frac{d_k}{K-1} = \frac{2}{K-1} \sum_{k=1}^K d_k \]

where

\[ \forall k=1,2,...,K : p_k = \frac{x_k}{\sum_{j=1}^K x_j} \]

\[ \forall k=1,2,...,K : c_k = \sum_{j=1}^k p_j \]

\[ c_k \leq 0.5 \Rightarrow d_k = c_k \]

\[ c_k > 0.5 \Rightarrow d_k = 1 - c_k \]

66.25 Coefficient of Dispersion

66.25.1 Definition

\[ CD = \frac{MAD}{Med(x)} = \frac{1}{n} \sum_{i=1}^{n} \left\| \frac{x_i - Med(x)}{Med(x)} \right\| \]

where Med() is the function that computes the median of the variable.

66.26 Index of Diversity (Simpson 1949)

66.26.1 Definition

\[ ID = 1 - p_1^2 - p_2^2 - ... - p_K^2 = 1 - \sum_{k=1}^K p_k^2 \]

where

\[ p_k = \frac{x_k}{\sum_{j=1}^{K} x_j} \]

Note: \(0 \leq D \leq \frac{K-1}{K}\).

66.27 Index of Qualitative Variation

66.27.1 Definition

\[ IQV = \frac{ID}{\frac{K-1}{K}} = \frac{K}{K-1} \left( 1 - \sum_{k=1}^K p_k^2 \right) \]

where

\[ ID = 1 - p_1^2 - p_2^2 - ... - p_K^2 = 1 - \sum_{k=1}^K p_k^2 \]

\[ \frac{1 - \sum_{k=1}^K p_k^2}{\frac{K-1}{K}} = \frac{K}{K-1} \left( 1 - \sum_{k=1}^K p_k^2 \right) \]

\[ p_k = \frac{x_k}{\sum_{j=1}^{K} x_j} \]

Note: \(0 \leq IQV \leq 1\).

66.28 Mean Squared Deviation from the Mean

66.28.1 Definition

\[ MSD = Aver \left( x - Aver(x) \right)^2 = Aver \left( x - \bar{x} \right)^2 = \frac{1}{n} \sum_{i=1}^n \left( x_i - \bar{x} \right)^2 \]

where Aver() is the function that computes the average (Arithmetic Mean) of the variable.

66.29 Mean Squared Deviation from the Median

66.29.1 Definition

\[ MSDMed = Aver \left( x - Med(x) \right)^2 = \frac{1}{n} \sum_{i=1}^n \left( x_i - Med(x) \right)^2 \]

where Aver() and Med() are the functions that compute the arithmetic mean and median of the variable.

66.30 Relationship between MAD, QD, and s

66.30.1 Relationship 1

If the data are unimodal and (reasonably) symmetric then

\[ MAD \approx \frac{4}{5} s \]

\[ QD \approx \frac{2}{3} s \]

66.30.2 Relationship 2

For a normal distribution the following relationships hold

\[ MAD = 0.7979 s \]

\[ QD = 0.6745 s \]

66.31 R Module

66.31.1 Public website

The Variability module can be found on the public website:

• https://compute.wessa.net/varia.wasp

66.31.2 RFC

The Variability module is available in RFC under the menu item “Descriptive / Variability”.

If you prefer to compute the measures of Variability on your local machine, the following script can be used in the R console:

x <- rnorm(150)

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 <- (1-f)*data[i] + 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)))
}

num <- 50
res <- array(NA,dim=c(num,2))
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))
cat_counts <- c(18, 24, 15, 27, 21, 19, 12, 14)
kcat <- length(cat_counts)
sumx <- sum(cat_counts)
pk <- cat_counts / sumx
ck <- cumsum(pk)
dk <- array(NA,dim=kcat)
for (i in 1:kcat) {
  if (ck[i] <= 0.5) dk[i] <- ck[i] else dk[i] <- 1 - ck[i]
}
bigd <- sum(dk) * 2 / (kcat-1)
iod <- 1 - sum(pk*pk)

