Introduction
Two statistical plots Ad- and Ud-plots derived from the empirical cumulative average deviation function (ecadf), \(U_n(t)\), which is introduced by Wijesuriya (2025) will be illustrated with examples. Suppose that \(X_1,X_2,...,X_n\) is a random sample from a unimodal distribution. Then for a real number \(t\), the ecadf computes the sum of the deviations of the data that are less than or equal to \(t\), divided by the sample size \(n\). The Ad-plot detects critical properties of the distribution such as symmetry, skewness, and outliers of the data. The Ud-plot, a slight modification of the Ad-plot, is prominent on assessing normality. To create these visualizations, let adplots package be installed, including stats and ggplot2 in R.
Ad-Plot
The first exhibit, Ad-plot consists of \(n\) ordered pairs of the form \((x_i,U_n(x_i))\) for \(i=1,2,...,n\).
Example 1
set.seed(2025)
X<-matrix(rnorm(100, mean = 2 , sd = 5))
adplot(X, title = "Ad-plot", xlab = "x", lcol = "black", rcol = "grey60")
Figure 1. Both left and right points from the sample average in Ad-plot are evenly distributed and hence, it indicates the symmetric property of the data distribution. Further, the leftmost data point appears to be an outlier.
Ud-Plot
Suppose that the random sample is from a normal distribution with mean \(\mu\) and variance \(\sigma^2\). Then the second illustration, Ud-plot consists of \(n\) ordered pairs of the form \((x_i,U_n(x_i)/[(1-n^{-1})s^{2}])\) for \(i=1,2,...,n\), where \(s^2\) is the sample variance.
Example 4
set.seed(2030)
X<-matrix(rnorm(30, mean = 2, sd = 5))
udplot(X, npdf = FALSE, lcol = "black", rcol = "grey60", pdfcol = "red")
Figure 4. The points in Ud-plot follow a bell-shaped curve evenly distributed about the sample average. Thus, it captures the symmetric property of the data distribution and confirms normality.
Example 5
set.seed(2030)
X<-matrix(rnorm(30, mean = 2, sd = 5))
udplot(X, npdf = TRUE, lcol = "black", rcol = "grey60", pdfcol = "red")
Figure 5. The points in the Ud-plot closely follow the estimated normal density curve, indicating that the data are from normal distribution with mean \(\mu\) and variance \(\sigma^2\) estimated by sample average \(\bar{X}\) and variance \(s^2\), respectively. Further, the \(d\)-value confirms the degree of proximity of Ud-plot to the estimated normal density curve.
