Lyne’s (1985) graphical technique for the evaluation of dispersion measures

corpus linguistics

dispersion

In this blog post, I revisit the graphical strategy used by Lyne (1985) to study the behavior of the dispersion measures D, D₂, and S, and then apply it to a number of other parts-based indices that have been proposed more recently.

Author

Affiliation

Lukas Sönning

University of Bamberg

Published

October 27, 2025

R setup

library(tlda)
library(lattice)
library(knitr)
library(kableExtra)
library(gtools)
library(tidyverse)


source("C:/Users/ba4rh5/Work Folders/My Files/R projects/my_utils_website.R")

Lyne (1985)’s graphical technique

In his dissertation, Lyne (1985) presents a corpus-based analysis of The vocabulary of French business correspondence. The book includes a chapter that examines the behavior of three dispersion measures: D, D₂, and S. The question of main interest is whether Carroll’s D₂ (Carroll 1970) and Rosengren’s S (Rosengren 1971) can really be considered an improvement over Juilland’s D, as claimed by these authors. To shed light on the way these measures respond to different subfrequency patterns, Lyne (1985) considers an item that occurs 10 times in a corpus that is divided into five parts. He then looks at all possible ways in which this item could be distributed over the five parts.

In R, we can use the function combinations() in the package {gtools} (Warnes et al. 2023) to find the number of ways in which 10 occurrences can be distributed across 5 corpus parts. It turns out that there are 30 possible ways, which are here listed from the most uneven distribution (dispersion pessimum) to the most even distribution (dispersion optimum).

R code: Create all possible combinations of subfrequencies

comb <- gtools::combinations(
    n = 5, 
    r = 10, 
    v = 1:10, 
    repeats.allowed = TRUE)

comb_tbl <- matrix(
    NA, 
    nrow = nrow(comb), 
    ncol = ncol(comb))

for(i in 1:nrow(comb)){
    comb_tbl[i,] <- sort(table(factor(comb[i,], levels = 1:5)), decreasing = TRUE)
}

combs <- str_split(
    unique(paste(
        comb_tbl[,1],
        comb_tbl[,2],
        comb_tbl[,3],
        comb_tbl[,4],
        comb_tbl[,5],
        sep = "-")),
    "-", simplify = TRUE)

combs <- data.frame(combs)

for(i in 1:5){
    combs[,i] <- as.numeric(combs[,i])
}

combs$n_zeros <- rowSums(combs[,1:5] == 0)

# print all combinations

distr_matrix <- (t(combs[,-6]))

write.table(
  format(
    distr_matrix, 
    justify="right"),
  row.names=F, col.names=F, quote=F)

10  9  8  8  7  7  7  6  6  6  6  6  5  5  5  5  5  5  4  4  4  4  4  4  4  3  3  3  3  2
 0  1  2  1  3  2  1  4  3  2  2  1  5  4  3  3  2  2  4  4  3  3  3  2  2  3  3  3  2  2
 0  0  0  1  0  1  1  0  1  2  1  1  0  1  2  1  2  1  2  1  3  2  1  2  2  3  2  2  2  2
 0  0  0  0  0  0  1  0  0  0  1  1  0  0  0  1  1  1  0  1  0  1  1  2  1  1  2  1  2  2
 0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  1  0  0  0  0  1  0  1  0  0  1  1  2

To reproduce Lyne (1985)’s graphical analyses of the three indices, we calculate dispersion scores for these 30 distributions. We will make use of the function disp() in the R package {tlda} (Soenning 2025). Further below, we will extend Lyne (1985)’s approach to a total of six dispersion measures:

D (Juilland and Chang-Rodriguez 1964)
D₂ (Carroll 1970)
S (Rosengren 1971)
D_P (Gries 2008)
D_A (Burch, Egbert, and Biber 2017)
D_KL (Gries 2021)

