Environmental Health Criteria 214

HUMAN EXPOSURE ASSESSMENT
The normal distribution, also known as the Gaussian
distribution, is one of the most important statistical
distributions. It is characterized by a symmetric, bell-shaped
frequency distribution and is commonly used as a basis for analysis of
environmental exposure data. Usually, a random variable (X) that
follows a normal distribution with mean µ and variance rho 2 is
denoted by X ~ N(µ, rho 2 ). The probability density function of the
normal distribution with parameters µ and rho 2 is given in Table 10.
Since the cumulative distribution function cannot be integrated
in a closed form, the best we can do is to numerically compute the
integral. The values µ = 0 and rho = 1 specify the standard normal
distribution. The values of the CDF for the standard normal
distribution have been tabulated and are available from most
statistical textbooks and computer packages. The capital letter Z is
usually reserved to denote a standard normal random variable, i.e.,
Z ~ N(0,1). The normal distribution ranges from positive infinity to
negative infinity and is symmetric. Equation 4.7 can be used to
transform any normal random variable X to a standard normal random
variable (Table 10). Standardized normal random variables are useful
for computing the probability of an event occurring, e.g., the
likelihood that someone in Malta has a blood lead concentration
greater than 384 µg/litre. Assuming the Maltese blood lead data
presented earlier are representative of the general population and the
blood lead concentrations are approximately normally distributed, the
standard normal distribution can be used to calculate the desired
probability.
4.3.2 Lognormal distribution
Many exposure measurements are strictly positive and right skewed
(i.e., asymmetric). Examples include the size distribution of
suspended particulate matter, personal exposures to various air
pollutants and human time-activity patterns. The lognormal
distribution is one possible model for describing data with these
characteristics. The natural log (ln) transform of a lognormally
distributed random variable has the properties of a normally
distributed random variable. In other words, the distribution defined
by the mean (µ ln x ) and standard deviation (rho ln x ) of the
ln-transformed values is bell-shaped and symmetric and can be
standardized according to the procedure outlined in the previous
section. Exponentiation of µ ln x and rho ln x gives values termed the
geometric mean (GM) and geometric standard deviation (GSD),
respectively. The GM and GSD can also be used to define a lognormally
distributed exposure measure.
A histogram of the blood faeces data from the Maltese sample
population is presented in Fig. 16a. The data depart from normality as
they are clearly right skewed. The histogram in Fig. 16b shows that
the ln-transformed values are approximately symmetric and indicates
that the data approximate a lognormal distribution rather than a
normal distribution. In this data set, µ ln x = 2.5 and rho ln x = 0.7
with corresponding GM = e 2.5 = 11.8 µg/litre and GSD = e 0.7 = 2.0. The
degree to which the lognormal distribution accurately describes the
data can be evaluated by plotting the raw data on lognormal
probability paper. This procedure is identical to that described in
relation in to Fig. 13, except that the y axis is expressed on a
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