The absolute measures of variation describe the dispersion of data and are expressed in the units of the variable. They can be grouped into three types. The first set is based on the distance between two extreme points, defined by order statistics: minimum (min), maximum (max), and percentiles (P10, P25, P75, and P90) [1]. These are known as ranges (R), which include the total range (R = max − min), the percentile range (PR = P90 − P10), the interquartile range (IQR = P75 − P25), and the semi-interquartile range (SIQR = (P75 − P25)/2).

A second set is based on the mean or median of the absolute or squared distances of the scores from a measure of central tendency such as the mean, median, or mode [2]. This group includes the variance (S²), the standard deviation (S), the mean absolute deviation about the mean (MAD), the median absolute deviation (MEAD), and the standard deviation from the mode (SDM). The latter was proposed by Kvalseth [3].

A third set of absolute variation measures is based on the mean or median of the differences between each sample data point and all other data points in the sample, also known as interindividual differences [4]. Given a sample of n data points, the n × (n − 1)/2 positive pairwise differences are computed. If their mean is calculated, the resulting statistic is the Mean Interindividual Difference (MID). If the median is computed, the result is the Median Interindividual Difference (MEID). If the differences are squared and their mean is taken, the resulting measure is the Interindividual Variance (IVAR). Taking the square root of the interindividual variance yields the Interindividual Standard Deviation (ISD). These measures typically produce values that are greater than or equal to those obtained from variation measures based on the distances from a central tendency statistic. A correspondence can also be established between the two groups: MID and MAD, MEID and MEAD, IVAR and S², as well as ISD and S.

Regarding notation, these measures are written in uppercase when used in descriptive statistics or when referring to the population parameter or value, and in lowercase or uppercase with a circumflex accent when used in inferential statistics to denote the point estimate obtained from a sample of size n [5].

It is worth noting that variation statistics based on interindividual differences are seldom used as descriptive measures of variability [4]. Their limited use is mainly due to the fact that they are not widely known, are not included in standard statistical software, and are labor-intensive to compute. Nonetheless, these statistics can be calculated using Excel [21] or programmed in the R language [22]. As indicators of intraindividual variability [23], they have been applied in fields such as personality research [24] and population genetics [25].

The present methodological study does not conceptualize these statistics as field-specific indicators tied to particular areas of applied research. Rather, it considers them as general descriptive statistical measures applicable to any quantitative variable, whether continuous (e.g., reaction time, age, height, weight) or discrete (e.g., frequency of a behavior, number of correct answers). This perspective underscores the importance of having practical tools for their computation and a more detailed understanding of their statistical properties. Moreover, MEID and ICV are proposed as two new statistics of this type.

Addressing this gap in descriptive and inferential statistics, the objective of this study is to develop an R script that computes point and interval estimates for five measures of interindividual variation: MID, MEID, IVAR, ISD, and ICV. For interval estimation, both asymptotic and bootstrap approaches are implemented. For MEID, the jackknife method is included to estimate bias, standard error, and confidence intervals. Because MEID is based on the median of interindividual differences, the generation of bootstrap distributions can be problematic when applied to data drawn from a discrete distribution, such as the binomial distribution. Consequently, an alternative to the bootstrap method is considered. In addition, the bootstrap-based sampling distributions are examined in terms of asymmetry, mesokurtosis, and normality.

The R script is illustrated using two random samples. One sample consists of 1000 observations drawn from a left-skewed PERT distribution, which may simulate scores on a knowledge-ability test. The other sample consists of 100 observations drawn from a right-skewed binomial distribution, which may simulate the recording of a behavior. The R environment was selected because it is a comprehensive statistical platform widely endorsed by the statistical and mathematical communities and is freely available [26].

To provide greater assurance in the interpretation of the presented examples, two well-known distributions with finite moments that are widely used in simulation studies were selected [27]. One distribution is continuous and based on a large sample size, allowing the examination of convergence between asymptotic and bootstrap standard errors and confidence intervals. The other distribution is discrete and based on a medium-sized sample, serving to demonstrate the difficulty of generating the sampling distribution for the MEID [28][29] and to observe how the distributional behavior of other statistics manifests, with particular attention to the relative measure. Both distributions were non-normal.

A simulation study, presented in a second article, evaluates whether the bootstrap and asymptotic approaches exhibit equivalent performance by comparing their standard errors and confidence interval widths and by testing whether the sampling distributions of interindividual variation statistics converge to normality.

Confidence intervals for five variation statistics based on interindividual comparisons (MID, MEID, IVAR, ISD, and ICV) were computed using a bootstrap approach, specifically the biascorrected and accelerated (BCa) percentile method. This method draws 2000 bootstrap samples of n data points with replacement from the original random sample of size n [30]. In addition, the asymptotic approach is included for the first four statistics because it is applicable, which is not the case for ICV.

The bootstrap approach was selected because it does not rely on parametric assumptions about the population distribution and, under regularity conditions, provides consistent estimates of standard errors and confidence intervals by approximating the sampling distribution of the statistic through resampling from the empirical distribution [31]. In addition, as sample size increases, the bootstrap distribution converges to the true sampling distribution for a broad class of smooth statistics. The BCa method was employed because it adjusts for both median bias and skewness in the sampling distribution via the bias-correction and acceleration parameters, thereby improving interval accuracy under asymmetric conditions [32]. The use of 2000 bootstrap resamples follows the recommendations of Efron and Narasimhan [33] and represents a practical compromise between computational cost and the stability of confidence interval estimation.

For the MEID statistic, bootstrap percentile (perc) method was included, which computes confidence interval more easily than BCa method. The jackknife method was also incorporated to estimate bias, standard error, and TypeT confidence intervals as an alternative approach in cases where bootstrap estimates are difficult to obtain. This issue may arise with large samples drawn from discrete distributions (e.g., binomial, hypergeometric, or Poisson), because MEID is based on the median of interindividual differences. As a result, the variability of the estimates can be greatly reduced, in some cases even yielding a constant value (degenerated distribution). The jackknife is a more stable and straightforward resampling method; however, it does not guarantee greater variability in median-based estimates [34].