R code: Calculate dispersion scores for the 30 combinations

results <- matrix(
  NA, 
  nrow = nrow(combs),
  ncol = 6
)

for(i in 1:nrow(combs)){
  results[i,] <- disp(
    subfreq = as.numeric(combs[i, 1:5]), 
    partsize = rep(1000, 5),
    verbose = FALSE,
    print_score = FALSE)[-1]
}
colnames(results) <- c("D", "D2", "S", "DP", "DA", "DKL")

combs <- cbind(combs, results)

combs$D_x_location <- as.numeric(factor(rank(-combs$D, ties.method = "min")))
combs$D2_x_location <- as.numeric(factor(rank(-combs$D2, ties.method = "min")))
combs$S_x_location <- as.numeric(factor(rank(-combs$S, ties.method = "min")))
combs$DP_x_location <- as.numeric(factor(rank(-combs$DP, ties.method = "min")))
combs$DA_x_location <- as.numeric(factor(rank(-combs$DA, ties.method = "min")))
combs$DKL_x_location <- as.numeric(factor(rank(-combs$DKL, ties.method = "min")))

combs$disp_avg <- rowMeans(combs[,7:12])
combs$disp_avg_x_location <- as.numeric(factor(rank(-combs$disp_avg, ties.method = "min")))

Lyne (1985) uses an elegant graphical technique to evaluate the behavior of D, D₂ and S. In his study, Juilland’s D serves as a point of reference, so he starts by ordering the 30 possible combinations according to D. Then he draws this distribution of D scores into a scatterplot. In Figure 1 this sequence of D scores appears as a curved grey reference line, which runs from the top left corner (dispersion optimum: 2 2 2 2 2) to the bottom right corner (dispersion pessimum: 10 0 0 0 0). In a few cases, two combinations produce the same D score (e.g. 4 2 2 1 1 and 3 3 2 2 0), so there are only 22 D scores for the total set of 30 possible combinations.

In a second step, Lyne (1985) adds the D₂ scores for all 30 distributions to the scatterplot. He groups them visually according to the number of zeros they contain. The D₂ scores therefore form five traces, which are connected using lines. The top-most trace in Figure 1, for instance, connects D₂ scores for those patterns that do not contain any zeros. This graphical arrangement allows Lyne (1985, 107) to make the following observations when comparing D₂ against D:

“D₂ gives consistently higher values, except at the two end points, where they coincide”
“D₂ favours the absence of zeros”

Draw Figure

combs <- combs[order(combs$D),]

zeros_labels <- c("no zeros",
                  "one zero",
                  "two zeros",
                  "three zeros",
                  "four zeros")

p1 <- xyplot(1~1, xlim = c(0, length(unique(combs$D))), type = "n", ylim = c(0,1),
       par.settings = my_settings, axis = axis_L,
       scales = list(x = list(draw = FALSE),
                     y = list(at = c(0, .25, .5, .75, 1), label = c("(uneven) 0", ".25", ".50", ".75", "(even) 1"))),
       ylab = "\n\nDispersion", xlab = list(label = "\n\n\nDistribution\n(ranked from most to least even (according to D)", lineheight = .85, cex = .9),
       panel = function(x,y){
         panel.segments(x0 = 9, x1 = 9, y0 = 0, y1 = .75, col = "grey", lty = "13")
        panel.points(x = combs$D_x_location, y = combs$D, type = "l", col = "grey")
        panel.points(x = combs$D_x_location, y = combs$D, pch = 21, col = "grey", fill = "white")
        panel.points(x = combs$D_x_location, y = combs$D2, cex = .8)
        panel.text(x = 3, y = .6, label = "D", col = "grey40")
        panel.text(x = 10, y = .95, label = expression(D[2]))
        
        for ( i in 0:4){
            data_subset <- subset(combs, n_zeros == i)
            panel.points(x = data_subset$D_x_location, y = data_subset$D2, type = "l")
            panel.text(x = max(data_subset$D_x_location + 1), 
                       y = data_subset$D2[which.max(data_subset$D_x_location)]+.02,
                       label = zeros_labels[i+1],
                       adj = 0, cex = .8)
            }
        })

print(p1, position = c(-.04,.08,.8,1))

Figure 1: Figure 3 in Lyne (1985, 106), comparing D₂ against D for all possible combinations of subfrequencies for 10 occurrences in 5 parts.

The fact that D₂ favors zeros is evident from the layered structure of the traces. Thus, in cases where two distributions receive the same value of D, D₂ assigns a higher score to the pattern with fewer zeros. The dotted vertical line in Figure 1 points out one such pair of distributions: A 5 3 1 1 0 and B 4 4 2 0 0. Both have a D score of .55, but A (one zero), has a D₂ score of .66 compared with .73 for B (two zeros).

