approximating the normal tail

8/3/2019 Approximating the Normal Tail

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Approximating the Normal Tail

Alan G. Hawkes

The Statistician, Vol. 31, No. 3. (Sep., 1982), pp. 231-236.

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Tile Statisticiai~,Vol. 31, No. 3 0039-0526j82/00760231 $02.00;Q 1982 Institute of Statisticians

Approximating the Normal TailALAN G. HAWKESDepartmetzt of'S tnt isti cs, Utrioersitj~College oJ'S~ vnn sen,Swnrzsen SA2 8PPA number of simple approximate formulae for the upper tailprobability of the norm al distribution are compared. Some newmore accurate, bu t only s lightly m ore complicated, formulae areintroduced. A ll are suitable for use wi th a pocket calculator.IntroductionIt is very useful t o have simple approximations to the cu n~ ula tive ormaldistributioil. About every statistician carries a pocket calculator, buthow m any of us carry a bo ok of tables everywhere we go ? When workingwith a desk-top microcomputer it would be inefficient to have to stop tolook up tables frequently. O ne could store a look-up table, b ut this woulduse up a lot of storage and the accuracy of interpolation in the table maynot be good . The appro xiination does not have to be all that simple fo r themicro or programmable calculator but clearly it should be very simplefor manual operation of a calc~~lator,nd simplicity would also help ~ O L Ito remember the formula. All of the formulae used in this paper werecalculated very easily on a Casio 502-P programmable pocket calculator.

A number of approximations have appeared in the literature. Wecompare the merits of some of them and introduce some new and veryaccurate formulae. The problem, formally stated, is that if Z has the unitnormal distribution we require approx imations to the uppe r tail probabilityQ(z )=P(Z > z )= +(x) dx4

where $(.u) = (27r)-llZ exp ( - .~2 /2 ) .We deal only with the case z>O. Th e modificatiojls needed fo r z < O willbe obvious.

Som e A lternative Approxinmtions2.1. Pnge's FormrihePage (1977) offers three ap proxim ations of the form

Q d z ) = 1 /{I +exp (2.~):


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wherey = alz( l+ azz2)

There are three possibilities suggested for the choice of constants ai, a2.These are, in increasing order of complexity,(i) a1= 2/(2/i7), aa = 0

(iii) a1=0.7988, a2= 0.04417The first of these, given by Tocher (1963), is not very accurate and isnot considered further. There is little to choose between the other two.Alternative (ii) is slightly simpler in requiring only one numerical constant,

but both are extremely easy to calculate. Alternative (iii) is very slightlymore accurate overall but (ii) is better in several places. We tabulate version(iii) in the third column of Table 1. The exact values given in column 2are taken from Pearson and Hartley's (1954) Biometrika Tables. It has amaximum absolute error of 0.00014. The percentage error is less than1 per cent up to z=2.3. Thereafter it rises steadily to 9 per cent at z= 3and 22 per cent at z = 3 . 5 .Either of these versions is good, being simple and accurate, but I would

not recommend them above z= 2.5.2.2. Harnalcer's FormulaA number of approximations are essentially modifications of a resultgiven by PoIya (1946)

Q(z)= i l l - {I-exp (- 2z"n)}l/z] (3)QHAJ{(Z) f[l - {I- exp ( - t2)11/zl (4)where

t= 0.806z(1- 0.01 82) (5)Note that 0.806 is close to (2/n)l/2=0.7979, so it represents quite a smallchange from equation (3). This function is tabulated in column 4 of Table 1.It is just as simple to calculate as Page's version (iii). It is not quite asaccurate as Q p for z


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Table 1Exact arzd various approximate tail probabilities of'the normal distributiott

Exact Page LewQ Q P


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2.3. Lew's FormulaeIt is difficult for one simple approx imation to be satisfactory fo r all valuesof z. Lew (1981) overcomes this problem by suggesting two formulae,one fo r small z and for large z . H e was also concerned to produce ultra-simple, easy to remember, formulae. T o this end he suggests

Q L I ( Z )4- (2n)-lla(z- z8/7) (6)andQ L Z ( Z )= (1+ @ ( z ) / ( l +z+ za) (7)

Lew recommends the use of (6 ) for z 6 1 and (7 ) fo r z > 1 , although theformer actually remains superior up to z= 1.14. They are tabulated incolumns 5 and 6 of Table 1 . Q L ~ ( z ) ,hich is again a modification of (3),has a maxim um absolute error of 0.00183 an d relative erro r below 0.9 percent in the range O < z < 1.1. F or larger z the erro r rises rapidly.In the range z , 1.1, Q ~ z ( z )as a maximum e rror of 0.00254.Th e relativeerror is abou t 2 per cent over the range 1.16 z< 2, drops to 1 per cent forall z>3 . It is very accurate for very large z , as shown in Table 2, withrelative err or less than 0.6 per cent fo r all z > 4.Q L I is less accurate than Q p and, in view also of the relatively poorperformance of Q L ~or z < 2 , I suggest a better approach would be t o

Table 2Norm al tail probabilities for large z

Q Q L ~ Q H ~ Negativeexponent

Note: The final column gives the negativepower of 10 by which the figures under Q,Q L ~r Q H ~must be multiplied to give theprobability. Thus, for example, Q(9)= 1129 x10-22.


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use Qp for zG2.4 and Q L ~or z > 2.4. This dual procedure would havea maximum absolute error of 0.00014 and maximum relative error 1.4percent, compared with values of 0.00265 and 2 per cent respectively for theprocedure suggested by Lew .Some New FormulaeThe procedures discussed above will be adequate for many purposes, butit is possible to achieve much greater accuracy with formulae only mar-ginally more complicated and well within the scope of a programmablepocket calculator. They are certainly to be preferred for use on a micro-computer.

For small z I proposeQxi(z)= 911- {I- exp (- 2t2/x)}llz] (8)where

t = z - (7.5166E- 3)z3+(3.1737E- 4)zj- (2.9657E- 6)zi (9)This may be thought of as a generalization of Hamaker's formula andis closely related to Bailey's (1981) solution to the incerse problem ofcom puting quantiles. It is tabulated in column 7 of Table 1. It has a maxi-

lnum error of 0.000017 fo r z


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REFERENCESBAILEY,B. 3. R. (1981). Alternatives to Hastings' approximation to the inverseof the normal cumulative distribution function. Applied Statistics, 30,275-6.'HAMAKER, H . C. (1978). Approximating the cumulative nonnal distribution andits inverse. Applied Statistics, 27,767.LEW,R . A. (1981). An approximation to the cumulative normal distributionwith simple coefficients. Applied Statistics. 30,299-301.PAGE,E. (1977). Approximations to the cumulative normal function and itsinverse for use o n a pocket calculator. Applied Statistics, 26,756.PEARSON,.S. and J~ARTLEY, H. 0. (1954). Biometrika ~ablesor Statisticians.Cambridge University Press.POLYA,G. (1946). Remarks on computing the probability integral in one an d twodimensions. Proceedings of the 1st Berkeley Symposium on Math. Statist.Prob., pp. 63-78.Tocm~,K.D. (1963). The Art of Simulation. English Universities Press, London.

approximating the normal tail

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