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 5
results in this case) is increased and the class interval reduced, it can be
seen that the histogram will resemble a smooth curve; in the limit when the
number of observations becomes infinite, the histogram becomes a smooth
curve, known as the probability density curve or the probability distribution
curve (Fig. 1.3-2). The area under the distribution curve
between any two points Xt and x 2 (i.e. the integral of the probability
density function f(x) between them) represents the probability that the
tensile strength of the concrete will lie between these two values. For a
very small increment dx, the function f(x) may be considered to be
constant from x to x + dx, and the probability that the variate will have a
value lying in this small interval is very nearly f(x) dx, which is the area of
the rectangle with heightf(x) and width dx. f(x) may therefore be thought
of as representing the probability density at x. Note that the probability of
the variate having exactly a particular value xis zero; we can only consider
the probability of its value lying in the interval x to x + dx. Also, the total
area under the entire curve of the probability function is unity.
Characteristics of distributions
The sum of a set of n numbers Xt. x 2 , ... Xn divided by n is called the
arithmetic mean, or simply the mean of the set of numbers. If .X denotes
the mean, then
_ Xt + Xz + X3 + ... Xn LX
x=
n
=
n
(1.3-1)
lfthe numbers Xt. x2 , ... xj have frequenciesft,f2 , ..• [j respectively, the
mean may equally be calculated from
_ !tXt + fzxz + ... frj Lfx
X - - - (1.3-2)
- ft + fz + · · · [j - n
......
~
....
~
~
II)
c:
~
~
:0
t'O
.0
0
'-
Q.
.------X
Fig. 1.3-2 Probability distribution curve