This leads Lyne (1985, 107) to consider the question of whether it is appropriate for a dispersion measure to be overly sensitive to the presence of zeros. He compares two sets of subfrequencies,

A: 8 3 3 3 3
B: 0 5 5 5 5

for which D yields the same score (.75) while D₂ produces different scores (A .95, B .86). Since, in his eyes, the two distributions are “mirror images of each other”, he argues that D treats zeros impartially, while D₂ penalizes them.

Lyne (1985, 110) makes the same graphical comparison for S (vs. D) and observes similar patterns. Figure 2 reproduces his Figure 4, which shows that “S penalizes zeros to an even greater extent than D₂”.

Draw Figure

p1 <- xyplot(1~1, xlim = c(0, length(unique(combs$D))), type = "n", ylim = c(0,1),
       par.settings = my_settings, axis = axis_L,
       scales = list(x = list(draw = FALSE),
                     y = list(at = c(0, .25, .5, .75, 1), label = c("(uneven) 0", ".25", ".50", ".75", "(even) 1"))),
       ylab = "\n\nDispersion", xlab = list(label = "\n\n\nDistribution\n(ranked from most to least even according to D)", lineheight = .85, cex = .9),
       panel = function(x,y){
        panel.points(x = combs$D_x_location, y = combs$D, type = "l", col = "grey")
        panel.points(x = combs$D_x_location, y = combs$D, pch = 21, col = "grey", fill = "white")
        panel.points(x = combs$D_x_location, y = combs$S, cex = .8)
        panel.text(x = 3, y = .6, label = "D", col = "grey40")
        panel.text(x = 10, y = .95, label = "S")
        
        for ( i in 0:4){
            data_subset <- subset(combs, n_zeros == i)
            panel.points(x = data_subset$D_x_location, y = data_subset$S, type = "l")
            panel.text(x = max(data_subset$D_x_location + 1), 
                       y = data_subset$S[which.max(data_subset$D_x_location)]+.02,
                       label = zeros_labels[i+1],
                       adj = 0, cex = .8)
            }
        })

print(p1, position = c(-.04,.08,.8,1))

Figure 2: Figure 4 in Lyne (1985, 110), comparing S against D for all possible combinations of subfrequencies for 10 occurrences in 5 parts.

In the course of this evaluation, Lyne (1985, 107) raises an important question:

[I]s it desirable that a dispersion measure for word frequency counts should penalize zeros in this way? This question is of a different order […], since it is about the real world rather than the world of statistical models.

He later returns to this question, when discussing an example given by Rosengren (1971, 115), for which, she argues, it is inappropriate for a dispersion measure to give the same score:

A: 3 2 1 1 1
B: 2 2 2 2 0

As Lyne (1985, 115) rightly notes, it is essential to state in the first place why B should receive a lower score than A. To him, the reason why D₂ and S give lower scores to B is the fact that they penalize zeros. And here, Lyne (1985, 115) clearly states his point of view:

But surely it is not the business of a dispersion measure to discriminate against particular distributions in this way. In the above example, […] it seems to us quite proper that B should be rated as highly as A, because the presence of a single low sub-frequency, 0, in B is balanced by the perfectly even distribution across the remaining four sections.

Even though I will not pursue this point further here, I am not sure whether we are able to answer this question so confidently. Rather, the answer would seem to depend primarily on the linguistic purpose underlying a dispersion analysis. It is easy to imagine settings where the question of whether an item is used at all (i.e. the difference between 0 and 1 occurrence) is more important than the question of whether the item occurs repeatedly (i.e. the difference between 1 and 2 occurrences). In such settings, the feature of main interest may be pervasiveness rather than evenness of distribution, and penalization of zeros may be an attractive feature of a dispersion measure. Further, it seems that the answer would also depend on the kinds of linguistic units that are represented by the corpus parts (e.g. texts, genres, or arbitrary corpus chunks of the same length). The smaller the corpus parts, the more zeros there will be in the resulting set of subfrequencies, which could make penalization, if it is indeed considered an issue, even more of a concern.

Application of Lyne (1985)’s graphical technique to other parts-based dispersion measures

Seeing that Lyne (1985)’s technique provides insights into the behavior of dispersion measures, I will now apply it to the other parts-based measures listed above. I will modify the strategy in one important regard: Instead of using D as a baseline, I will rely on a consensus opinion: The 30 combinations will be ordered according to the average score they receive across the six dispersion measures. For illustration, Figure 3 compares S against the consensus profile, which is again shown in grey. We also introduce color coding for signaling the number of zeros in a pattern, ranging from dark blue (no zeros) to dark red (four zeros).

