F. K. Kong MA, MSc, PhD, CEng, FICE, FIStructE, R. H. Evans CBE, DSc, D ès Sc, DTech, PhD, CEng, FICE, FIMechE, FIStructE (auth.)-Reinforced and Prestressed Concrete-Springer US (1987)

Statistical concepts 7
an important probability distribution is the normal probability distribution,
defined by the equation
Y = a~t2n) exp{ -!(x- x) 2 /cr} (1.3-7)
where a is the standard deviation, the exponential e( = 2.71828) is the base
of the natural logarithm, and xis the mean of the variable x.
Suppose in eqn (1.3-7) the variable x is expressed in terms of another
variable z defined by
z=--
x-x
a
(1.3-8)
that is to say, z is the deviation from the mean expressed in multiples of the
standard deviation. Equation (1.3-7) is then replaced by the following
so-called standard form [8, 9]:
1 ( l 2)
y = ~(2 n) exp -2z (1.3-9)
(See also Example 1.3-5.) Figure 1.3-3 shows the graph of the
standardized eqn (1.3-9), where the range of z is from -oo to + oo. Of
course, the total area bounded by this curve and the x-axis is equal to unity;
the area under the curve between z = z 1 and z = z 2 represents the
probability that z lies between z 1 and z 2 • For example, 68.26% of the total
area is included between z = -1 and + 1; 95.44% between z = -2 and + 2;
y
~~----4-----~--~--~~~--+---~~z
-3 -2 -1
-30' -20' -ff
-h;.~J
2
20"
3
30' X-X
Fig. 1.3-3 Areas under the normal probability distribution curve