REVIEW 4 major objections 5 minor 9 references
A New Approach to Determine the Coefficient of Skewness and An Alternative Form of Boxplot
T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read A rank-based skewness coefficient built on the midrange is claimed to beat classical measures on skewed data.
desk verdict The paper's own Weibull simulations contradict its headline claim, and the estimator is never actually defined in the text, so the central contribution is not verifiable. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is Rank Skewness, a rank-based coefficient: sort the data, insert the midrange, assign standard competition ranks, then combine the ranks of all observations with the rank of the midrange into a normalized value in $[-1, +1]$. The rank transform removes the influence of extreme magnitudes and irregular gaps between consecutive values, leaving only information about how many observations sit on each side of the midrange. The Four Point Summary Graph is the same idea made visual: it plots the minimum, maximum, median, and midrange on one horizontal line, and uses the median's position relative to the midrange to indicate the direction and approximate amount of skewness.
What would settle it
Take one right-skewed sample and replace only the largest value with an extremely large value (for example, 100 becomes 100,000), leaving all other observations unchanged; if the Rank Skewness coefficient moves substantially, the claim that it is unaffected by extreme values fails, because the midrange and hence the ranks shift.
Extended reading notes
Core claim
The paper's central claim is that skewness is better read from the ranks of observations relative to the midrange than from their numerical distances to a center. The proposed Rank Skewness coefficient is constructed by ordering the sample, inserting the midrange into the ordered values, ranking with standard competition ranking so ties share a rank and later ranks are skipped, and then forming a normalized combination of the ranks of the observations and the rank of the midrange so that the coefficient lies in $[-1, +1]$. The authors report that across repeated resamples for normal, gamma, Weibull, and lognormal distributions, Rank Skewness has the smallest variability for lognormal data at every sample size, becomes competitive or best for gamma at larger sample sizes, and generally beats Pearson's and Fisher's moment measures on skewed distributions, while the median-based predecessor remains best for normal and negatively skewed Weibull data. In the three real datasets, Rank Skewness gives values near 0.94 to 0.99 for positively skewed samples, which the authors interpret as performing well. The companion Four Point Summary Graph declares skewness by the median's position relative to the midrange: median left of midrange indicates positive skewness, median right indicates negative skewness, and equality indicates symmetry.
Load-bearing premise
The whole approach rests on treating the midrange as the true center of a skewed dataset, so that the side of the midrange holding more observations reliably indicates the direction and amount of skewness.
Editorial extensions
If this is right
- Analysts get a skewness coefficient that is bounded in $[-1,1]$, requires no moments, and is computable from ranks alone.
- On lognormal data, the smaller standard deviation and mean deviations reported for Rank Skewness imply more stable skewness estimates across sample sizes than Pearson or moment coefficients.
- A simpler skewness display than the boxplot becomes available: the four-point summary graph uses all observations and indicates skewness by the median's position relative to the midrange.
- For skewed distributions such as gamma and Weibull, Rank Skewness at larger sample sizes matches or beats the quartile-based Bowley measure, giving a rank-only alternative that keeps information from the tails.
Reading between the lines
- A natural extension is to replace the midrange with a trimmed or winsorized midpoint; the paper's own logic suggests the coefficient would then resist outliers even when the extremes themselves are contaminated, a variant that is directly testable on the paper's simulated distributions.
- The four-point graph's decision rule is essentially a sign test on the median relative to the extremes, so a permutation null distribution could turn the graph into a formal symmetry test with calibrated significance levels.
- Because the simulation benchmark compares every estimator to moment-based population skewness, the test setup structurally favors moment estimators; comparing each estimator to its own population limit under the same distributions would be a fairer re-analysis and could change the reported rankings.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proposes a new skewness coefficient, called "Rank Skewness", based on the rank of the midrange among the ranked observations, and a four-point summary graph intended as a simpler alternative to the boxplot. The authors compare the sampling variability of five skewness measures through bootstrap simulations for Normal, Gamma, Weibull (both positive and negative), and Lognormal distributions, and illustrate the measures on three real datasets. The abstract and conclusion claim that the proposed measure performs better than existing measures, especially for skewed distributions, and that the four-point graph gives better results than the five-point summary boxplot.