The operation of the R script was illustrated using two random samples, which were displayed graphically by means of box plots. One sample consisted of 100 observations drawn from a binomial distribution (n = 10, p = 0.25), which could represent 10 independent records of bullying behavior (present or absent) among adolescents in compulsory secondary education. The second sample consisted of 1000 observations drawn from a PERT (0, 8, 10) distribution, which could represent scores on a knowledge-ability test ranging from 0 to 10 collected from university students. A single seed (123) was set to generate the random sample and ensure reproducibility.

For the five variation measures based on interindividual differences, point and bootstrap estimates, bootstrap bias, bootstrap standard errors, and the lower limits, upper limits, and widths of the 95% bootstrap confidence intervals were calculated. The symmetry of the corresponding bootstrap distribution was examined using the D’Agostino test [35]; mesokurtosis was evaluated using the Anscombe-Glynn test [36]; and goodness-of-fit to the normal distribution was assessed with the Shapiro-Francia W test [37][38], the K² test [39], and a normal Q-Q plot. The bootstrap distribution was graphically represented using a histogram. The asymptotic standard error and the 95% Wald-type confidence interval were provided for the MID, MEID, IVAR, and ISD statistics.

The R script is provided in the Appendix and is also available as a Word file in the following GitHub repository: https://github.com/josemoraldelarubia579/R_script_for_IV_Measures.git.

By default, the script renders the graphics in the console. To save the graphics as JPEG files, the hash symbols (#) must be removed from the commands: #jpeg (…), #par (…), and #dev.off (). The script can be executed online (with graphics displayed in the console) at https://rdrr.io/snippets/. When run with a sample of 1000 observations drawn from a PERT distribution (Example 1), the user execution time is 55.130 s, the system time is 0.340 s, and the elapsed time is 55.789 s. When the same script and sample are executed on a desktop computer (with graphics saved as JPEG files) equipped with an Intel® Celeron® J4025 CPU @ 2.00 GHz, 12.0 GB of RAM (11.8 GB usable), and a 64-bit operating system (x64-based processor), the user execution time is 65.92 s, the system time is 5.91 s, and the elapsed time is 91.91 s, based on a single execution.

The script can be run directly in R (version 3.0 or later; preferably the most recent version, 4.5.3) or using the RStudio environment (R version 3.6 or later). Three packages are required: boot [40], moments [41], and nortest [42]. R can be downloaded from https://cran.r-project.org/bin/windows/base/, and RStudio from https://posit.co/download/rstudio-server/. The mc2d package [43] was used to generate the sample drawn from the PERT distribution.

To measure the execution time of the script applied to a sample of 100 observations drawn from a binomial distribution (Example 2), the BCa confidence interval codes for MEID were disabled: #boot_ci_MEID_bca < -…, #boot_width_MEID_bca < -..., and #cat ("95% BCa confidence interval…). Otherwise, an error message was generated because the BCa method could not be applied due to a degenerate bootstrap distribution. Under these conditions, online execution yields a user time of 1.960 s, a system time of 0.140 s, and an elapsed time of 2.173 s. When executed locally on the desktop computer, the user execution time is 1.22 s, the system time is 0.58 s, and the elapsed time is 1.84 s, based on a single execution.

The four absolute statistics based on interindividual differences are primarily intended for quantitative variables and are most naturally interpreted when the underlying measurement scale is continuous. When the variables are ordinal, particularly with a limited number of ordered categories—such as Likert-type items with five to seven response options—the median interindividual difference can be computed without strong metric assumptions, and the mean interindividual difference is often used as an approximation under an assumed interval structure. In contrast, the interindividual variance and the interindividual standard deviation, which rely on squared differences and thus on stronger assumptions about equal intervals and metric properties, are less straightforward to interpret for such ordinal variables and should therefore be used with caution [48][49].

The most novel measure proposed is the interindividual coefficient of variation (ICV), defined as the ratio of the interindividual standard deviation to the sample range. As with other coefficients of variation, it is multiplied by 100. This coefficient belongs to the class of measures proposed by Merce [6], with which a direct correspondence can be established. Like other coefficients of variation, its meaningful application presupposes quantitative variables with a well-defined metric structure. Consequently, it is not appropriate for ordinal variables, particularly those with a very limited number of categories—such as Likert-type items—where both the standard deviation and the range are strongly constrained by the response scale.

Unlike classical coefficients of variation, the ICV does not, under general conditions, attain a theoretical upper bound of 100. For bounded variables, the ratio between the standard deviation and the sample range is itself bounded. Extremely bimodal discrete distributions, with all probability mass concentrated at the boundaries—such as a symmetric Bernoulli distribution (p = 0.5)—reach this bound and therefore provide a natural extreme benchmark for the interindividual coefficient of variation: I C V = 100 × I S D / R a n g e = 100 × 2 × s n − 1 / R a n g e = 100 / 2 . For bounded distributions, the statistic could be adjusted by multiplying it by 2 so that its potential range extends from 0 to 100. This rescaling ( I C V r = √ 2 × 100 × I S D / R a n g e = 100 × 2 × s n − 1 / R a n g e ) does not alter the relative ordering of variability across samples but facilitates interpretability and comparability with other normalized dispersion indices, such as the quartile coefficient of variation.

In contrast, for unbounded distributions no such bound exists at the sample level. In finite samples drawn from distributions with unbounded support, the sample range may become arbitrarily small due to sampling variability, while the sample standard deviation remains positive, allowing the ratio s_n−1/Range to exceed the Bernoulli benchmark. Consequently, it is not possible to define an absolute upper limit, and the rescaling obtained by multiplying by 2 lacks universal validity, particularly in samples from unknow distributions.

With large samples drawn from continuous distributions with finite moments, at least up to the fourth order, asymptotic and bootstrap-based approaches for the four absolute measures of interindividual variation are expected to yield similar results, as illustrated by the first example with 1000 observations, where the conditions for asymptotic approximations are well satisfied [50]. For relatively small random samples (approximately 30 ≤ n ≤ 50), bootstrap methods may offer a useful alternative for estimating standard errors and confidence intervals, although their performance depends on the smoothness of the statistic and the underlying distribution. When sample sizes fall below this range, bootstrap estimates may become unstable and their inferential accuracy may deteriorate [51]. Similar considerations apply to samples of any size drawn from distributions lacking finite fourth-order or higher moments, such as Student’s t distributions with fewer than four degrees of freedom, for which both asymptotic and bootstrap-based inference should be interpreted with caution.