Draw Figure

combs <- combs[order(combs$disp_avg),]
my_cols <- rev(scales::pal_brewer(palette = "RdBu")(8)[c(1,2,3,6,7)])

p1 <- xyplot(1~1, xlim = c(0, length(unique(combs$disp_avg_x_location))), type = "n", ylim = c(0,1),
       par.settings = my_settings, axis = axis_L,
       scales = list(x = list(draw = FALSE),
                     y = list(at = c(0, .25, .5, .75, 1), label = c("(uneven) 0", ".25", ".50", ".75", "(even) 1"))),
       ylab = "\n\nDispersion", xlab = list(label = "\n\n\n\nDistribution\n(ranked from most to least even based\non the average across all measures)", lineheight = .85, cex = .9),
       panel = function(x,y){
        panel.points(x = combs$disp_avg_x_location, y = combs$disp_avg, type = "l", col = "grey")
        panel.points(x = combs$disp_avg_x_location, y = combs$disp_avg, pch = 21, col = "grey", fill = "white")
        #panel.points(x = combs$disp_avg_x_location, y = combs$D2, cex = .8)
        panel.text(x = 8, y = .5, label = "All six\ndispersion\nmeasures", col = "grey40", lineheight = .85, cex = .8)

        for ( i in 0:4){
            data_subset <- subset(combs, n_zeros == i)
            panel.points(x = data_subset$disp_avg_x_location, y = data_subset$S, type = "l", col = my_cols[i+1])
            panel.points(x = data_subset$disp_avg_x_location, y = data_subset$S, col = my_cols[i+1], cex = .8, pch = 16)
            panel.text(x = max(data_subset$disp_avg_x_location + 1), 
                       y = data_subset$S[which.max(data_subset$disp_avg_x_location)]+.02,
                       label = zeros_labels[i+1],
                       adj = 0, cex = .8, col = my_cols[i+1])
            }
        panel.text(x = 15.5, y = 1.1, label = expression(S))
        })

print(p1, position = c(-.04,.15,.8,.925))

Figure 3: Comparison of S vs. baseline for all possible combinations of subfrequencies for 10 occurrences in 5 parts.

Figure 4 compares each of the six dispersion measures against the consensus benchmark. A number of instructive insights emerge:

The only measure that shows a clear “penalization” of zeros is S.
Following the change in reference distribution (consensus opinion instead of D only), D₂ no longer appears to penalize zeros.
Compared against the average behavior over the six indices, D favors the presence of zeros: Compared to S, it produces the reversed layering.
There is also some indication that D_P favors zeros.

Draw Figure

combs <- combs[order(combs$disp_avg),]

p_D <- xyplot(1~1, xlim = c(0, length(unique(combs$disp_avg_x_location))), type = "n", ylim = c(0,1),
       par.settings = my_settings, axis = axis_L,
       scales = list(x = list(draw = FALSE),
                     y = list(at = c(0, 1))),
       ylab = " ", xlab = "",
       panel = function(x,y){
        panel.points(x = combs$disp_avg_x_location, y = combs$disp_avg, type = "l", col = "grey")
        panel.points(x = combs$disp_avg_x_location, y = combs$disp_avg, pch = 21, col = "grey", fill = "white")

        for ( i in 0:4){
            data_subset <- subset(combs, n_zeros == i)
            panel.points(x = data_subset$disp_avg_x_location, y = data_subset$D, type = "l", col = my_cols[i+1])
            panel.points(x = data_subset$disp_avg_x_location, y = data_subset$D, cex = .8, pch = 16, col = my_cols[i+1])
            }
        panel.text(x = 15.5, y = 1.1, label = expression(D))
        })

p_D2 <- xyplot(1~1, xlim = c(0, length(unique(combs$disp_avg_x_location))), type = "n", ylim = c(0,1),
       par.settings = my_settings, axis = axis_L,
       scales = list(x = list(draw = FALSE),
                     y = list(at = c(0, 1))),
       ylab = " ", xlab = "",
       panel = function(x,y){
        panel.points(x = combs$disp_avg_x_location, y = combs$disp_avg, type = "l", col = "grey")
        panel.points(x = combs$disp_avg_x_location, y = combs$disp_avg, pch = 21, col = "grey", fill = "white")