Significance. A rank-based skewness coefficient that is insensitive to extreme values could be of practical use, and the simulation setup with a fixed seed and 500,000 resamples is transparent. However, the paper does not establish its central claims: the explicit formula is not typeset, no formal properties are derived, and the paper's own simulation tables contradict the claimed superiority for skewed distributions. The four-point summary graph is purely descriptive and no criterion for "better" is supplied. As it stands, the contribution is not supported by the evidence presented.
major comments (4)
- [§2.5, Tables I–III; Abstract; §5] The paper's own comparison criterion, smaller standard deviation or mean deviation, shows the proposed Rank Skewness to be worse than both Bowley's and Forhad-Adnan's measures for the positively skewed Weibull(2,2) distribution at every reported sample size. For example, in Table I at n=20 the standard deviation is 0.2385 for Rank Skewness versus 0.1686 for Bowley and 0.1393 for Forhad-Adnan; at n=100 the corresponding values are 0.1733 versus 0.0928 and 0.0821. This directly contradicts the abstract and conclusion that the proposed measure performs better specially for skewed distributions. The text itself reports that Forhad-Adnan is best for Normal and negatively skewed Weibull, leaving only Lognormal as a uniformly favorable case. The superiority claim must either be removed or supported by a consistent pattern across skewed distributions.
- [§2.5] The comparison metric is incomplete: comparing standard deviations or mean deviations of statistics that estimate different population quantities is not a valid way to rank estimators unless bias is taken into account. A low-variance measure can be far from the quantity of interest, and no bias-adjusted comparison such as mean squared error relative to a common target is provided. The second comparison criterion is also not independent, since the population skewness values are computed with the moment formula that the authors criticize elsewhere in the paper.
- [§2.1] The explicit formula for Rank Skewness is not actually typeset; the text shows only placeholder text where the expression should be. This makes the core object of the paper unavailable for verification and reproduction. In addition, the claimed range of -1 to +1 is asserted without proof, and the procedure for inserting the midrange and using standard competition ranking needs to specify how ties are handled, since the midrange can coincide with an existing observation.
- [§2.2 and §4] The four-point summary graph compares only the median with the midrange, so the assertion that it is based on all observations and gives better results than the traditional boxplot is not supported by any quantitative definition or comparison. The real-data applications provide no ground truth for skewness, and statements such as the proposed measure "performs well" are qualitative judgments made after inspecting the computed values rather than objective evidence.
minor comments (5)
- [General] The section numbering is inconsistent: the introduction is numbered 1.1 and the next section is 2.1, with no Section 1 or Section 2 headers.
- [§4] Table V appears twice with different content, and the second occurrence uses rows labeled Sample-1 and Sample-2 under a column header that reads 'Form'; this needs to be corrected.
- [References] The text cites Groeneveld, Meeden and Gin (1984) and Hossain (2007) for the EUPP outlier method, but these references are not included in the reference list.
- [Abstract and text] There are numerous typographical and grammatical issues, including 'scatterdness', 'Laondon', and 'middle most fifty percent observation'; Dataset-3 also appears to have a missing comma ('9 7').
- [Figure 1] Figure 1 is difficult to read: the axis labels are garbled and the labels Median, Midrange, and Mid are repeated without clear connection to the plotted points.