With medium and large samples drawn from discrete distributions, the computation of the standard error and confidence interval of the MEID using resampling methods is problematic or impossible [11][28]. The resulting sampling distribution is degenerate because it converges to a constant with a null standard error [29]. This suggests that the central limit theorem does not apply, as demonstrated by the sample of 100 observations in the second example, and it is therefore preferable to report only the point estimate.

In the discrete-data MEID case, it should be clarified that the nearly degenerate bootstrap distribution results from the limited set of possible pairwise differences and the resampling process. With discrete data containing few distinct values, interindividual differences cluster around a small number of outcomes, restricting the population median to a narrow range. Bootstrap resamples repeatedly reproduce these same configurations, yielding identical or nearly identical MEID estimates and thus a nearly degenerate distribution.

Practically, this means the bootstrap offers too little variability for meaningful interval estimation. A useful diagnostic is the proportion of unique MEID bootstrap values: if only a very small fraction (e.g., < 1 - 5%) of resamples produce distinct estimates, the distribution can be considered degenerate. In such cases, interval estimates should be suppressed or reported as undefined because the bootstrap cannot approximate the statistic’s sampling variability.

The sampling distribution of the relative measure, for which the delta method cannot define the asymptotic standard error [52], can be accessed via the bootstrap procedure [20]. This procedure is valid for estimating both the standard error and the bias, as well as for computing confidence intervals using either the BCa or percentile method [33]. These intervals can be computed for both continuous and discrete distributions, as demonstrated by the two examples presented.

The present article is limited to introducing an R script for the complete computation of interindividual variation measures and illustrating its performance with two samples: a large sample drawn from a continuous distribution and a medium-sized sample drawn from a discrete distribution. A more comprehensive evaluation of these measures requires dedicated simulation studies, such as those reported in three complementary articles accompanying the present proposal.

The derivation of the asymptotic standard errors was based on general statistical principles. However, further methodological work grounded in formal statistical theory would facilitate more rigorous derivations under restrictive assumptions, particularly for the MEID and, more specifically, for the Merce-type relative measure (ICV).

It is concluded that the R script is appropriate for its intended purposes and functions correctly with samples drawn from both continuous and discrete distributions. The large sample drawn from a non-normal continuous distribution with finite moments allows verification of the convergence of asymptotic and bootstrap standard errors and confidence intervals. The medium-sized sample drawn from a discrete distribution illustrates the difficulty of generating the sampling distribution for the MEID, which in this case becomes nearly degenerate (i.e., almost constant), a problem that did not arise with the continuous data sample. This discrete-data example also shows approximate convergence to normality for several absolute measures of interindividual variation.

In the two empirical illustrations presented here, the relative measure based on the range (ICV) does not exhibit convergence to normality. This behavior is consistent with the fact that the statistic depends on the sample range, whose sampling distribution is governed by extreme-order statistics and may remain highly skewed, especially in moderate samples or in the presence of discrete data. Nevertheless, approximate normality may be plausible under more regular conditions, such as large samples drawn from smooth continuous distributions with stable extremes. Therefore, the present observation should be interpreted as illustrative of the examples analyzed rather than as a general property of the ICV.

It is recommended to apply these measures of variation in the descriptive analysis of quantitative variables using the R script developed for this purpose and to explore their extension to ordinal variables. Likewise, it is advisable to deepen their study by employing simulated samples drawn from known continuous and discrete probability distributions, as well as empirical samples derived from hypothesized or unknown distributions. The type of distribution, the type of bounds (bounded, semi-bounded, and unbounded), the nature of the data (continuous quantitative, discrete quantitative, or ordinal), the degree of skewness, and the level of kurtosis may be considered design factors for examining the convergence between the asymptotic and bootstrap approaches in estimating the standard error and constructing the confidence interval, as well as for assessing the approximation to normality of the sampling distribution of the statistics. Moreover, this type of study would allow for the establishment of more precise interpretive intervals for the relative measure (ICV).

The author thanks the reviewers and the editor for their valuable comments and suggestions, which substantially improved the quality of the manuscript.

# This R script is also available in the following GitHub repository: https://github.com/josemoraldelarubia579/R_script_for_IV_Measures.git

# Generate the random sample of the first example presented in the article.

library(mc2d) # Required package to generate a sample from the PERT distribution.

set.seed(123) # A fixed seed is used to ensure reproducibility of the random sample.

x <- rpert(n = 1000, min = 0, mode = 8, max = 10) # PERT (0, 8, 10) distribution.

# Generate the random sample of the second example presented in the article.

# set.seed(123) # A fixed seed is used to ensure reproducibility of the random sample.

# x <- rbinom(n = 100, size = 10, prob = 0.25) # Binomial (10, 0.25) distribution.

# Replacing the example data with your own sample

# Option 1: Define your sample manually. For example: x = {8.097046, 7.485967, ..., 6.630036} ⊂ X = "Test scores".

# x <- c(8.097046, 7.485967, ..., 6.630036)

# Option 2: Load a plain text file (e.g., college_data.csv).

# data <- read.csv("C:/Users/YourName/Documents/college_data.csv")

# x <- data$test_score # Extract the random variable test_score and rename it x.

# Option 3: Load an Excel file (e.g., mydata.xlsx) using the ‘readxl’ library.

# library(readxl)

# data <- read_excel("C:/Users/YourName/Documents/college_data.xlsx")

# x <- data$test_score # Extract the random variable test_score and rename it x.

# Option 4: Load an SPSS file (e.g., college_data) using the ‘haven’ library.

# library(haven)

# data <- read_sav("C:/Users/YourName/Documents/college_data.sav")

# x <- data$test_score # Extract the random variable test_score and rename it x.

# Load the required R packages.

# If you don’t have one, install it from a nearby Comprehensive R Archive Network (CRAN) mirror. Select a mirror from the R toolbar (e.g. Mexico (Mexico City)).

