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Engineering, Technology & Applied Science Research Vol. 7, No. 1, 2017, 1382-1386 1382  
  

www.etasr.com Hamasha: A Mathematical Approximation of the Left-sided Truncated Normal Distribution Using the … 
 

A Mathematical Approximation of the Left-sided 
Truncated Normal Distribution Using the Cadwell 

Approximation Model 
 

Mohammad M. Hamasha   
Department of Engineering Management  

Prince Sultan University  
Riyadh, Saudi Arabia 

mhamash1@binghamton.edu  
 

 

Abstract—In the case that life distribution of new devices follows 
the normal distribution, the life distribution of the same brand 
used devices follows left-sided truncated normal distribution. In 
spite of many mathematical models being available to 
approximate the normal distribution density functions, there is a 
few work available on modeling/approximating the density 
functions of left-sided truncated normal distribution. This article 
introduces a high accuracy mathematical model to approximate 
the cumulative density function of left-sided truncated standard 
normal distribution defined on the range of [truncation point 
(ZL): ∞]. The introduced model is derived from the Cadwell 
approximation of the normal cumulative density. The accuracy 
level change with Z score is discussed in details. The maximum 
deviation of the model results, from the real results for the whole 
region of [-∞<ZL<-2: ∞], is 0.006877.  

Keywords-truncated normal distribution; normal distribution; 
mathematical approximation   

I. INTRODUCTION  
If is a continuous random variable X having a normal 

distribution with mean of μ and standard deviation of σ, the 
normal curve can be graphed using (1)  

2

2

( )

2
1

( )
2

x

f x e
m

s

s p

- -

=       (1) 

The normal distribution can take any mean and any positive 
standard deviation. These two parameters determine the shape 
of the normal curve. The mean provides the location of middle 
of distribution, and the standard deviation defines how much 
the data are spread from the mean. 

Normal distribution is used to model many sets of 
measurements in industry, business and nature. For instance, 
the human blood sugar level, the lifetime of radio sets, and 
even automobile costs are all normally distributed. A normal 
distribution has many unique properties including that 1) the 
value for mean, median, and mode are equal, 2) the distribution 
curve is bell shaped and symmetric about the mean, 3) the total 
area under the normal curve equals to 1 (all probability 

distributions have this property), 4) the normal curve 
approaches to zero with X going to -∞ or ∞, and finally, 5) 
around the center of the curve, the graph curves downward 
(concave). Further, the graph curves upward at the right and the 
left sides. The inflection points are the points at which the 
curve changes from curving downward to upward. The normal 
distribution with σ equals to 1 and μ equals to 0 is called the 
standard normal distribution. The standard normal distribution 
function is shown in (2).  

2

2

2

1
)(

z

ez





        (2) 

The standard normal distribution horizontal scale is well 
known as Z score. Any  not standard normal distribution can be 
transferred into Z score with using of the following 
transformation formula: ( ) /z x m s= - . 

There are many situation where the interest is to compute 
the probability that the observed value X is less than or equal to 
some real number x. Therefore, the cumulative distribution 
function is defined as F(x) = P(X ≤ x). For every real number x, 
F(x) is defined, as shown in (3).  

F(x) = P(X  x) ( )
x

f t dt
-¥

£ = ò    (3) 
The cumulative distribution function for the normal 

distribution is denoted by Φ(z) as addressed in the following 
equation: 

dtez
z t






 2
2

2

1
)(


     (4) 

The solution of the above mentioned equation is complex 
and cannot be solved manually. Therefore, the statisticians 
prepared table to estimate the cumulative distribution function 
values for the standard distribution. This table is called a Z-
table.     

The normal distribution can be truncated in various ways as 
needed. For example, if the distribution of the GRE scores is 



Engineering, Technology & Applied Science Research Vol. 7, No. 1, 2017, 1382-1386 1383  
  

www.etasr.com Hamasha: A Mathematical Approximation of the Left-sided Truncated Normal Distribution Using the … 
 

normally distributed and a graduate school has a minimum 
acceptable score, then the distribution of the admitted student 
scores is basically left-sided truncated normal distribution. This 
is because the left tail of the normal distribution (e.g., scores 
lower than minimum acceptable GRE scores) is truncated. 
Another example, if the life in years, of a television set is 
normally distributed, then the distribution of a two years used 
television set from the same brand is left-sided truncated 
normal distribution. The tail of distribution at P(X<2) is 
truncated. The following figure (Figure 1) is an example on the 
left-sided truncated normal distribution.  