        for ( i in 0:4){
            data_subset <- subset(combs, n_zeros == i)
            panel.points(x = data_subset$disp_avg_x_location, y = data_subset$D2, type = "l", col = my_cols[i+1])
            panel.points(x = data_subset$disp_avg_x_location, y = data_subset$D2, cex = .8, pch = 16, col = my_cols[i+1])
            }
        panel.text(x = 15.5, y = 1.1, label = expression(D[2]))
        })

p_S <- xyplot(1~1, xlim = c(0, length(unique(combs$disp_avg_x_location))), type = "n", ylim = c(0,1),
       par.settings = my_settings, axis = axis_L,
       scales = list(x = list(draw = FALSE),
                     y = list(at = c(0, 1))),
       ylab = " ", xlab = "",
       panel = function(x,y){
        panel.points(x = combs$disp_avg_x_location, y = combs$disp_avg, type = "l", col = "grey")
        panel.points(x = combs$disp_avg_x_location, y = combs$disp_avg, pch = 21, col = "grey", fill = "white")

        for ( i in 0:4){
            data_subset <- subset(combs, n_zeros == i)
            panel.points(x = data_subset$disp_avg_x_location, y = data_subset$S, type = "l", col = my_cols[i+1])
            panel.points(x = data_subset$disp_avg_x_location, y = data_subset$S, cex = .8, pch = 16, col = my_cols[i+1])
            }
        panel.text(x = 15.5, y = 1.1, label = expression(S))
        })

p_DP <- xyplot(1~1, xlim = c(0, length(unique(combs$disp_avg_x_location))), type = "n", ylim = c(0,1),
       par.settings = my_settings, axis = axis_L,
       scales = list(x = list(draw = FALSE),
                     y = list(at = c(0, 1))),
       ylab = " ", xlab = "",
       panel = function(x,y){
        panel.points(x = combs$disp_avg_x_location, y = combs$disp_avg, type = "l", col = "grey")
        panel.points(x = combs$disp_avg_x_location, y = combs$disp_avg, pch = 21, col = "grey", fill = "white")

        for ( i in 0:4){
            data_subset <- subset(combs, n_zeros == i)
            panel.points(x = data_subset$disp_avg_x_location, y = data_subset$DP, type = "l", col = my_cols[i+1])
            panel.points(x = data_subset$disp_avg_x_location, y = data_subset$DP, cex = .8, pch = 16, col = my_cols[i+1])
            }
        panel.text(x = 15.5, y = 1.1, label = expression(D[P]))
        })

p_DA <- xyplot(1~1, xlim = c(0, length(unique(combs$disp_avg_x_location))), type = "n", ylim = c(0,1),
       par.settings = my_settings, axis = axis_L,
       scales = list(x = list(draw = FALSE),
                     y = list(at = c(0, 1))),
       ylab = " ", xlab = "",
       panel = function(x,y){
        panel.points(x = combs$disp_avg_x_location, y = combs$disp_avg, type = "l", col = "grey")
        panel.points(x = combs$disp_avg_x_location, y = combs$disp_avg, pch = 21, col = "grey", fill = "white")

        for ( i in 0:4){
            data_subset <- subset(combs, n_zeros == i)
            panel.points(x = data_subset$disp_avg_x_location, y = data_subset$DA, type = "l", col = my_cols[i+1])
            panel.points(x = data_subset$disp_avg_x_location, y = data_subset$DA, cex = .8, pch = 16, col = my_cols[i+1])
            }
        panel.text(x = 15.5, y = 1.1, label = expression(D[A]))
        })

p_DKL <- xyplot(1~1, xlim = c(0, length(unique(combs$disp_avg_x_location))), type = "n", ylim = c(0,1),
       par.settings = my_settings, axis = axis_L,
       scales = list(x = list(draw = FALSE),
                     y = list(at = c(0, 1))),
       ylab = " ", xlab = "",
       panel = function(x,y){
        panel.points(x = combs$disp_avg_x_location, y = combs$disp_avg, type = "l", col = "grey")
        panel.points(x = combs$disp_avg_x_location, y = combs$disp_avg, pch = 21, col = "grey", fill = "white")