Circularity Check
No significant circularity: Rank Skewness is a direct definition evaluated by simulation, and the author self-citations are not load-bearing.
full rationale
The proposed Rank Skewness coefficient is introduced as a stand-alone definition based on the rank of the midrange relative to the ranks of all observations, with no fitted parameters and no derivation that presupposes the comparison outcomes. The Four Point Summary Graph is likewise defined directly from the minimum, maximum, median and midrange. The simulation study in Sections 2.4-3 is an external empirical evaluation rather than an algebraic consequence of the definition: the paper chooses a comparison criterion (smaller standard deviation or mean deviation is better) and reports the resulting tables. The only self-citations are Hossain and Adnan (2007), used as the FA comparator in the simulations and as a previously proposed skewness formula, and Hossain (2007), used for EUPP outlier detection in the real-data examples. Neither citation supplies the content of the proposed measure or forces its claimed superiority; the Rank Skewness formula would stand unchanged without these references. The abstract's claim that Rank Skewness 'performs better specially for skewed distribution' is contradicted by the paper's own Tables I-III for Weibull(2,2), where FS Rank has higher standard deviation and mean deviations than both Bowley and FA at every reported sample size. That is a correctness and evidence problem, not a circularity problem, because the result is not true by construction. No step in the paper reduces, by the paper's own equations or by a load-bearing self-citation chain, to its own inputs.
Assumptions & free parameters
assumptions (3)
- domain assumption Midrange is an appropriate measure of central location for skewed data, so that the rank of the midrange can serve as the symmetry reference point.
- ad hoc to paper The sampling distribution of a skewness statistic is better when its variance is smaller, independent of bias.
- domain assumption Standard competition ranking ('1224') is the appropriate ranking scheme for the proposed measure.
Cite this review
Pith. "Pith review of A New Approach to Determine the Coefficient of Skewness and An Alternative Form of Boxplot." pith.science (2026). https://pith.science/paper/6ZZ5ROLK
@misc{pith2026190806400,
author = {Pith},
title = {Pith review of: A New Approach to Determine the Coefficient of Skewness and An Alternative Form of Boxplot},
year = {2026},
howpublished = {\url{https://pith.science/paper/6ZZ5ROLK}},
note = {Machine review of arXiv:1908.06400}
}
read the original abstract
To solve the problems in measuring coefficient of skewness related to extreme value, irregular distance from the middle point and distance between two consecutive numbers, "Rank skewness" a new measure of the coefficient of skewness has been proposed in this paper. Comparing with other measures of the coefficient of skewness, proposed measure of the coefficient of skewness performs better specially for skewed distribution. An alternative of five point summary boxplot, a four point summary graph has also been proposed which is simpler than the traditional boxplot. It is based on all observation and give better result than the five point summary.
Figures
Reference graph
Works this paper leans on
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[1]
Bowley, A. L. (1901): Elements of Statistics, P.S. King and Son, Laondon
work page 1901
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[2]
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Probability and Statistics for Engineering and Sciences
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work page 2000
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[4]
Nonparametric Statistical Inference
Gibbons, J.D. and Chakraborti, S. (2003): “Nonparametric Statistical Inference”, 4 th edition, Marcel Dekkar, Inc
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[5]
Comparing Measures of Sample Skewness and Kurtosis,
Joanest, D.N. and Gill, C.A (1998): “Comparing Measures of Sample Skewness and Kurtosis,” Journal of the Royal Statistician Society, Series D (The Statistician), Vol. 47, No. 1, pp. 183-189
work page 1998
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[6]
Biometrika Tables for Statisticians
Pearson, E.S. and Hartley, H.O. (1966): “Biometrika Tables for Statisticians”, V ols. I and II. Cambridge University Press, Cambridge
work page 1966
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[7]
A New Approach to Determine the Asymmetry of a Distribution,
Hossain, M.F. and Adnan, M.A.S.A (2007): “A New Approach to Determine the Asymmetry of a Distribution,” Journal of Applied St atistical Science, Vol.15, pp. 127-134
work page 2007
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[8]
Kendall’s Advanced Theory of Statistics
Stuart, A. and Ord, J.K. (1987): “Kendall’s Advanced Theory of Statistics”, Psychological Bulletin, Vol. 114, pp. 510-532
work page 1987
Show all 9 references
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[9]
(1912): An introduction to the theory of statistics
Yule, Udny, G. (1912): An introduction to the theory of statistics. C. Griffin, limited
1912
Reviewed August 14, 2026 · model on record in the stance chip above.
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