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  i1   N

• Standard deviation ( ): N 2 (xi  )   i1 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  i1  n %CV  %CV

n • Sample standard deviation (S): 2 (xi  X ) S  i1 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   74.36100 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 gg g gg 2 gg 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  g ) and GSD (  g ): – GM: The GM is the antilog of the arithmetic mean of the

logtransformed values (  l ). N

ln(xi ) l i1 l  GM  e N

– GSD: N 2 [ln( x )   ]   i l l i1 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  i1 l n n  2 [ln(xi )  X l ] Sl  GSD  e i1 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 X95% is:

1.645 X95% 1502.5  677.18

• More equations: ln(X )  ln(GM ) p% Z p%  ln(GSD)

GM X GSD   84% X16% GM 17