        for ( i in 0:4){
            data_subset <- subset(combs, n_zeros == i)
            panel.points(x = data_subset$disp_avg_x_location, y = data_subset$DKL, type = "l", col = my_cols[i+1])
            panel.points(x = data_subset$disp_avg_x_location, y = data_subset$DKL, cex = .8, pch = 16, col = my_cols[i+1])
            }
        panel.text(x = 15.5, y = 1.1, label = expression(D[KL]))
        })

cowplot::plot_grid(
  NULL, NULL, NULL,
  p_D, p_D2, p_S, 
  NULL, NULL, NULL,
  p_DP, p_DA, p_DKL, nrow = 4, 
  rel_heights = c(1,8,1,8))

Figure 4: Comparison of all dispersion measures against baseline for all possible combinations of subfrequencies for 10 occurrences in 5 parts. See Figure 3 for a key.

The graphical technique developed by Lyne (1985) proves to be a useful tool for studying the behavior of indices. Considering the fact that Anthony Lyne drew all of his graphs by hand makes his methodological contribution even more impressive. In general, this examination is instructive for understanding how dispersion measures behave in settings where there are few corpus parts. Unfortunately, the technique does not scale well, meaning that it cannot be applied to study distributions across a much larger number of corpus parts. This is because the number of possible combinations grows exponentially as we increase the number of corpus parts and/or the number of occurrences of the item. Nevertheless, Lyne (1985)’s graphical technique has managed to shed more light on the performance of indices and, perhaps more importantly, it has confronted us with a deeper question, which is “beyond the world of statistical models” [p. 107] and needs to be answered from a linguistic perspective: Should dispersion measures deal with subfrequencies of 0 in a special way, i.e. should a drop from 1 to 0 depress a score more noticeably than a drop from 2 to 1?

References

Burch, Brent, Jesse Egbert, and Douglas Biber. 2017. “Measuring and Interpreting Lexical Dispersion in Corpus Linguistics.” Journal of Research Design and Statistics in Linguistics and Communication Science 3 (2): 189–216. https://doi.org/10.1558/jrds.33066.

Carroll, John B. 1970. “An Alternative to Juilland’s Usage Coefficient for Lexical Frequencies and a Proposal for a Standard Frequency Index.” Computer Studies in the Humanities and Verbal Behaviour 3 (2): 61–65. https://doi.org/10.1002/j.2333-8504.1970.tb00778.x.

Gries, Stefan Th. 2008. “Dispersions and Adjusted Frequencies in Corpora.” International Journal of Corpus Linguistics 13 (4): 403–37. https://doi.org/10.1075/ijcl.13.4.02gri.

———. 2021. “A New Approach to (Key) Keywords Analysis: Using Frequency, and Now Also Dispersion.” Research in Corpus Linguistics 9 (2): 1–33. https://doi.org/10.32714/ricl.09.02.02.

Juilland, Alphonse, and E. Chang-Rodriguez. 1964. Frequency Dictionary of Spanish Words. The Hague: De Gruyter. https://doi.org/10.1515/9783112415467.

Lyne, Anthony A. 1985. The Vocabulary of French Business Correspondence. Paris: Slatkine-Champion.

Rosengren, Inger. 1971. “The Quantitative Concept of Language and Its Relation to the Structure of Frequency Dictionaries.” Etudes de Linguistique Appliquee (Nouvelle Serie) 1: 103–27.

Soenning, Lukas. 2025. Tlda: Tools for Language Data Analysis. https://github.com/lsoenning/tlda.

Warnes, Gregory R., Ben Bolker, Thomas Lumley, Arni Magnusson, Bill Venables, Genei Ryodan, and Steffen Moeller. 2023. Gtools: Various r Programming Tools. https://doi.org/10.32614/CRAN.package.gtools.

Citation

BibTeX citation:

@online{sönning2025,
  author = {Sönning, Lukas},
  title = {Lyne’s (1985) Graphical Technique for the Evaluation of
    Dispersion Measures},
  date = {2025-10-27},
  url = {https://lsoenning.github.io/posts/2025-10-25_dispersion_lyne/},
  langid = {en}
}

For attribution, please cite this work as:

Sönning, Lukas. 2025. “Lyne’s (1985) Graphical Technique for the Evaluation of Dispersion Measures .” October 27, 2025. https://lsoenning.github.io/posts/2025-10-25_dispersion_lyne/.