A Simple Approximation to the Area Under Standard Normal Curve

Mathematics and Statistics 2(3): 147-149, 2014 http://www.hrpub.org DOI: 10.13189/ms.2014.020307

Amit Choudhury

Department of Statistics, Gauhati University, Guwahati 781014, India *Corresponding Author: [email protected]

Abstract Of all statistical distributions, the standard approximations or bounds have been for the standard normal normal is perhaps the most popular and widely used. Its use variety. often involves computing the area under its probability curve. Various approximations and bounds are placed below. 2kx 2kx Unlike many other statistical distributions, there is no closed i. Tocher (1963) : Φ(x) ≈ e /(1+ e ) where k = 2 form theoretical expression for this area in case of the normal π ii. Zelen and Severo (1964): distribution. Consequently it has to be approximated. While 2 3 there are a number of highly complex but accurate Φ(x) ≈ 1- (0.4361836 t – 0.1201676 t + 0.9372980 t ) −1   2 algorithms, some simple ones have also been proposed in  2  e−x 2 , literature. Even though the simple ones may not be very  π  accurate, they are nevertheless useful as accuracy has to be where t = (1+0.33267x)-1 gauged vis-à-vis simplicity. In this short paper, we present iii. A popular bound found in many probability texts (for another simple approximation formula to the cumulative example, Feller 1968). For x > 0 −1 distribution function of standard normal distribution. This -1 -3   2 (x – x )  2  e−x 2 < 1 - Φ(x) < x-1 new formula is fairly good when judged vis-à-vis its  π  −1 simplicity.   2  2  e−x 2  π  Keywords Cumulative Distribution Function, Normal 2 Distribution, Approximations iv. Page (1977): Φ(x) ≈ 0.5{1+ tanh(y)} where y= π x(1+0.044715x2)

0.5   2  v. Hammakar (1978): 1- Φ(x) ≈ 0.51− 1−e−y  , y     1. Introduction = 0.806x(1-0.018x) In the collection of statistical distributions, the most well vi. Abernathy (1988): Φ(x) ≈ 0.5+ ∞ n 2n+1 known and most frequently used across all domains is the 1 (−1) x standard normal distribution. In doing so, the area under the ∑ , x > 0 2π n ( + ) standard normal probability curve is calculated. This is often n =0 2 n! 2n 1 expressed as a function of the cumulative distribution  2 vii. Lin(1989): 1- Φ(x) ≈ 0.5e−0.717x−0.416x  , x > function (cdf) of the distribution. Since a closed form   expression of this cdf does not exist, it becomes necessary to 0 approximate it. These days, many softwares like Ms-Excel viii. Lin(1990): 1- Φ(x) ≈ 1/(1+ey) where y = 4.2 and statistical softwares like SAS, SPSS provide for π{x (9 − x)} , x > 0 computation of this cdf using highly sophisticated ix. (ix) Bagby(1995): algorithms. 0.5   1  − 2 2 − 2 2− 2 π 2 − 2 φ(x) ≈ 0.5 + 0.51− 7e x +16e x ( )+(7+ x )e x  Even with the advent of computers, there has been an   30  4  interest in providing simple approximating this cdf. A review , x>0 of the literature points to three approaches to the construction x. (x) Szarek bounds (1999): For x > -1 of an approximation. The most popular among them is the 2 ∞ 2 construction of approximation formulas. The second 0.5 e−x 2 2 0.25 e−x 2 ≤ ∫ e−t 2dt ≤ approach uses distributions that closely resemble the normal 0.5 0.5 x +(x2+4) x 3x+(x2+8) distribution under specific conditions. The third approach involves construction of bounds. In all these approaches, the xi. Byrc (2002A): 148 A Simple Approximation to the Area Under Standard Normal Curve

2 (4 − π)x + 2π (π − 2) −x a function. φ(x)≈1− e 2 (4 − π)x2 2π + 2πx + 2 2π (π − 2) xii. Byrc (2002B):Φ(x) ≈ 1- 2 2 4. Appendix x + 5.575192695x +12.77436324 −x e 2 x3 2π +14.38718147 x2 + 31.53531977x + 25.548726 Subroutine Std_Normal_CDF(x : float as input, z : float as xiii. Standard Logistic cdf (Johnson and Kotz, 1994) : output) Φ(x) ≈ F(x) = {1+ exp(- πx/√3)}-1 variables: pi,x,y,z : float 1+ g(−x) − g(x) indi : integer xiv. Shore(2005): φ(x) ≈ 2 pi = 3.14159265358979 where begin case    (λ S )   case: x < 0 g(x) = exp− log(2)exp[α (λ S1)](1+ S1x) 1 −1 + S2x      indi = 1, λ =−=− 0.61228883,S1 0.11105481, x = (-1)*x

