🔭 astrophysics

Simple approximations of some statistical functions

This paper proposes simple and sufficiently accurate approximation expressions for the quantiles of the inverse normal distribution, Student's t-distribution, and the outlier rejection criterion to simplify the computation of these statistical functions in hypothesis testing.

Original authors: Zinovy Malkin

Published 2026-04-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Zinovy Malkin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a chef trying to bake a perfect cake for a thousand different parties. In the world of statistics, the "cake" is a complex calculation used to decide if your data is real or just a fluke. Usually, professional statisticians use "super-computers" (or heavy software) to bake these cakes with microscopic precision. They measure every grain of sugar to the billionth of a gram.

But what if you are baking in a small kitchen, or you need to bake a thousand cakes in an hour, and you don't have a super-computer? You don't need a grain-of-sugar measurement; you just need a cake that tastes good enough to serve.

This paper by Zinovy Malkin is essentially a cookbook for "Good Enough" Statistical Recipes.

Here is the breakdown of the three main "dishes" (statistical functions) the author simplifies, using everyday analogies:

1. The "Normal Distribution" (The Bell Curve)

The Problem: Imagine you are measuring the height of people in a city. Most people are average height, with fewer very tall or very short people. This forms a "Bell Curve."
Sometimes, you need to answer the reverse question: "If I want to find the top 1% of tallest people, what height do I need to cut off?"
Mathematically, finding this "cut-off point" (called a quantile) is like trying to solve a complex maze. The standard way to do it involves heavy, slow math.

Malkin's Solution:
Instead of solving the whole maze, Malkin says, "Let's just look at the most common paths people take."
He created a simple shortcut formula (a "back-of-the-napkin" calculation) that works perfectly for the confidence levels we actually use in real life (like 90%, 95%, or 99%).

The Analogy: Instead of calculating the exact distance to the moon using complex orbital mechanics, Malkin gives you a ruler that says, "It's about 240,000 miles." It's not perfect, but for planning a picnic, it's accurate enough and takes one second to read.

2. The "Student's t-Distribution" (The Small Sample Detective)

The Problem: The Bell Curve works great when you have millions of data points. But what if you only have 10 measurements? The math gets wobbly and unpredictable. Statisticians use a different shape called the "Student's t-distribution" to handle these small groups.
Calculating the "cut-off" for these small groups is usually a slow, heavy process.

Malkin's Solution:
Malkin realized that for most practical uses, the shape of this curve settles down quickly as you add more data. He created a simple formula that acts like a "smart guess."

The Analogy: Imagine you are trying to guess the average weight of a bag of apples. If you have 500 bags, it's easy. If you only have 5 bags, it's harder. Malkin's formula is like a seasoned grocer who looks at the 5 bags and says, "I know the math is tricky, but based on these 5, the answer is roughly X." It's not a supercomputer simulation, but it gets the job done fast.

3. The "Outlier Rejection" (The Bad Apple)

The Problem: Sometimes, in a pile of data, one number is just weird. Maybe you measured a temperature of 200°F in a room that's usually 70°F. That's an "outlier." You need a rule to decide: "Is this a real measurement, or is the thermometer broken?"
The standard rule involves a complex calculation to see if that "bad apple" is far enough away from the rest to be thrown out.

Malkin's Solution:
He created a simple formula to calculate the "throw-away line."

The Analogy: Imagine a line of people waiting for a bus. Most are between 5'5" and 6'0". One guy is 7'2". You need a rule to decide if he's a giant or just a guy standing on a box. Malkin's formula is a simple tape measure. If the guy is more than X inches taller than the average, you say, "Okay, that's an outlier, let's check his shoes." The formula tells you exactly what X is without needing a PhD in math.

Why Does This Matter?

The author argues that while "perfect" math exists, it is often overkill.

Speed: These simple formulas are like driving a bicycle instead of a rocket ship. You get to the destination (the answer) much faster.
Simplicity: They are easy to program into simple devices, like a handheld sensor or an old computer, that can't handle heavy math.
Accuracy: The paper proves that for 99% of real-world jobs (astronomy, engineering, quality control), these "bicycle" formulas are accurate enough. You don't lose any real information; you just save a lot of time.

The Bottom Line

Zinovy Malkin is saying: "Stop using a sledgehammer to crack a nut."
He has provided a set of simple, fast, and reliable tools for scientists and engineers to make quick decisions about their data without needing a supercomputer. It's about efficiency: getting a result that is "good enough" to be trusted, in a fraction of the time.

1. Problem Statement

Statistical computations are fundamental to processing observational data, particularly for testing hypotheses and assessing significance. While advanced statistical packages exist, they often rely on complex proprietary algorithms that may be computationally expensive.

The Challenge: In automated, large-scale, or real-time data processing (often under limited computing resources), the high complexity of exact statistical function calculations creates a bottleneck.
The Gap: Existing literature offers highly accurate approximation formulas for statistical distributions, but these are often designed to cover the entire mathematical domain of the function. This results in excessive complexity and computational time for practical applications.
The Need: There is a need for simplified algorithms that maintain sufficient accuracy specifically for the narrow range of confidence probabilities (typically $0.9$ to $0.999$) used in the vast majority of applied observation problems.