# Remove the hash symbol (#) to this line to enable the code.

library(boot)

# install.packages("boot")

# install.packages("moments")

library(moments)

# install.packages("nortest")

library(nortest)

# Round results to 4 decimal places.

# Change 4 to the desired decimals, such as 2, 3, or 6.

rnd <- function(z, k = 4) round(z, k)

# Represent sample distribution using a box plot

# The graph is displayed in the console by default.

# Remove the next three hash symbols to save it as a JPG file.

# jpeg("Boxplot_sample_distribution.jpg", width = 3, height = 2, units = "in", res = 300)

# par(mar = c(4, 1, 1, 1))

boxplot(x, horizontal = TRUE, notch = FALSE, col = "darkolivegreen2", xlab = "X values", cex.axis = 0.7, cex.lab = 0.7)

# dev.off()

# ------------------------------------------------------------------------------------------------------------

# Point estimates of the variation statistics based on interindividual differences

# ------------------------------------------------------------------------------------------------------------

# Compute the interindividual differences

diffs <- as.vector(dist(x))

# Compute the variation statistics based on interindividual differences

# Mean Interindividual Difference (MID)

MID <- mean(diffs)

# Median Interindividual Difference (MEID)

MEID <- median(diffs)

# Interindividual Variance (IVAR)

IVAR <- mean(diffs^2)

# Interindividual Standard Deviation (ISD)

ISD <- sqrt(IVAR)

# Interindividual Coefficient of Variation (ICV)

ICV = 100 * ISD /(max(x) - min(x))

# ------------------------------------------------------------------------------------------------------------

# Asymptotic Standard Errors and Wald-type 95% confidence intervals

# ------------------------------------------------------------------------------------------------------------

alpha <- 0.05 # Significance level. This value can be changed to 0.1 for small samples.

z <- qnorm(1 - alpha/2) # Critical value

n <- length(x) # Sample size

## MID = mean(diffs)

# Kernel h(x_i, x_j) = |x_i - x_j|, a symmetric second-order U-statistic

# Estimation of the influence function:

# phi_i = E[h(X_i, X_j) | X_i = x_i] - theta ≈ (1 / (n - 1)) * sum_{j != i} |x_i - x_j| - MID

abs_row_means <- vapply(seq_len(n), function(i) {

mean(abs(x[i] - x[-i]))

}, numeric(1))

phi <- abs_row_means - MID

# Asymptotic variance: Var(MID) ≈ 4 * Var(phi) / n

asympt_se_MID <- sqrt(4 * var(phi) / n)

asympt_ci_MID <- c(MID - z * asympt_se_MID, MID + z * asympt_se_MID)

asympt_width_MID = asympt_ci_MID[2] - asympt_ci_MID[1]

## MEID = median(diffs)

den <- density(diffs, bw = "nrd0", kernel = "gaussian")

fhat_MEID <- approx(den$x, den$y, xout = MEID, rule = 2)$y

# Symmetric second-order U-quantile.

# Influence function estimator: psi_i = P(|x_i - X’| <= MEID | X_i = x_i) - 1/2

psi <- vapply(seq_len(n), function(i) {

mean(abs(x[i] - x[-i]) <= MEID) - 0.5

}, numeric(1))

asympt_se_MEID <- sqrt(4 * var(psi) / (n * fhat_MEID^2))

asympt_ci_MEID <- c(MEID - z * asympt_se_MEID, MEID + z * asympt_se_MEID)

asympt_width_MEID = asympt_ci_MEID[2] - asympt_ci_MEID[1]

## IVAR = mean(diffs^2) = 2 * var(x)

# SE(var) ≈ sqrt((m4 - (n - 3)/(n - 1) * s^4) / n)

m4 <- mean((x - mean(x))^4)

se_var <- sqrt(max(m4 - (n - 3)/(n - 1) * var(x)^2, 0) / n)

asympt_se_IVAR <- 2 * se_var

asympt_ci_IVAR <- c(IVAR - z * asympt_se_IVAR, IVAR + z * asympt_se_IVAR)

asympt_width_IVAR = asympt_ci_IVAR[2] - asympt_ci_IVAR[1]

## ISD = √IVAR = √[mean(diffs^2)] = sqrt[2*sd(x)]

asympt_se_ISD <- asympt_se_IVAR / (2 * ISD)

asympt_ci_ISD <- c(ISD - z * asympt_se_ISD, ISD + z * asympt_se_ISD)

asympt_width_ISD = asympt_ci_ISD[2] - asympt_ci_ISD[1]

# ------------------------------------------------------------------------------------------------------------

# Bootstrap standard error and 95% BCa confidence intervals

# ------------------------------------------------------------------------------------------------------------

# Compute all pairwise absolute differences

boot_stats <- function(data, index) {

xb <- data[index]

d <- as.vector(dist(xb))

c(

MID = mean(d),

MEID = median(d),

IVAR = 2 * var(xb),

ISD = sqrt(2) * sd(xb),

ICV = 100 * (sqrt(2) * sd(xb)) / (max(xb) - min(xb))

)}

# Generate the bootstrap distribution of each statistic

set.seed(123) # # A fixed seed is used to ensure reproducibility of the results.

bd <- boot(data = x, statistic = boot_stats, R = 1000)

## ----------------------------------------------------------------------------------------------------------

# Jackknife standard error and jackknife confidence interval (for discrete distributions)

## ----------------------------------------------------------------------------------------------------------

# Leave-one-out MEID

T_loo <- numeric(n)

for (i in seq_len(n)) {

xi <- x[-i]

T_loo[i] <- median(as.vector(dist(xi)))

}

# Pseudovalues and t-type CI

pseudo <- n * MEID - (n - 1) * T_loo

jack_MEID <- mean(pseudo)

jack_se_MEID <- sd(pseudo) / sqrt(n)

t_crit <- qt(1 - alpha/2, df = n - 1) # Hendi (2017)

jack_bias_MEID <- (n-1) * (mean(T_loo) - MEID)

jack_ci_MEID <- c(jack_MEID - t_crit * jack_se_MEID,

jack_MEID + t_crit * jack_se_MEID)

jack_width_MEID <- diff(jack_ci_MEID)