 

 
Fig. 1.  Left-sided truncated normal distribution (e.g.,μ=21.1, σ= 5.3, 

truncation point (XL)=10 ).  

The left-sided truncated standard normal distribution is 
defined on the region (ZL:∞). Since the final area under the 
curve of truncated distribution must equal to 1, the new curve 
is stretched up to compensate the lost truncated area over the 
region (-∞: ZL). The probability density function of the left-
sided truncated normal distribution is addressed in (5) 

,

)(

)(
)(




Lz

T

dtt

z
z




  zzL       (5) 

The cumulative distribution function for the left-sided 
truncated normal distribution can be defined as the cumulative 
density function divided by the area under the original 
distribution function over the region (zL: ∞). Equation (6) 
represents the cumulative distribution function of left-sided 
truncated standard normal distribution.  

dt

dtt

t
z

z

z

z

T

L

L





)(

)(
)(




,  zzL       (6) 

Equation (6) is a complex integration function, and need a 
numerical solution to be solved. Therefore, engineers usually 
use sophisticated computer programs or specialized software 
package to handle this kind of equations. However, in the case 
that φ(z) is approximated with function that is able to be 

differentiated and integrated, then the truncated normal 
distribution can be estimated in a function.  

II. REVIEW OF PRIOR RESEARCH 
Much research on mathematical approximation of normal 

distribution functions is available in the literature. At least 15 
highly accurate models were introduced. Cadwell (1949) [1-3] 
introduced a very nice approximating model, as shown in (7)  




































 



























 








0,
3

)3(22
exp(11

2

1
1

0,
3

)3(22
exp(11

2

1

)(
2

1

2

42

2

1

2

42

z
zz

z
zz

z










(7) 

The following 6 models are mentioned in [1, 2]: Polya 
(1945), 2) Hart (1963), 3) Zelen and Severo (1966), 4) 
Schucany and Gray (1968), 5) Hoyt (1968), 6) Hamaker 
(1978), 7) Lin (1988), 8) Lin (1989), and 9) Lin (1990). 
Further, there are few recent models such as Vazquez-Leal 
(2012) [4], Choudhury (2014) [5] and Luis (2015) [6]. 

Many authors discussed in details many aspect about the 
truncated  normal distribution.  For example, Ke et al. [7] used 
double-sided truncated normal distribution on action reliability 
for ammunuition swing device of large caliber gun.  Pender [8] 
provided an exact expression for the moments of the truncated 
normal distribution using Stein’s lemma. His moment 
expressions provide insight into the steady state skewness and 
kurtosis dynamics of single server queues with impatient 
customers. Sun et al. [9] worked on finite fault modelling for 
the Wenchuan earthquake using hybrid slip model with 
truncated normal distributed source parameters. Beucher and 
Renard [10] described a stochastic model called truncated 
Gaussian simulations (TGS), which distributes a collection of 
facies or lithotypes over an area of interest. This method is 
based on facies proportions, spatial distribution and 
relationships, which can be easily tuned to produce numerous 
different textures. Mukerjee and Ong [11] worked on variance 
and covariance inequalities for truncated joint normal 
distribution via monotone likelihood ratio and log-concavity. 
Horrace [12] developed probability statements and ranking and 
selection rules for independent truncated normal populations. 
Sharples and Pezzey [13] presented results pertaining to 
truncated multivariate normal distributions, some of which 
already appear in the mathematical literature. They focused to 
make these types of results more accessible to the 
environmental science community and to this end, they 
included a conceptually simple alternative derivation of an 
important result. Furthermore, they illustrated how the theory 
of truncated multivariate normal distributions is employed in 
the environmental sciences. 

