Download Exposure Distribution

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
0. 0.1 0.2 0.3 0.4
– 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
0.2
0.2
0.1
0.1
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.
• The value of the pth percentile is denoted by Xp.
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Terminology
– Equations:
N
• Mean (  ):

x
i
i 1
N
• Standard deviation ( ):
N

• Coefficient of Variation (CV):
2
(
x


)
 i
i 1
N

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 
S 
n
• Sample mean ( X ):
X
x
i 1
i

% CV  %CV
n
• Sample standard deviation (S):
n
S
 (x  X )
2
i
i 1
n 1

• Sample coefficient of variation ( CV):

CV 
S
X
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Normal Distribution
Z
a
r
e
a
=
0
.
6
8
4
x

If x values are normally distributed,
then the Z values are normally distributed,
have mean= 0 and standard deviation=1.
Z ~ N (0,1)
a
r
e
a
=
0
.
9
5
4







 





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?
Z
x

74.36  100

 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.
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Lognormal Distribution
• Lognormal distribution curve
A
r
e
a
=
0
.
6
8
3
A
r
e
a
=
0
.
9
5
4









gg
g
2
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
l  i 1
N
GM  e
– GSD:
N
GSD  e l
[ln( x )   ]
2
l 
i 1
i
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

GM  e

Xl
Xl 
 ln( x )
i
i 1
n
n
GSD  e
Sl
sl 
2
[ln(
x
)

X
]
 i
l
i 1
n1
• The GM and GSD can be used to estimate percentile of the
lognormal distribution.
X p %  GM  GSD
Z p%
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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:
X 95%  150  2.51.645  677.18
• More equations:
Z p% 
ln( X p % )  ln(GM )
ln(GSD)
GM
X 84%
GSD 

X 16% GM
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