The purpose of this paper is to broaden the knowledge of mean difference and, in particular, of an important distribution model known as truncated normal distribution, which is widely used in applied sciences and economics. In this work, we obtained the general formula of mean difference, which is not yet reported in literature, for the aforementioned distribution model and also for particular truncated cases.
Truncated normal distribution is a part of the important continuous distribution models that find wide application in various fields of scientific research (Johnson et al., 1994 [
Density function of standardized normal distribution is
ϕ ( x ) = e − ( x − μ ) 2 2 σ 2 2 π , − ∞ < x < ∞ , − ∞ < μ < ∞ , σ > 0. (1)
Cumulative distribution function of normal distribution is
Φ ( x ) = ∫ − ∞ x ϕ ( u ) d u . (2)
We can consider the case of μ = 0 and σ = 1, namely, of standardized normal distribution. Density function of truncated normal distribution between a and b corresponds to the formula
φ ( x ) = ϕ ( x ) Φ ( b ) − Φ ( a ) , a < x < b . (3)
It is clearly obtained
φ ( x ) = e − x 2 2 ∫ a b e − x 2 2 d x , a < x < b . (4)
Cumulative distribution function of truncated normal distribution is then
Ψ ( x ) = ∫ a x e − x 2 2 d x ∫ a b e − x 2 2 d x . (5)
Various formulas are available to calculate the mean difference of a continuous distribution, the simplest one uses the cumulative distribution function F ( x )
Δ = ∫ a b 2 F ( x ) [ 1 − F ( x ) ] d x , (6)
in which a and b are extreme values of the range of distribution. Dealing with truncated normal distribution, it is necessary to use the cumulative distribution function Ψ ( x ) :
Δ = ∫ a b 2 Ψ ( x ) [ 1 − Ψ ( x ) ] d x . (7)
By means of heavy integrations and simplifications, the following general formula of the mean difference of truncated normal distribution is obtained
Δ = 2 { 2 [ erf ( b ) − erf ( a ) ] − 2 [ e − a 2 2 + e − b 2 2 ] [ erf ( b 2 ) − erf ( a 2 ) ] } π [ erf ( b 2 ) − erf ( b 2 ) ] , (8)
in which the error function occurs
erf ( x ) = 2 π ∫ 0 x e − t d t . (9)
The complement of error function is
erfc ( x ) = 2 π ∫ x ∞ e − t d t = 1 − erf ( x ) . (10)
In the following
As can easily be seen, starting from 0, when a and b are both zeros, the mean difference grows with a decreasing and b increasing up to the maximum, equal to 2 π , which is obtained when a = − ∞ and b = ∞ , that is the case of complete normal distribution.
Let us consider mean difference formulas in some interesting special cases of truncated normal distribution.
When b = − a in the general formula, after a few steps it leads to
Δ = 2 2 e − a 2 2 erf ( a 2 ) − 2 erf ( a ) 2 π erf ( a π ) 2 . (11)
When a = 0 in the general formula, after a few steps it leads to
Δ = 2 [ 2 erf ( b ) − 2 erf ( b 2 ) ( 1 + e − b 2 2 ) ] 2 π erf ( b 2 ) . (12)
When b = 0 , it leads to a similar result
Δ = 2 [ 2 erf ( a 2 ) ( 1 + e − a 2 2 ) − 2 erf ( a ) ] 2 π erf ( a 2 ) . (13)
When a = 0 and b = ∞ in the general formula, after a few steps it leads to
Δ = 2 ( 2 − 2 ) π . (14)
The same result is obtained when a = − ∞ and b = 0 .
When a > 0 and b = ∞ in the general formula, after a few steps it leads to
Δ = 4 [ 1 − erf ( a ) ] − 2 2 e − a 2 2 [ 1 − erf ( a 2 ) ] π [ 1 − erf ( a 2 ) ] 2 . (15)
The same result is obtained with a = − ∞ and b < 0 in the general formula:
Δ = 4 [ 1 + erf ( b ) ] − 2 e − b 2 2 [ 1 + erf ( b 2 ) ] π [ 1 + erf ( b 2 ) ] 2 . (16)
When a = − ∞ and b = ∞ , it leads to the well-known result of mean difference of complete normal distribution.
Δ = 2 π . (17)
In this work, we do not deal with other special cases.
Truncated normal distribution is an important distribution model that can be applied in many experimental and observational fields, particularly in quantitative economic sciences. Variability indexes as range of variation, mean deviation and standard deviation and shape indexes (asymmetry and disnormality) have been already obtained, but the mean difference of the truncated normal distribution is not known in literature. This note fills the gap regarding the lack of knowledge of mean difference general formula of said model. Moreover, the obtained formula is proposed for any truncation and for some particular truncated cases.
The authors declare no conflicts of interest regarding the publication of this paper.
Girone, G., Massari, A., Manca, F. and Marin, C. (2020) Mean Difference of Truncated Normal Distribution. Applied Mathematics, 11, 1162-1166. https://doi.org/10.4236/am.2020.1111078