We can conclude from the above discussion that there is a 
lot on work on mathematical aspects and applications of the 
normal cumulative distribution function. Even so, 
mathematical approximations on the truncated normal 
distribution are not that frequent. In this paper, a mathematical 
approximation model of left-sided truncated normal cumulative 



Engineering, Technology & Applied Science Research Vol. 7, No. 1, 2017, 1382-1386 1384  
  

www.etasr.com Hamasha: A Mathematical Approximation of the Left-sided Truncated Normal Distribution Using the … 
 

distribution function is proposed. The model is derived from 
Cadwell’s model of approximating the normal cumulative 
distribution function. 

III. PROPOSED MODEL 
Since (7) is separated into two parts according to z value, 

the model will be divided on two parts. We are going to 
formulate a model for the left part (Z<0) first. The left part is 
recalled, as extracted and simplified in (8).  

0,
3

)3(22
exp(1

2

1

2

1
)(

2

1

2

42








 




 z
zz

z





    (8) 

The left-sided truncated standard normal distribution can be 
obtained through integration as explained in (6). Remember 
that φ(z) =dΦ(z)/dz. The following equation can be used to 
obtain the first part of our model, as addressed in (9).  

 )(zT  

dt

dt
tt

dt

d

tt

dt

d

z

z

z

L

L










































 












































 






0
2

1

2

42

2

1

2

42

3

)3(22
exp(11

2

1
15.0

3

)3(22
exp(11

2

1
1










, z<0 (9) 

 
The solution of the equation will give the following 

formula (written in the simplest form).  



























 




























 







2

1

2

42

2

1

2

42

3

)3(22
exp(11

3

)3(22
exp(11

1)(











LL

T

zz

zz

z  z<0   (10) 

If Z>0, then approximating ՓT(z) needs to extract the 
positive z part of (7) and substitute it in (6), exactly as we did 
in (9).  The result is addressed in (11).  



























 





























 







2

1

2

42

2

1

2

42

3

)3(22
exp(11

1
3

)3(22
exp(1

1)(











LL

T

zz

zz

z , z≥0    (11) 

IV. NUMERICAL EXAMPLES 
Example 1: If the life of a television set is normally 

distributed with a mean of 22 years and standard deviation of 7. 
The interest of this question is to find the 6 year reliability of 6 
years used television from the same brand.  In other words, 
what is the chance that the 6 years old television will be 
survived for another 6 years?  

Solution: Reliability is defined as the probability that a 
device, part or system will perform their function for a given 

period of time when operated under stated conditions [14]. The 
requirement can be written in statistics notation as P(X> 6+6). 
Since this distribution is not standard, the transformation 
formula, z=(x-µ)/σ will be used to find zL and z that are 
corresponding to x=6 and x=12, respectively.  

286.2
7

226



Lz  

249.1
7

2212



z  

The probability P(X>12) is corresponding to P(Z>-1.249) 
or 1-ՓT(-1.249). By using (11), the result is 0.905713. The 
model result is very close to the actual result which is 
0.904229. The deviation of the model result from the actual 
result (i.e., error) is only 0.00148 

Example 2: Let’s assume that the distribution of scores of 
GRE analytical part test out of 800 is normally distributed with 
a mean of 540 points and standard deviation of 40 points, and 
let’s assume that the acceptable score for some graduate school 
is 460. If an admitted student is selected randomly, what is the 
probability that his score is below 620?  

Solution: Since the value of probability density at any X 
greater than 800 (more than is 6.5 sigma) is very close to zero, 
we will not assume that the distribution is truncated from the 
right side at 800.  The distribution of the admitted students is 
left-sided truncated normal distribution at the acceptable level 
(i.e., 460). zL = -2 is corresponding to xL = 460 by using the 
transformation formula (i.e., z=(x-µ)/σ). Further, z=2 is 
corresponding to x=620. In statistic notation, the requirement is 
P(X<620) which is corresponding to P(Z<2) or  Փ T(-1.249). By 
using (11), the result is 0.9826. The model result is very close 
to the actual result for this example. The actual result is 
0.97672, and deviation of the model result from the actual 
result is only 0.00588.  

