On a Logistic Approximation to the Normal and Other Cumulative Distribution Functions


  • Roberto N. Padua
  • Mark S. Borres University of San Jose-Recoletos
  • Efren O. Barabat University of San Jose-Recoletos




logistic approximation, cumulative distribution functions, normal distribution, chi-square distribution, estimation of parameters


This paper proposes an approach that maintains the simplicity of various
approximations as well as having a closed- form algebraic inverse yet easily generalizable to the approximation of other cumulative distribution functions. The proposed logistic approximation to the normal cumulative distribution function has a maximum error of 0.000560, lower than the maximum error reported by Aludaat et al (2007). Furthermore, the logistic approximation is more flexible in that it can be used to approximate the cumulative distribution functions of other distribution. A Chi-square approximation was also investigated and reported a maximum error of 0.00494.

Author Biographies

Roberto N. Padua

Dr. Roberto Natividad Padua, scientist, received his PhD in Mathematical Statistics from Clemson University, South Carolina, USA under the Fullbright-Hays scholarship grant. He is an accomplished author, a multi-awarded researcher and an internationally acclaimed lecturer. Dr. Padua is a former Commissioner of CHED and currently a consultant to several state and private universities. He is also conducting lectures on research and Fractal Statistics. Dr. Padua is a Summa Cum Laude, BS in Mathematics Teaching graduate from the Philippine Normal College under the National Science Development Board (NSDB) program. He obtained his MS in Mathematics Education Degree from the Centro Escolar University as a Presidential Scholar.

Mark S. Borres, University of San Jose-Recoletos

graduated Bachelor of Science in Mathematics–major in Pure Mathematics at the University of the Philippines, Cebu College. Since 2009, he worked for the University of San Jose- Recoletos as a faculty member of the College of Arts and Sciences and handled Mathematics subjects such as College Algebra, Advanced Algebra, Abstract Algebra, Analytical Geometry, Euclidean geometry, Trigonometry, Business Mathematics, Linear Programming, Mathematics of Investment, Discrete Structure, and Statistics across colleges.

Efren O. Barabat, University of San Jose-Recoletos

is an Electronics Engineer, graduated from the University of San Jose-Recoletos in 2010, Cum Laude honors. He ranked as top 9 examinee in the April 2011 ECE Licensure Examination. He worked as Field Engineer in SMART Communications, Inc. from 2011 to 2012. Currently, a full-time faculty member of the Electronics Engineering Department of USJ-R College of Engineering, handling Mathematics and Major Subjects of ECE.


Aludaat, K. M & Alodat, M. T (2008). A Note on Approximating the Normal Distribution Function. Applied Mathematical Sciences, Vol.2, 2008.

Cook, R. D. & Weisberg, S. (1999). Applied Regression Including Computing and Graphics, New York: Wiley.

Cardano, G. (1663). Opera Omnia Hieronymi Cardani, Mediolanensis: Ars Magna (The Great Art), Lyons, Europe.

Craig, W. & Hogg (2000), An Introduction to Mathematical Statistics, (Wiley and Sons, New York)

Dudley, R. M. (1999). “Uniform Central Limit Theorems”, Cambridge University Press. ISBN 0 521 46102.

Durrett, R. (1991). Probability: Theory and Examples. Pacific Grove, CA: Wadsworth & Brooks/Cole.

Esseen, C. G. (1956). “A moment inequality with an application to the central limit theorem”. Skand. Aktuarietidskr. 39: 160–170.

Feller, W. (1972). An Introduction to Probability Theory and Its Applications, Volume II (2nd ed.). New York: John Wiley & Sons.

Graybill, J.(1987) AnIntroductoryCourseinMathematical Statistics (Wiley Series, New York)

Huber, P. (1985). Projection pursuit. The annals of Statistics, 13(2):435 – 475.

Johnson, R and Wichern (2000) Applied Multivariate Statistical Analysis (Wiley and Sons, New York)

Manoukian, E. B. (1986). Modern Concepts and Theorems of Mathematical Statistics. New York: Springer-Verlag.

Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. New York: John Wiley & Sons.

Shevtsova, I. G. (2007). “Sharpening of the upper bound of the absolute constant in the Berry– Esseen inequality”. Theory of Probability and its
Applications 51 (3): 549–553.

Shevtsova, I. G. (2008). “On the absolute constant in the Berry-Esseen inequality”. The Collection of Papers of Young Scientists of the Faculty of Computational Mathematics and Cybernetics Theory of Probability
and its Applications (5): 101-110.

Shiganov, I.S. (1986). “Refinement of the upper bound of a constant in the remainder term of the central limit theorem”. Journal of Soviet mathematics 35: 109–115.

Shorack, G.R., Wellner J.A. (1986) Empirical Processes with Applications to Statistics, Wiley.

Tyurin, I.S. (2009). “On the accuracy of the Gaussian approximation”. Doklady Mathematics 80 (3): 840-843.

Van der Vaart, A. W. (1998), Asymptotic Statistics. Cambridge Series in Probabilistic Mathematics.

Vapnik, V.N. and Chervonenkis, A. Ya (1971). On uniform convergence of the frequencies of events to their probabilities. Theor. Prob. Appl. 16, 264-280.

Yeo, In-Kwon and Johnson, Richard (2000). A new family of power transformations to improve normality or symmetry. Biometrika, 87, 954-959.




How to Cite

Padua, R. N., Borres, M. S., & Barabat, E. O. (2014). On a Logistic Approximation to the Normal and Other Cumulative Distribution Functions. Recoletos Multidisciplinary Research Journal, 2(1). https://doi.org/10.32871/rmrj1402.01.08




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