# -------------------------------------------------------------------------------------------------------

# Mean Interindividual Difference (MID)

# -------------------------------------------------------------------------------------------------------

# Tests for symmetry, kurtosis, and normality for the bootstrap distribution of MID

# D’Agostino skewness test

sq_b1_MID <- agostino.test(bd$t[,1])

# Anscombe–Gynn kurtosis test

b2_MID <- anscombe.test(bd$t[,1])

# Shapiro–Francia normality test

w_MID <- sf.test(bd$t[,1])

# D’Agostino–Belanger–D’Agostino normality test

k2_MID <- sq_b1_MID$statistic[2]^2 + b2_MID$statistic[2]^2

p_value_MID <- pchisq(k2_MID, df = 2, lower.tail = FALSE)

# Display or save the normal QQ plot to evaluate normality graphically

# QQ-plot normal. The graph is displayed in the console by default.

# Remove the next three hash symbols to save it as a JPG file.

# jpeg("QQ_plot_MID.jpg", width = 3, height = 2, units = "in", res = 300)

# par(mar = c(4, 4, 1, 1), cex.main = 0.8, cex.axis = 0.75, cex.lab = 0.7)

qqnorm(bd$t[,1], main = "Normal QQ plot for MID", xlab = "Theoretical quantiles: normal", ylab = "Sample quantiles: boot", cex.main = 0.8, cex.axis = 0.7, cex.lab = 0.7)

qqline(bd$t[,1], col = "darkblue")

# dev.off()

# Display or save density histogram with overlaid density curve.

# Remove the next three hash symbols to save the graph as a JPG file.

# jpeg("Histogram_MID.jpg", width = 3, height = 2, units = "in", res = 300)

# par(mar = c(4, 4, 1, 1))

hist(bd$t[,1], breaks = 50, probability = TRUE, col = "lightyellow", main = "", xlab = "MID", cex.axis = 0.8, cex.lab = 0.7)

lines(density(bd$t[,1]), col = "orange", lwd = 2)

x_values <- seq(mean(bd$t[,1]) - 4 * sd(bd$t[,1]), mean(bd$t[,1]) + 4 * sd(bd$t[,1]), length = 1000)

y_values <- dnorm(x_values, mean = mean(bd$t[,1]), sd = sd(bd$t[,1]))

lines(x_values, y_values, col = "darkred", lwd = 2)

legend("topright", title = "Density curves", legend = c("Empiric", "Normal"), fill = c("orange", "darkred"), cex = 0.6, bty = "n")

# dev.off()

# 95% BCa confidence interval for MID

boot_ci_MID <- boot.ci(bd, type = "bca", index = 1)

boot_width_MID <- boot_ci_MID$bca[5] - boot_ci_MID$bca[4]

# Print the results

cat("\nSample size: n =", n, ".\n")

cat("Sample minimum: min =", rnd(min(x)), ".\n")

cat("Sample maximum: max =", rnd(max(x)), ".\n")

# Print the results about the MID on the console.

cat("\nMean Interindividual Difference: MID = mean(|xi - xj|):\n")

cat(" MID =", rnd(MID), ".\n")

cat(" Asymptotic standard error:", rnd(asympt_se_MID), ".\n")

cat(" 95% Wald-type confidence interval: (", rnd(asympt_ci_MID[1]), "," , rnd(asympt_ci_MID[2]), "); width =", rnd(asympt_width_MID), ".\n")

cat(" Bootstrap standard error:", rnd(sd(bd$t[,1])), ".\n")

cat(" Bootstrap bias:", rnd(mean(bd$t[,1]) - MID), ".\n")

cat(" 95% BCa confidence interval (", rnd(boot_ci_MID$bca[4]), ",", rnd(boot_ci_MID$bca[5]), "); width =", rnd(boot_width_MID), ".\n")

cat(" D’Agostino skewness test for the bootstrap distribution of MID:\n")

cat(" √b1 =", rnd(sq_b1_MID$statistic[1]), ", z =", rnd(sq_b1_MID$statistic[2]), ", p-value =", rnd(sq_b1_MID$p.value), ".\n")

cat(" Anscombe-Glynn kurtosis test for the bootstrap distribution of MID:\n")

cat(" b2 =", rnd(b2_MID$statistic[1]), ", z =", rnd(b2_MID$statistic[2]), ", p-value =", rnd(b2_MID$p.value), ".\n")

cat(" Shapiro-Francia normality test for the bootstrap distribution of MID:\n")

cat(" W’ =", rnd(w_MID$statistic[1]), ", z =", rnd(qnorm(w_MID$p.value, mean = 0, sd = 1, lower.tail = FALSE)), ", p-value =", rnd(w_MID$p.value), ".\n")

cat(" D’Agostino-Belanger-D’Agostino (DBD) normality test for the bootstrap distribution of MID:\n")

cat(" K^2 =", rnd(k2_MID), ", p-value =", rnd(p_value_MID), ".\n")

# ------------------------------------------------------------------------------------------------------------

# Median Interindividual Difference (MEID)

# ------------------------------------------------------------------------------------------------------------

# Tests for symmetry, kurtosis, and normality for the bootstrap distribution of MEID

# D’Agostino skewness test

sq_b1_MEID <- agostino.test(bd$t[,2])

# Anscombe–Gynn kurtosis test

b2_MEID <- anscombe.test(bd$t[,2])

# Shapiro–Francia normality test

w_MEID <- sf.test(bd$t[,2])

# D’Agostino–Belanger–D’Agostino normality test

k2_MEID <- sq_b1_MEID$statistic[2]^2 + b2_MEID$statistic[2]^2

p_value_MEID <- pchisq(k2_MEID, df = 2, lower.tail = FALSE)

# Display or save the normal QQ plot to evaluate normality graphically

# Remove the next three hash symbols to save the graph as a JPG file.

# jpeg("QQ_plot_MEID.jpg", width = 3, height = 2, units = "in", res = 300)

# par(mar = c(4, 4, 1, 1))

qqnorm(bd$t[,2], main = "Normal QQ plot for MEID", xlab = "Theoretical quantiles: normal", ylab = "Sample quantiles: boot", cex.main = 0.8, cex.axis = 0.7, cex.lab = 0.7)

qqline(bd$t[,2], col = "darkblue")