2. Methodology

The author proposes simplified approximation expressions for three specific statistical functions. The methodology relies on restricting the approximation domain to practical confidence levels rather than the full theoretical range, allowing for simpler polynomial or rational functions with fewer coefficients.

The study utilizes data from Bol'shev and Smirnov (1983) and Owen (1962) as the ground truth for generating and testing the approximations.

A. Inverse Normal Distribution ( $\Psi(p)$ )

Function: Calculates the quantile $x$ for a given confidence probability $p$ in a standard normal distribution ( $N(0,1)$ ).
Approximation Formula:
$\Psi(p) = a_1 + a_2 t + a_3 \sqrt{t} - a_4$
Where $t = -\ln(1-p)$ .
Implementation: The coefficients ( $a_1$ to $a_4$ ) are optimized for specific intervals of $p$ (e.g., $0.95 \le p \le 0.999$ ). The paper provides distinct sets of coefficients for different intervals to ensure maximum accuracy within those ranges.
Optimization: For standard confidence levels (e.g., 0.95, 0.99), the author suggests returning exact tabular values rather than computing the formula to further reduce error and time.

B. Student's t-Distribution ( $t(Q, m)$ )

Function: Calculates the quantile for the Student's t-distribution based on significance level $Q$ and degrees of freedom $m$ .
Approximation Formula:
$t(Q, m) = a_1 + \frac{a_2}{m + a_3}$
Implementation: Coefficients are provided for various significance levels (ranging from 10% down to 0.05% for one-tailed tests).
Convergence: The formula is designed such that as $m \to \infty$ , the result converges to the inverse normal distribution value ( $a_1 \approx \Psi(1-Q/100)$ ), ensuring consistency with the normal distribution limit. The approximation remains valid for $m > 500$ .

C. Outlier Rejection Criterion ( $\zeta(Q, n)$ )

Function: Determines the threshold for rejecting outliers in a dataset of size $n$ based on the maximum residual relative to the sample standard deviation.
Approximation Formula:
$\zeta(Q, n) = \Psi(p) \cdot \left( a_1 + \frac{a_2}{n} + \frac{a_3}{a_4 + n} \right)$
Where $p = 1 - Q/(200n)$ and $\Psi(p)$ is the inverse normal function defined in Section 1.1.
Data Adjustment: The author adjusted standard tables (Bol'shev and Smirnov) to account for the specific definition of sample variance used in the paper ( $s = \frac{1}{n-1}\sum v_i^2$ ) rather than the population variance definition found in some tables.
Implementation: Two sets of coefficients are provided for each significance level: one optimized for $n \in [6, 500]$ and a more precise one for $n \in [6, 100]$ .

3. Key Contributions

Domain-Specific Simplification: The paper shifts the focus from "universal" approximations to "practical" approximations. By limiting the scope to the $0.9 $–$ 0.999$ confidence range, the author derives significantly simpler formulas than those found in general literature.
Explicit Coefficient Tables: The paper provides ready-to-use tables of coefficients for:
- Inverse Normal Distribution (Table 1).
- Student's t-Distribution (Table 2).
- Outlier Rejection Criterion (Table 3).
Error Quantification: The author explicitly calculates and reports the maximum approximation error for each interval, demonstrating that the errors are negligible for practical purposes (e.g., $< 0.007$ for the outlier criterion).
Software Availability: The author notes that Fortran routines implementing these algorithms are available via the Pulkovo Observatory website, facilitating immediate adoption.

4. Results

Accuracy vs. Complexity Trade-off: The proposed formulas achieve accuracy sufficient for almost all observational data processing tasks while drastically reducing computational complexity compared to exact solutions or high-precision algorithms (like Odeh and Evans, 1974, or Wichura, 1988).
Performance:
- The Inverse Normal approximation achieves errors as low as $5 \times 10^{-5}$ for the $0.95-0.999$ interval.
- The t-distribution approximation maintains errors below $0.05$ for $m > 2$ and below $0.001$ for $m > 6$ (depending on the significance level).
- The Outlier Criterion approximation yields errors $< 0.007$ for $n$ up to 500.
Comparison: While more complex algorithms exist with higher precision (e.g., $10^{-6}$ ), the author argues that such precision is often unnecessary for observational data, making the proposed simpler formulas more efficient for large-scale processing.

5. Significance

This paper is significant for the fields of astronomy, astrophysics, and general data science where large volumes of observational data require rapid statistical analysis.

Efficiency: The simplified algorithms enable faster processing speeds, which is critical for real-time data analysis and automated pipelines where computational resources may be constrained.
Accessibility: By providing simple, closed-form expressions, the paper allows developers to implement robust statistical tests without relying on heavy external libraries or complex numerical integration methods.
Practicality: The work bridges the gap between theoretical statistical rigor and engineering constraints, offering a "good enough" solution that prioritizes speed without sacrificing the reliability required for scientific hypothesis testing.

In conclusion, Malkin provides a pragmatic toolkit for statistical computation, demonstrating that by narrowing the scope of approximation to practical confidence intervals, one can achieve a highly efficient balance between speed and accuracy.