df = data.frame(Statistic = c("Absolute range                                             ",
                              "Relative range (n-1 denominator)                           ",
                              "Relative range (biased)                                    ",
                              "Variance (unbiased)                                        ",
                              "Variance (biased)                                          ",
                              "Standard Deviation (n-1 denominator)                       ",
                              "Standard Deviation (biased)                                ",
                              "Coefficient of Variation (n-1 denominator)                 ",
                              "Coefficient of Variation (biased)                          ",
                              "Mean Squared Error (MSE versus 0)                          ",
                              "Mean Squared Error (MSE versus Mean)                       ",
                              "Mean Absolute Deviation from Mean (MAD Mean)               ",
                              "Mean Absolute Deviation from Median (MAD Median)           ",
                              "Median Absolute Deviation from Mean                        ",
                              "Median Absolute Deviation from Median                      ",
                              "Mean Squared Deviation from Mean                           ",
                              "Mean Squared Deviation from Median                         ",
                              "Interquartile Difference (Weighted Average at Xnp)         ",
                              "                     (Weighted Average at X(n+1)p)         ",
                              "                 (Empirical Distribution Function)         ",
                              "     (Empirical Distribution Function - Averaging)         ",
                              " (Empirical Distribution Function - Interpolation)         ",
                              "                             (Closest Observation)         ",
                              "        (True Basic - Statistics Graphics Toolkit)         ",
                              "                         (MS Excel (old versions))         ",
                              "Semi Interquartile Difference (Weighted Average at Xnp)    ",
                              "                          (Weighted Average at X(n+1)p)    ",
                              "                      (Empirical Distribution Function)    ",
                              "          (Empirical Distribution Function - Averaging)    ",
                              "      (Empirical Distribution Function - Interpolation)    ",
                              "                                  (Closest Observation)    ",
                              "             (True Basic - Statistics Graphics Toolkit)    ",
                              "                              (MS Excel (old versions))    ",
                              "Coefficient of Quartile Variation (Weighted Average at Xnp)",
                              "                              (Weighted Average at X(n+1)p)",
                              "                          (Empirical Distribution Function)",
                              "              (Empirical Distribution Function - Averaging)",
                              "          (Empirical Distribution Function - Interpolation)",
                              "                                      (Closest Observation)",
                              "                 (True Basic - Statistics Graphics Toolkit)",
                              "                                  (MS Excel (old versions))",
                              "Number of all Pairs of Observations                        ",
                              "Squared Differences between all Pairs of Observations      ",
                              "Mean Absolute Differences between all Pairs of Observations",
                              "Gini Mean Difference                                       ",
                              "Leik Measure of Dispersion                                 ",
                              "Index of Diversity                                         ",
                              "Index of Qualitative Variation                             ",
                              "Coefficient of Dispersion                                  ",
                              "Observations                                               "), 
                Value = c(range,
                          range/sd(x),
                          range/sqrt(varx*biasf),
                          varx,
                          bvarx,
                          sdx,
                          bsdx,
                          sdx/mx,
                          bsdx/mx,
                          mse0,
                          msem,
                          sum(axmm)/lx,
                          sum(axmmed)/lx,
                          median(axmm),
                          median(axmmed),
                          msem,
                          msemed,
                          qarr[1,1],
                          qarr[2,1],
                          qarr[3,1],
                          qarr[4,1],
                          qarr[5,1],
                          qarr[6,1],
                          qarr[7,1],
                          qarr[8,1],
                          qarr[1,2],
                          qarr[2,2],
                          qarr[3,2],
                          qarr[4,2],
                          qarr[5,2],
                          qarr[6,2],
                          qarr[7,2],
                          qarr[8,2],
                          qarr[1,3],
                          qarr[2,3],
                          qarr[3,3],
                          qarr[4,3],
                          qarr[5,3],
                          qarr[6,3],
                          qarr[7,3],
                          qarr[8,3],
                          lx*(lx-1)/2,
                          sdpo,
                          adpo,
                          gmd,
                          bigd,
                          iod,
                          iod*kcat/(kcat-1),
                          sum(axmmed)/lx/medx,
                          lx))

print(df)