# dev.off()

# Display or save density histogram with overlaid density curve.

# Remove the next three hash symbols to save the graph as a JPG file.

# jpeg("Histogram_MEID.jpg", width = 3, height = 2, units = "in", res = 300)

# par(mar = c(4, 4, 1, 1))

hist(bd$t[,2], breaks = 50, probability = TRUE, col = "lightyellow", main = "", xlab = "MEID", cex.axis = 0.8, cex.lab = 0.7)

lines(density(bd$t[,2]), col = "orange", lwd = 2)

x_values <- seq(mean(bd$t[,2]) - 4 * sd(bd$t[,2]), mean(bd$t[,2]) + 4 * sd(bd$t[,2]), length = 1000)

y_values <- dnorm(x_values, mean = mean(bd$t[,2]), sd = sd(bd$t[,2]))

lines(x_values, y_values, col = "darkred", lwd = 2)

legend("topright", title = "Density curves", legend = c("Empiric", "Normal"), fill = c("orange", "darkred"), cex = 0.6, bty = "n")

# dev.off()

# 95% bootstrap and jackknife confidence interval for MEID

# Add the hash symbol (#) to the beginning of the following two lines to disable the code in case of an error caused by a degenerate bootstrap distribution. This error typically occurs when samples are drawn from discrete distributions or ordinal variables.

boot_ci_MEID_bca <- boot.ci(bd, type = "bca", index = 2)

boot_width_MEID_bca <- boot_ci_MEID_bca$bca[5] - boot_ci_MEID_bca$bca[4]

# 95% bootstrap PERC confidence interval for MEID

boot_ci_MEID_perc <- boot.ci(bd, type = "perc", index = 2)

boot_width_MEID_perc <- boot_ci_MEID_perc$perc[5] - boot_ci_MEID_perc$perc[4]

# Print the results about the MEID on the console.

cat("\nMedian Interindividual Difference: MEID = median(|xi - xj|):\n")

cat(" MEID =", rnd(MEID), ".\n")

cat(" Asymptotic standard error:", rnd(asympt_se_MEID), ".\n")

cat(" 95% Wald-type confidence interval: (", rnd(asympt_ci_MEID[1]), "," , rnd(asympt_ci_MEID[2]), "); width =", rnd(asympt_width_MEID), ".\n")

cat(" Bootstrap standard error:", rnd(sd(bd$t[,2])), ".\n")

cat(" Bootstrap bias:", rnd(mean(bd$t[,2]) - MEID), ".\n")

cat(" 95% BCa confidence interval (", rnd(boot_ci_MEID_bca$bca[4]), ",", rnd(boot_ci_MEID_bca$bca[5]), "); width =", rnd(boot_width_MEID_bca), ".\n")

# Prepend the hash symbol (#) to the previous line to disable the code in case of an error.

cat(" 95% bootstrap percentile confidence interval (", rnd(boot_ci_MEID_perc$perc[4]), ",", rnd(boot_ci_MEID_perc$perc[5]), "); width =", rnd(boot_width_MEID_perc), ".\n")

# Prepend the hash symbol (#) to the previous to disable the code in case of an error.

cat(" Jackknife standard error:", rnd(jack_se_MEID), ".\n")

cat(" Jackknife bias:", rnd(jack_bias_MEID), ".\n")

cat(" 95% jackknife T-type confidence interval: (", rnd(jack_ci_MEID[1]), "," , rnd(jack_ci_MEID[2]), "); width =", rnd(jack_width_MEID), ".\n")

cat(" D’Agostino skewness test for the bootstrap distribution of MEID:\n")

cat(" √b1 =", rnd(sq_b1_MEID$statistic[1]), ", z =", rnd(sq_b1_MEID$statistic[2]), ", p-value =", rnd(sq_b1_MEID$p.value), ".\n")

cat(" Anscombe-Glynn kurtosis test for the bootstrap distribution of MEID:\n")

cat(" b2 =", rnd(b2_MEID$statistic[1]), ", z =", rnd(b2_MEID$statistic[2]), ", p-value =", rnd(b2_MEID$p.value), ".\n")

cat(" Shapiro-Francia normality test for the bootstrap distribution of MEID:\n")

cat(" W’ =", rnd(w_MEID$statistic[1]), ", z =", rnd(qnorm(w_MEID$p.value, mean = 0, sd = 1, lower.tail = FALSE)), ", p-value =", rnd(w_MEID$p.value), ".\n")

cat(" DBD normality test for the bootstrap distribution of MEID:\n")

cat(" K^2 =", rnd(k2_MEID), ", p-value =", rnd(p_value_MEID), ".\n")

# ------------------------------------------------------------------------------------------------------------

# Interindividual Variance (IVAR)

# ------------------------------------------------------------------------------------------------------------

# Tests for symmetry, kurtosis, and normality for the bootstrap distribution of IVAR

# D’Agostino skewness test

sq_b1_IVAR <- agostino.test(bd$t[,3])

# Anscombe–Gynn kurtosis test

b2_IVAR <- anscombe.test(bd$t[,3])

# Shapiro–Francia normality test

w_IVAR <- sf.test(bd$t[,3])

# D’Agostino–Belanger–D’Agostino normality test

k2_IVAR <- sq_b1_IVAR$statistic[2]^2 + b2_IVAR$statistic[2]^2

p_value_IVAR <- pchisq(k2_IVAR, df = 2, lower.tail = FALSE)

# Display or save the normal QQ plot to evaluate normality graphically

# Remove the next three hash symbols to save the graph as a JPG file.

# jpeg("QQ_plot_IVAR.jpg", width = 3, height = 2, units = "in", res = 300)

# par(mar = c(4, 4, 1, 1))

qqnorm(bd$t[,3], main = "Normal QQ plot for IVAR", xlab = "Theoretical quantiles: normal", ylab = "Sample quantiles: boot", cex.main = 0.8, cex.axis = 0.7, cex.lab = 0.7)

qqline(bd$t[,3], col = "darkblue")