S2 = 0.44334159,α = − 6.37309208 case: x ≥ 0 indi = 0 xv. Aludaat and Alodat (2008): π 2 − x end case Φ(x) ≈ 0.5 + 0.5 1− e 8 y =1- (1/(2*sqrt(2*pi))) * exp(-sqr(x)/2) / (0.226+0.64*x xvi. Winitzki (2008): + 0.33 *sqrt(x*x+3)) 1    2   2  2 begin case 1+ 1− exp − 2 2  4 + 0.147 x  1+ 0.147 x    (x ) π 2 2        case: indi=1 φ(x)≈ 2 z = 1 – y It is therefore clear that the topic has been of interest for more case: indi = 0 than half a decade. z = y end case return (z) 2. A Simple Approximation Formula End of Subroutine Std_Normal_CDF We first note that it is enough to present an approximation formula for x > 0. We concentrate on the tail area probability and suggest the following approximation formula for it. REFERENCES 2 2 2 [1] Abernathy, R.W. (Nov 1988). Finding Normal Probabilities 2 du = , x > 0 with Hand held Calculators. Mathematics Teacher,81, 0.226+0.64−𝑥𝑥+�0.33 2+3 ∞ −𝑢𝑢 � 𝑒𝑒 651-652. Then a ∫simple𝑥𝑥 𝑒𝑒 approximation to the𝑥𝑥 standard�𝑥𝑥 normal cdf is [2] Aludaat,K.M. and Alodat,M.T.(2008).A Note on Approximating Normal Distribution Function. Applied 2 1 2 Mathematical Statistics,Vol.2, 2008, no.9, 425-429. Φ(x) ≈ 1- , x > 0 2 0.226+0.64−𝑥𝑥+�0.33 2+3 𝑒𝑒 [3] Bagby, R.J. (1995).Calculating Normal Probabilities. √ 𝜋𝜋 𝑥𝑥 �𝑥𝑥 American Math Monthly, 102, 46-49. [4] Byrc, W. (2002A and 2002B). A uniform approximation to 3. Conclusion the right normal tail integral. Applied Mathematics and In this paper we have proposed a very simple Computation, 127, 365-374. approximation to the cdf of standard normal distribution. We [5] Feller. W. (1968). An Introduction to Probability Theory and endeavored to present a simplistic approximation formula its Applications, Vol. 1, Wiley Eastern Limited. with reasonable accuracy. Our formula is so simple that it [6] Hammakar, H. C. (1978). Approximating the Cumulative can be implemented in any hand held calculator. Accuracy of Normal Distribution and its Inverse. Applied Statistics, 27, any approximation has to be seen vis a vis the simplicity of 76-77. the approximation. The maximum absolute error of our [7] Johnson, N.L. and Kotz, S. (1994). Continuous Univariate approximation is 0.0001922. In other words, even though Distribution Vol. 1. John Wiley and Sons. our approximation is very simple, it provides a minimum accuracy of three digits. We calculated mean absolute error [8] Page, E. (1977). Approximations to the cumulative normal of our formula for x=0.00005 (0.00005) 3.0 and it was found function and its inverse for use on a pocket calculator. Applied Statistics, 26, 75-76. to be 5.54339E-05. An algorithm for constructing a function for cdf of standard normal distribution can now be designed. [9] Lin, J.T. (1989). Approximating the Normal Tail Probability We place in the appendix a pseudocode for constructing such and its Inverse for use on a Pocket Calculator. Applied Mathematics and Statistics 2(3): 147-149, 2014 149

Statistics, 38, 69-70. University Press, London. [10] Lin, J.T. (1990). A Simpler Logistic Approximation to the [14] Wall, H.(1948).Analytic Theory of Continued Fractions, Van Normal Tail Probability and its Inverse. Applied Statistics, 39, Nostrand,Princeton, N.J. pp 358 255-257. [15] Winitzki.S. (Feb 2008) A handy approximation for the error [11] Shenton, L.R. (1954).Inequalities for the normal integral. function and its inverse. Downloaded from Biometrika, 41,177-189. http://homepages.physik.uni-muenchen.de/~Winitzki/erf-app rox.pdf on Nov 18, 2008. [12] Shore,H.(2005).Accurate RMM-based Approximations for the CDF of the Normal Distribution. Communications in [16] Zelen, M. and Severo, N.C. (1964). Probability Function (pp Statistics-Theory and Methods,34,507-513. 925-995) in Handbook of Mathematical Functions, Edited by M. Abramowitz and I. A. Stegun, Applied Mathematics [13] Tocher, K. D. (1963). The Art of Simulation. English Series, 55.