                                                     Statistic        Value
1  Absolute range                                              6.065044e+00
2  Relative range (n-1 denominator)                            5.665120e+00
3  Relative range (biased)                                     5.684099e+00
4  Variance (unbiased)                                         1.146172e+00
5  Variance (biased)                                           1.138530e+00
6  Standard Deviation (n-1 denominator)                        1.070594e+00
7  Standard Deviation (biased)                                 1.067019e+00
8  Coefficient of Variation (n-1 denominator)                  1.097662e+01
9  Coefficient of Variation (biased)                           1.093997e+01
10 Mean Squared Error (MSE versus 0)                           1.148043e+00
11 Mean Squared Error (MSE versus Mean)                        1.138530e+00
12 Mean Absolute Deviation from Mean (MAD Mean)                8.629014e-01
13 Mean Absolute Deviation from Median (MAD Median)            8.606067e-01
14 Median Absolute Deviation from Mean                         7.706503e-01
15 Median Absolute Deviation from Median                       7.319228e-01
16 Mean Squared Deviation from Mean                            1.138530e+00
17 Mean Squared Deviation from Median                          1.145465e+00
18 Interquartile Difference (Weighted Average at Xnp)          1.589987e+00
19                      (Weighted Average at X(n+1)p)          1.606575e+00
20                  (Empirical Distribution Function)          1.584776e+00
21      (Empirical Distribution Function - Averaging)          1.584776e+00
22  (Empirical Distribution Function - Interpolation)          1.573038e+00
23                              (Closest Observation)          1.584776e+00
24         (True Basic - Statistics Graphics Toolkit)          1.606575e+00
25                          (MS Excel (old versions))          1.584776e+00
26 Semi Interquartile Difference (Weighted Average at Xnp)     7.949936e-01
27                           (Weighted Average at X(n+1)p)     8.032877e-01
28                       (Empirical Distribution Function)     7.923878e-01
29           (Empirical Distribution Function - Averaging)     7.923878e-01
30       (Empirical Distribution Function - Interpolation)     7.865189e-01
31                                   (Closest Observation)     7.923878e-01
32              (True Basic - Statistics Graphics Toolkit)     8.032877e-01
33                               (MS Excel (old versions))     7.923878e-01
34 Coefficient of Quartile Variation (Weighted Average at Xnp) 4.683934e+00
35                               (Weighted Average at X(n+1)p) 4.446692e+00
36                           (Empirical Distribution Function) 4.596883e+00
37               (Empirical Distribution Function - Averaging) 4.596883e+00
38           (Empirical Distribution Function - Interpolation) 4.413111e+00
39                                       (Closest Observation) 4.596883e+00
40                  (True Basic - Statistics Graphics Toolkit) 4.446692e+00
41                                   (MS Excel (old versions)) 4.596883e+00
42 Number of all Pairs of Observations                         1.117500e+04
43 Squared Differences between all Pairs of Observations       2.292343e+00
44 Mean Absolute Differences between all Pairs of Observations 1.209083e+00
45 Gini Mean Difference                                        1.209083e+00
46 Leik Measure of Dispersion                                  5.104762e-01
47 Index of Diversity                                          8.668444e-01
48 Index of Qualitative Variation                              9.906794e-01
49 Coefficient of Dispersion                                   6.034353e+01
50 Observations                                                1.500000e+02

To compute the Variability measures, the R code uses several standard and custom-made functions.

66.32 Purpose of Variability in general

Variability measures are mainly used to summarize univariate variables. As such they are used as a descriptive statistic of the underlying probability distribution. They are extensively used in a wide variety of other statistical methods such as Bootstrap Plots, Mean Plots, Hypothesis Testing, and many types of statistical modeling.

From the Explorative Data Analysis point of view, Variability is used as a measure of uncertainty that is associated with a variable or a predictive model. The quality of a prediction model will, generally speaking, be better if the variability of the prediction errors is smaller.

Gini, Corrado. 1912. “Variabilità e Mutabilità.” Studi Economico-Giuridici Della Facoltà Di Giurisprudenza Dell’Università Di Cagliari 3: 3–159.

Leik, Robert K. 1966. “A Measure of Ordinal Consensus.” Pacific Sociological Review 9 (2): 85–90. https://doi.org/10.2307/1388242.

Simpson, Edward H. 1949. “Measurement of Diversity.” Nature 163 (4148): 688. https://doi.org/10.1038/163688a0.