It is noticeable from both examples, that the error level is 
very low. Indeed, this level of error is very ignorable for most 
applications of reliability engineering as well as other 
probability fields.  

V. ACCURACY ANALYSIS 
In this section, the accuracy of the model is presented in 

term of approximating the left-sided truncated normal 
distribution. Mainly, the change of deviation of the model 
results from the real results (i.e., error) with Z and ZL is 
focused. To make sure that the introduced model is good at all 
value Z and ZL, we are concerning about the maximum error 
over the whole defining range, (ZL:∞) at any ZL  (-∞:-1.5).  

Figure 2 presents the error versus Z at ZL= -4, -3.5, -3, -2.5, 
-2, and -1.5. We can clearly note that there are two maximum 
peaks (i.e., maximum deviation or error) for every curve. The 
first peak is noticed at somewhere between Z=-1.6 and Z=-1.7, 
and the second peak is noticed at somewhere between Z=1.6 
and Z=1.7 for all curves. The maximum error for each curve is 
as follows: for ZL= -4, the maximum absolute error is about 
0.006452, for ZL= -3.5, the maximum absolute error is about 
0.006456, for ZL= -3, the maximum absolute error is about 
0.0065, for ZL= -2.5, the maximum absolute error is about 



Engineering, Technology & Applied Science Research Vol. 7, No. 1, 2017, 1382-1386 1385  
  

www.etasr.com Hamasha: A Mathematical Approximation of the Left-sided Truncated Normal Distribution Using the … 
 

0.00663, for ZL= -2, the maximum absolute error is about 
0.006887, and for ZL= -1.5, the maximum absolute error is 
about 0.0071251. Usually, the literature does not focus on any 
truncation point higher that ZL=-2 as it is rarely needed in most 
applications. In this study, we took ZL=-1.5 as the final 
truncation point. 

In Figure 3, the error versus Z is constructed for the original 
approximation of standard normal distribution that our model is 
based on (i.e.,  Cadwell approximation). We can see that the 
maximum error is 0.006466 at Z= -1.655 and at Z=1.655. The 
accuracy of the Cadwell model is close to the accuracy of the 
truncated Cadwell model at any considered ZL. The current 
model is more accurate than the logistic function-based 
truncated normal distribution which introduced in [1]. The 
current model leads to a maximum absolute error of about 
0.0071, while the maximum absolute error in the logistic 
function-based model is close to 0.02 at the range of -∞<ZL<-
1.5.  Furthermore, the current model is easier to implement 

using simple calculator. Figure 4 refers to a comparison graph 
of the model and real results of ZL=-2 left-sided truncated 
normal cumulative distribution function. We can notice that 
they are very close to each other. We can’t distinguish the 
difference between the two curves with the normal paper scale. 

VI. CONCLUSIONS  
In this article, an approximation to the left-sided truncated 

cumulative distribution function is introduced. The model can 
be used by reliability and quality engineers in order to avoid 
the sophisticated computer programs or software when having 
such distributions in their work. The model was build based on 
Cadwell approximation to the normal distribution. This study is 
unique because it provided model with a higher accuracy than 
any similar study in the literature. It provided a maximum error 
of 0.006877 for Z  (-2:∞)  while logistic function-based 
approximation [1], for example,  provides a maximum error 
0.017 on the same range of Z. 

 

 
Fig. 2.  Error (i.e., different between model and real results) versus Z at ZL=-4, -3.5, -3, -2.5, -2, and -1.5.  

 

 
Fig. 3.  Error versus Z for Cadwell approximation (i.e., untruncated 

standard normal distribution). 

 
Fig. 4.  Truncated normal distribution approximation versus the real 

truncated normal distribution at ZL= -2. 

