Environmental Health Criteria 214

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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 http://www.inchem.org/documents/ehc/ehc/ehc214.htm Page 64 of 284 6/1/2007
HUMAN EXPOSURE ASSESSMENT logarithmic scale. The Maltese faecal lead data are plotted on lognormal probability paper in Fig. 17. The nearly straight line formed by the faecal lead measurements displayed on a logarithmic scale versus Z indicates that the data are approximately lognormally distributed. 4.3.3 Binomial distribution In some situation, the analyst may be interested in characterizing the frequency of a binary exposure outcome (e.g., yes/no; true/false). The binomial distribution is useful for modelling FIGURE 16a;V214EH21.BMP FIGURE 16b;V214EH22.BMP http://www.inchem.org/documents/ehc/ehc/ehc214.htm Page 65 of 284 6/1/2007
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Environmental Health Criteria 214

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