# dev.off()

# Display or save density histogram with overlaid density curve.

# Remove the next three hash symbols to save the graph as a JPG file.

# jpeg("Histogram_IVAR.jpg", width = 3, height = 2, units = "in", res = 300)

# par(mar = c(4, 4, 1, 1))

hist(bd$t[,3], breaks = 50, probability = TRUE, col = "lightyellow", main = "", xlab = "IVAR", cex.axis = 0.8, cex.lab = 0.7)

lines(density(bd$t[,3]), col = "orange", lwd = 2)

x_values <- seq(mean(bd$t[,3]) - 4 * sd(bd$t[,3]), mean(bd$t[,3]) + 4 * sd(bd$t[,3]), length = 1000)

y_values <- dnorm(x_values, mean = mean(bd$t[,3]), sd = sd(bd$t[,3]))

lines(x_values, y_values, col = "darkred", lwd = 2)

legend("topright", title = "Density curves", legend = c("Empiric", "Normal"), fill = c("orange", "darkred"), cex = 0.6, bty = "n")

# dev.off()

# 95% BCa confidence interval for IVAR

boot_ci_IVAR <- boot.ci(bd, type = "bca", index = 3)

boot_width_IVAR <- boot_ci_IVAR$bca[5] - boot_ci_IVAR$bca[4]

# Print the results about the IVAR on the console.

cat("\nInterindividual Variance: IVAR = mean((xi - xj)^2) = 2 * var(x):\n")

cat(" IVAR =", rnd(IVAR), ".\n")

cat(" Asymptotic standard error:", rnd(asympt_se_IVAR), ".\n")

cat(" 95% Wald-type confidence interval: (", rnd(asympt_ci_IVAR[1]), ", ", rnd(asympt_ci_IVAR[2]), "); width =", rnd(asympt_width_IVAR), ".\n")

cat(" Bootstrap standard error:", rnd(sd(bd$t[,3])), ".\n")

cat(" Bootstrap bias:", rnd(mean(bd$t[,3]) - IVAR), ".\n")

cat(" 95% BCa confidence interval (", rnd(boot_ci_IVAR$bca[4]), ",", rnd(boot_ci_IVAR$bca[5]), "); width =", rnd(boot_width_IVAR), ".\n")

cat(" D’Agostino skewness test for the bootstrap distribution of IVAR:\n")

cat(" √b1 =", rnd(sq_b1_IVAR$statistic[1]), ", z =", rnd(sq_b1_IVAR$statistic[2]), ", p-value =", rnd(sq_b1_IVAR$p.value), ".\n")

cat(" Anscombe-Glynn kurtosis test for the bootstrap distribution of IVAR:\n")

cat(" b2 =", rnd(b2_IVAR$statistic[1]), ", z =", rnd(b2_IVAR$statistic[2]), ", p-value =", rnd(b2_IVAR$p.value), ".\n")

cat(" Shapiro-Francia normality test for the bootstrap distribution of IVAR:\n")

cat(" W’ =", rnd(w_IVAR$statistic[1]), ", z =", rnd(qnorm(w_IVAR$p.value, mean = 0, sd = 1, lower.tail = FALSE)), ", p-value =", rnd(w_IVAR$p.value), ".\n")

cat(" DBD normality test for the bootstrap distribution of IVAR:\n")

cat(" K^2 =", rnd(k2_IVAR), ", p-value =", rnd(p_value_IVAR), ".\n")

# ------------------------------------------------------------------------------------------------------------

# Interindividual Standard Deviation (ISD)

# ------------------------------------------------------------------------------------------------------------

# Tests for symmetry, kurtosis, and normality for the bootstrap distribution of ISD

# D’Agostino skewness test

sq_b1_ISD <- agostino.test(bd$t[,4])

# Anscombe–Gynn kurtosis test

b2_ISD <- anscombe.test(bd$t[,4])

# Shapiro–Francia normality test

w_ISD <- sf.test(bd$t[,4])

# D’Agostino–Belanger–D’Agostino normality test

k2_ISD <- sq_b1_ISD$statistic[2]^2 + b2_ISD$statistic[2]^2

p_value_ISD <- pchisq(k2_ISD, df = 2, lower.tail = FALSE)

# Display or save the normal QQ plot to evaluate normality graphically

# Remove the next three hash symbols to save the graph as a JPG file.

# jpeg("QQ_plot_ISD.jpg", width = 3, height = 2, units = "in", res = 300)

# par(mar = c(4, 4, 1, 1))

qqnorm(bd$t[,4], main = "Normal QQ plot for ISD", xlab = "Theoretical quantiles: normal", ylab = "Sample quantiles: boot", cex.main = 0.8, cex.axis = 0.7, cex.lab = 0.7)

qqline(bd$t[,4], col = "darkblue")

# dev.off()

# Display or save density histogram with overlaid density curve.

# Remove the next three hash symbols to save the graph as a JPG file.

# jpeg("Histogram_ISD.jpg", width = 3, height = 2, units = "in", res = 300)

# par(mar = c(4, 4, 1, 1))

hist(bd$t[,4], breaks = 50, probability = TRUE, col = "lightyellow", main = "", xlab = "ISD", cex.axis = 0.8, cex.lab = 0.7)

lines(density(bd$t[,4]), col = "orange", lwd = 2)

x_values <- seq(mean(bd$t[,4]) - 4 * sd(bd$t[,4]), mean(bd$t[,4]) + 4 * sd(bd$t[,4]), length = 1000)

y_values <- dnorm(x_values, mean = mean(bd$t[,4]), sd = sd(bd$t[,4]))

lines(x_values, y_values, col = "darkred", lwd = 2)

legend("topright", title = "Density curves", legend = c("Empiric", "Normal"), fill = c("orange", "darkred"), cex = 0.6, bty = "n")

# dev.off()

# 95% BCa confidence interval for ISD

boot_ci_ISD <- boot.ci(bd, type = "bca", index = 4)

boot_width_ISD <- boot_ci_ISD$bca[5] - boot_ci_ISD$bca[4]

