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arxiv: 2604.17094 · v1 · submitted 2026-04-18 · 🌌 astro-ph.IM · physics.data-an· stat.CO

Simple approximations of some statistical functions

Zinovy Malkin This is my paper

Pith reviewed 2026-05-10 06:17 UTC · model grok-4.3

classification 🌌 astro-ph.IM physics.data-anstat.CO
keywords statistical approximationsquantile functionsnormal distributionStudent's t-distributionoutlier rejectionhypothesis testingdata analysis
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The pith

Simple approximation expressions are proposed for the quantiles of the inverse normal distribution, Student's t-distribution, and outlier rejection criterion.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper considers ways to simplify computations for three statistical functions used in hypothesis testing when processing observations: the inverse normal distribution, the Student's t-distribution, and the criterion for rejecting outliers. It proposes simple approximation expressions for their quantiles. These are presented as sufficiently accurate for most practical applications. A reader might care because such approximations can make statistical analysis more accessible without specialized software or tables.

Core claim

For the inverse normal distribution, the Student's t-distribution, and the criterion for rejecting outliers, simple approximation expressions are proposed for the quantiles, which are accurate enough for most practical applications when testing statistical hypotheses.

What carries the argument

Simple closed-form approximation expressions for the quantiles of the inverse normal, Student's t, and outlier rejection distributions.

Load-bearing premise

That the proposed simple expressions provide sufficient accuracy for most practical uses in hypothesis testing without needing precise error specifications or broad validation.

What would settle it

Direct numerical comparison of the approximated quantiles against exact values at standard significance levels such as 0.05 or 0.01, checking whether deviations stay small enough for routine use.

Figures

Figures reproduced from arXiv: 2604.17094 by Zinovy Malkin.

Figure 1
Figure 1. Figure 1: Inverse function of the standard normal distribution. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Examples of the t-distribution for three significance levels: In the graph captions, the first number corresponds to the one-tailed region, the second to the two-tailed region. 1.2 Student’s t-distribution Another important distribution for statistical data analysis is the Student’s t-distribution, which is used, for example, to test the statistical significance of correlation coefficients, construct confi… view at source ↗
Figure 3
Figure 3. Figure 3: Examples of the ζ(Q, n) distribution for three significance levels. Suppose that the k-th data point has the maximum residual vk and it is necessary to check whether it is an outlier to be rejected. The rejection question is decided positively if the following condition is met: |vk| s > ζ(Q, n), (7) where Q is the significance level. The form of the function ζ(Q, n) for three significance levels is shown i… view at source ↗
read the original abstract

Possibilities are considered to simplify the computation of several statistical functions used to test statistical hypotheses when processing observations: the inverse normal distribution, the Student's t-distribution, and the criterion for rejecting outliers. For these three cases, simple approximation expressions are proposed for the quantiles of these statistical distributions, which are accurate enough for most practical applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript proposes simple approximation expressions for the quantiles of the inverse normal distribution, the Student's t-distribution, and the outlier rejection criterion. These are intended to simplify computations in statistical hypothesis testing when processing observations, with the assertion that the approximations are accurate enough for most practical applications.

Significance. If the approximations prove accurate within usable tolerances over relevant parameter ranges, they could offer practical value for quick statistical tests in data-intensive fields such as astro-ph.IM. However, the lack of any quantitative error analysis or validation means the claimed utility cannot be assessed and the work does not yet advance the state of practical methods.

major comments (1)
  1. [Abstract] Abstract: the central claim that the proposed approximations 'are accurate enough for most practical applications' is unsupported by any quantitative evidence. No maximum absolute or relative errors, no tested intervals for probability levels or degrees of freedom, and no comparisons to exact references (e.g., numerical inversion or standard libraries) are provided.
minor comments (1)
  1. The explicit functional forms of the three approximations should be stated clearly in the abstract or a dedicated section so readers can evaluate them without needing the full derivation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback. We agree that quantitative validation is necessary to support the practical utility of the approximations and will revise the manuscript to include it.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the proposed approximations 'are accurate enough for most practical applications' is unsupported by any quantitative evidence. No maximum absolute or relative errors, no tested intervals for probability levels or degrees of freedom, and no comparisons to exact references (e.g., numerical inversion or standard libraries) are provided.

    Authors: We acknowledge that the manuscript as submitted provides no explicit quantitative error analysis or comparisons to exact values. In the revised version we will add a new section (or appendix) presenting maximum absolute and relative errors for each approximation. This will cover the relevant ranges: probability levels from 0.9 to 0.999 for the normal and outlier criteria, and degrees of freedom from 1 to 100 (plus the limiting case) for the t-distribution. Errors will be computed against reference values obtained from standard numerical inversion routines and libraries (e.g., SciPy or R). We will also state the intervals over which the stated accuracy holds. revision: yes

Circularity Check

0 steps flagged

No circularity in proposed statistical approximations

full rationale

The manuscript proposes simple closed-form approximations for quantiles of the inverse normal distribution, Student's t-distribution, and an outlier rejection criterion. No derivation chain, fitting procedure, or self-citation is presented that reduces any claimed result to its own inputs by construction. The approximations are introduced directly as practical simplifications whose accuracy is asserted for most applications; this assertion does not rely on re-using fitted parameters as predictions or on uniqueness theorems imported from the author's prior work. The paper therefore remains self-contained against external benchmarks and exhibits no load-bearing circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities identifiable from abstract alone.

pith-pipeline@v0.9.0 · 5329 in / 827 out tokens · 30931 ms · 2026-05-10T06:17:49.830709+00:00 · methodology

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Reference graph

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9 extracted references · 9 canonical work pages

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    ENTRY address archive author booktitle chapter doi edition editor eid eprint howpublished institution journal key month note number organization pages publisher school series title type url volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all ...

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