Exposure Distribution 1 Terminology • Population/distribution – A population is a definable collection of individual elements (or units). • For example, if a workplace has only 10 workers and all of them are exposed to a chemical, then one-day exposure measurements for the chemical are conducted to the workers. These 10 workers’ 8-hour TWAs can be considered as a one-day population of the workers’ 8-hour TWA exposures for the chemical. • Assume this population consists of the set of values below: (1, 2, 2, 3, 3, 3, 3, 4, 4, 5) • This data set can also be termed a distribution which describes the situation of event occurrence. 2 Terminology • Probability distribution – A probability distribution is also a set of numbers, but for each number (or small subset of numbers) we can assign a probability (=relative frequency) of occurrence. 0.4 0.4 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0.0 1 2 3 4 5 The probability distribution for the example is: { 1(0.1), 2(0.2), 3(0.4), 4(0.2), 5(0.1)} 3 Terminology • Probability distribution (continued) – A probability distribution is a convenient way to describe a large population without listing all elements. – Under certain conditions, a probability distribution can be described by a single equation or mathematical function termed probability density function (pdf). – In industrial hygiene, three pdf’s commonly seen are normal, lognormal and exponential. 4 Terminology • Parameters – A parameter is a constant characteristic of a distribution. Every distribution has numerous parameters, including a mean, a median, a standard deviation, and percentiles. – Mean ( ): • The mean is the arithmetic average of the distribution. • It is also denoted in other text as E (for expectation). – Median: • The median is the value below and above which lies 50% of the elements in the distribution. • It can be thought of as the middle value in an ordered count of the elements in the distribution. – Standard deviation (): • It is a measure of dispersion or variability around the mean of the distribution. 5 Terminology • Parameters (continued) – Coefficient of Variation (CV): • Defined as the standard deviation divided by the mean. • It is a measure of relative variability. It is used to compare the variability in two or more distributions. – Percentile: • A percentile (=quantile) is a value at or below which lies a specified percent or proportion of the distribution. th • The value of the p percentile is denoted by Xp. 6 Terminology – Equations: N • Mean ( ): xi i1 N • Standard deviation ( ): N 2 (xi ) i1 N • Coefficient of Variation (CV): CV 7 Terminology – Sample: • A sample is a collection of elements from a population where the elements are chosen according to a specific scheme. For each element in the sample, we measure the value of some variables of interest, and use the sample data to estimate population parameters. X n S • Sample mean ( X ): xi X i1 n %CV %CV n • Sample standard deviation (S): 2 (xi X ) S i1 n 1 • Sample coefficient of variation ( CV ): S CV X 8 Normal Distribution x Z If x values are normally distributed, area=0.684 then the Z values are normally distributed, have mean= 0 and standard deviation=1. Z ~ N(0,1) area=0.954 9 Normal Distribution • Normal distribution curve – Symmetric – The mean equals to median. – When moving equal distances along the x-axis to the right or left away from the mean, equal proportions of the distribution are covered. – The proportion of the distribution lying between the lowest value and some higher value is termed cumulative probability, which happens to be the same thing as a percentile. 10 Normal Distribution • Example question: – For a normal distribution with =100, =20, what proportion of the distribution is less than the value 74.36? x 74.36100 Z 1.282 20 Look up Z = -1.282 in the Z table Area Z 0.0968 -1.30 0.1056 -1.25 Interpolate. Area = 0.1000 This means that 10% of the distribution 74.36. 11 Lognormal Distribution • Lognormal distribution curve Area=0.683 Area=0.954 2 gg gg g gg 2 gg 12 Lognormal Distribution • Lognormal distribution – The lognormal distribution is a nonsymmetrical curve skewed to the right when the actual values are plotted on an arithmetic x-axis. – When the logarithms of the values (=logtransformed values) are plotted on the arithmetic x-axis, the skewed curve becomes the familiar normal distribution. – The natural logarithm, that is, the logarithm to the base e=2.71828… is used to do the value transformation. 13 Lognormal Distribution • Lognormal distribution – It is common to describe the lognormal distribution by its geometric mean (GM or g ) and geometric standard deviation (GSD or ). g – The GM is the value (in typical units) below and above which lies 50% of the elements in the population. Hence, the GM is the population median. – The GSD is a unitless number and always greater than 1.0. – The GSD reflects variability in the population around the GM. 14 Lognormal Distribution • Calculation of GM (or ) and GSD ( ): – GM: g The GM is the antilog g of the arithmetic mean of the logtransformed values ( l ). N ln(xi ) l i1 GM e l N – GSD: N 2 [ln( x ) ] i l l i1 GSD e l N 15 Lognormal Distribution • Sample estimates of the GM and GSD are calculated in a similar fashion to those of X and S in the normal distribution. n ln(x ) X i GM e l X i1 l n n 2 [ln(xi ) X l ] Sl GSD e i1 sl n 1 • The GM and GSD can be used to estimate percentile of the lognormal distribution. Z p% X p% GM GSD 16 Lognormal Distribution For example, if there is a lognormal distribution of exposure levels with GM=150 ppm and GSD=2.5, the value of X is: 95% 1.645 X95% 1502.5 677.18 • More equations: ln(X ) ln(GM ) p% Z p% ln(GSD) GM X GSD 84% X16% GM 17 .