# Print the results about the ISD on the console.

cat("\nInterindividual Standard Deviation: ISD = √(IVAR) = √[mean((xi - xj)^2)] = √(2) * sd(x):\n")

cat(" ISD =", rnd(ISD), ".\n")

cat(" Asymptotic standard error:", rnd(asympt_se_ISD), ".\n")

cat(" 95% Wald-type confidence interval: (", rnd(asympt_ci_ISD[1]), "," , rnd(asympt_ci_ISD[2]), "); width =", rnd(asympt_width_ISD), ".\n")

cat(" Bootstrap standard error:", rnd(sd(bd$t[,4])), ".\n")

cat(" Bootstrap bias:", rnd(mean(bd$t[,4]) - ISD), ".\n")

cat(" 95% BCa confidence interval (", rnd(boot_ci_ISD$bca[4]), ",", rnd(boot_ci_ISD$bca[5]), "); width =", rnd(boot_width_ISD), ".\n")

cat(" D’Agostino skewness test for the bootstrap distribution of ISD:\n")

cat(" √b1 =", rnd(sq_b1_ISD$statistic[1]), ", z =", rnd(sq_b1_ISD$statistic[2]), ", p-value =", rnd(sq_b1_ISD$p.value), ".\n")

cat(" Anscombe-Glynn kurtosis test for the Bootstrap distribution of ISD:\n")

cat(" b2 =", rnd(b2_ISD$statistic[1]), ", z =", rnd(b2_ISD$statistic[2]), ", p-value =", rnd(b2_ISD$p.value), ".\n")

cat(" Shapiro-Francia normality test for the Bootstrap distribution of ISD:\n")

cat(" W’ =", rnd(w_ISD$statistic[1]), ", z =", rnd(qnorm(w_ISD$p.value, mean = 0, sd = 1, lower.tail = FALSE)), ", p-value =", rnd(w_ISD$p.value), ".\n")

cat(" DBD normality test for the Bootstrap distribution of ISD:\n")

cat(" K^2 =", rnd(k2_ISD), ", p-value =", rnd(p_value_ISD), ".\n")

# ------------------------------------------------------------------------------------------------------------

# Interindividual Coefficient of Variation (ICV)

# ------------------------------------------------------------------------------------------------------------

# Tests for symmetry, kurtosis, and normality for the bootstrap distribution of ICV

# D’Agostino skewness test

sq_b1_ICV <- agostino.test(bd$t[,5])

# Anscombe–Gynn kurtosis test

b2_ICV <- anscombe.test(bd$t[,5])

# Shapiro–Francia normality test

w_ICV <- sf.test(bd$t[,5])

# D’Agostino–Belanger–D’Agostino normality test

k2_ICV <- sq_b1_ICV$statistic[2]^2 + b2_ICV$statistic[2]^2

p_value_ICV <- pchisq(k2_ICV, df = 2, lower.tail = FALSE)

# Display or save the normal QQ plot to evaluate normality graphically

# Remove the next three hash symbols to save the graph as a JPG file.

# jpeg("QQ_plot_ICV.jpg", width = 3, height = 2, units = "in", res = 300)

# par(mar = c(4, 4, 1, 1))

qqnorm(bd$t[,5], main = "Normal QQ plot for ICV", xlab = "Theoretical quantiles: normal", ylab = "Sample quantiles: boot", cex.main = 0.8, cex.axis = 0.7, cex.lab = 0.7)

qqline(bd$t[,5], col = "darkblue")

# dev.off()

# Display or save density histogram with overlaid density curve.

# Remove the next three hash symbols to save the graph as a JPG file.

# jpeg("Histogram_ICV.jpg", width = 3, height = 2, units = "in", res = 300)

# par(mar = c(4, 4, 1, 1))

hist(bd$t[,5], breaks = 50, probability = TRUE, col = "lightyellow", main = "", xlab = "ICV", cex.axis = 0.8, cex.lab = 0.7)

lines(density(bd$t[,5]), col = "orange", lwd = 2)

x_values <- seq(mean(bd$t[,5])-4*sd(bd$t[,5]), mean(bd$t[,5])+4*sd(bd$t[,5]), length=1000)

y_values <- dnorm(x_values, mean = mean(bd$t[,5]), sd = sd(bd$t[,5]))

lines(x_values, y_values, col = "darkred", lwd = 2)

legend("topright", title = "Density curves", legend = c("Empiric", "Normal"), fill = c("orange", "darkred"), cex = 0.6, bty = "n")

# dev.off()

# 95% BCa confidence interval for ICV

boot_ci_ICV <- boot.ci(bd, type = "bca", index = 5)

boot_width_ICV <- boot_ci_ICV$bca[5] - boot_ci_ICV$bca[4]

# Print the results about the ICV on the console.

cat("\nInterindividual Coefficient of Variation: ICV = 100 × ISD / (max(x) - min(x)):\n")

cat(" ICV =", rnd(ICV), ".\n")

cat(" Bootstrap standard error:", rnd(sd(bd$t[,5])), ".\n")

cat(" Bootstrap bias:", rnd(mean(bd$t[,5]) - ICV), ".\n")

cat(" 95% BCa confidence interval (", rnd(boot_ci_ICV$bca[4]), ",", rnd(boot_ci_ICV$bca[5]), "); width =", rnd(boot_width_ICV), ".\n")

cat(" D’Agostino skewness test for the Bootstrap distribution of ICV:\n")

cat(" √b1 =", rnd(sq_b1_ICV$statistic[1]), ", z =", rnd(sq_b1_ICV$statistic[2]), ", p-value =", rnd(sq_b1_ICV$p.value), ".\n")

cat(" Anscombe-Glynn kurtosis test for the Bootstrap distribution of ICV:\n")

cat(" b2 =", rnd(b2_ICV$statistic[1]), ", z =", rnd(b2_ICV$statistic[2]), ", p-value =", rnd(b2_ICV$p.value), ".\n")

cat(" Shapiro-Francia normality test for the Bootstrap distribution of ICV:\n")

cat(" W’ =", rnd(w_ICV$statistic[1]), ", z =", rnd(qnorm(w_ICV$p.value, mean = 0, sd = 1, lower.tail = FALSE)), ", p-value =", rnd(w_ICV$p.value), ".\n")

cat(" DBD normality test for the Bootstrap distribution of ICV:\n")

cat(" K^2 =", rnd(k2_ICV), ", p-value =", rnd(p_value_ICV), ".\n")