REFERENCES 
[1] M. M. Hamasha, “Practitioner advice: approximation of the cumulative 

density of left-sided truncated normal distribution using logistic function 
and its implementation in Microsoft Excel”, Qual. Eng., 2016  

[2] S. R. Bowling, M. T. Khasawneh, S. Kaewkuekool, B. R. Cho, “A 
logistic approximation to the cumulative normal distribution”, J. Ind. 
Eng. Manage., Vol. 2, No. 1, pp. 114-127, 2009 

[3] J. H. Cadwell, “The bivariate normal integral”, Biometrika, Vol. 38, pp. 
475-479, 1951 

[4] H. Vazquez-Leal, R. Castaneda-Sheissa, U. Filobello-Nino, A. 
Sarmiento-Reyes, J. S. Orea, “High accurate simple approximation of 
normal distribution integral”, Math. Prob. Eng., Vol. 2012, Article 
ID 124029, pp. 1-22, 2016 

[5] A. Choudhury, “A simple approximation to the area under standard 
normal curve”, Math. Stat., Vol. 2, No. 3, pp. 147-149, 2014 

[6] A. J. Luis, “An Approximation to the Probability Normal Distribution 
and Its Inverse”, Ingeniería, Investigación y Tecnología, Vol. 16, No. 4, 
pp. 605-611, 2015 

[7] X. Ke, J. Hou, T. Chen, “Notice of retraction research on action 
reliability for ammunition swing device of large caliber gun based on 
doubly truncated normal distribution”, International Conference on 
Quality, Reliability, Risk, Maintenance, and Safety Engineering 
(QR2MSE), July 15-18, 2013 [retracted]  



Engineering, Technology & Applied Science Research Vol. 7, No. 1, 2017, 1382-1386 1386  
  

www.etasr.com Hamasha: A Mathematical Approximation of the Left-sided Truncated Normal Distribution Using the … 
 

[8] J. Pender, The truncated normal distribution: Applications to queues 
with impatient customers”, Oper. Res. Lett., Vol. 43 No. 1, pp. 40-45, 
2015 

[9] X. D. Sun, X. X. Tao, C. Q. Liu, “Finite fault modelling for the 
Wenchuan earthquake using hybrid slip model with truncated normal 
distributed source parameters”, Appl. Mech. Mater., Vol. 256-259, pp. 
2161-2167, 2013 

[10] H. Beucher, D. Renard, “Truncated Gaussian and derived methods”, In 
Press, C. R. Geosci., Vol. 348, No. 7, pp. 510–519, 2016  

[11] R. Mukerjee, S. H. Ong, “Variance and covariance inequalities for 
truncated joint normal distribution via monotone likelihood ratio and 
log-concavity”, J. Multi. Var. Anal., Vol. 139, pp. 1-6, 2015 

[12] W. C. Horrace, “On ranking and selection from independent truncated 
normal distributions”, J. Econometrics, Vol. 126, pp. 335-354, 2005 

[13] J. J. Sharples, J. C. V. Pezzey, Expectations of linear functions with 
respect to truncated multinormal distributions–With applications for 
uncertainty analysis in environmental modelling”, Environ. Modell. 
Software, Vol. 22, No. 7, pp. 915-923, 2007 

[14] C. E. Ebeling, An introduction to reliability and maintainability, 2nd Ed. 
Waveland Pr Inc, USA. 2009 

AUTHORS PROFILE 
Dr. Mohammad M. Hamasha received his PhD in Industrial and Systems 
Engineering from the State University of New York at Binghamton, New 
York, USA, in December 2011. He also received his MS in Industrial 
Engineering and BS in Biosystems Engineering from Jordan University of 
Science and Technology (JUST) in 2008 and 2005, respectively. He worked 
for two years from 2011 to 2013 as a postdoctoral associate and team leader 
with the Center for Autonomous Solar Power (CASP) in Binghamton NY 
doing a research related to reliability of solar materials and cell, and 
optimization of smart grid. Up to August 2016, Dr. Hamasha has published 
about 45 publications. Over the years, he has been involved in a variety of 
projects that range from experimental-based research to operations research, 
management, simulation, material science, thin films and photovoltaic 
research. His current research interests are in the areas of Simulation, 
operational research, reliability engineering, design and analysis of production 
systems, Stochastic and Simulation Optimization and Lean Six Sigma.