Summarizing Risk Analysis Results To quantify the risk of an output

variable, 3 properties must be estimated: A measure of central tendency (e.g. µ

) A measure of dispersion (e.g. σ or

range) The shape of the distribution

describes which values are more likely than others to occur

Histograms and proportions

Sources of Uncertainty Inherent risk of output variable

Measured by range and Determined by risk specified for input

variables Sampling risk

Associated with size of sample (e.g. number of iterations) and likelihood of error in parameter estimate

Estimation Sample Statistics are used to estimate

Population Parameters is used to estimate Population Mean,

Problem: Different samples provide different estimates of the population parameter

A sampling distribution describes the likelihood of different sample estimates that can be obtained from a population

_

X

Developing Sampling Distributions Assume there is a population … Population size N=4 Random variable, X,

is age of individuals Values of X: 18, 20,

22, 24 measured inyears A

B C

D

1

2

1

18 20 22 2421

4

2.236

N

ii

N

ii

X

N

X

N

.3

.2

.1

0 A B C D (18) (20) (22) (24)

Uniform Distribution

P(X)

X

Developing Sampling Distributions (continued

)

Summary Measures for the Population Distribution

1st 2nd Observation Obs 18 20 22 24

18 18,18 18,20 18,22 18,24

20 20,18 20,20 20,22 20,24

22 22,18 22,20 22,22 22,24

24 24,18 24,20 24,22 24,24

All Possible Samples of Size n=2

16 Samples Taken with Replacement

16 Sample Means1st 2nd Observation Obs 18 20 22 24

18 18 19 20 21

20 19 20 21 22

22 20 21 22 23

24 21 22 23 24

Developing Sampling Distributions

(continued)

1st 2nd Observation Obs 18 20 22 24

18 18 19 20 21

20 19 20 21 22

22 20 21 22 23

24 21 22 23 24

Sampling Distribution of All Sample Means

18 19 20 21 22 23 240

.1

.2

.3

P(X)

X

Sample Means

Distribution

16 Sample Means

_

Developing Sampling Distributions

(continued)

Properties of Summary Measures

i.e. is unbiased

Standard error (standard deviation) of the sampling distribution when sampling with replacement:

As n increases, decreases Sampling more decreases the uncertainty in

the estimate for

X

X

Xn

X

X

X

Comparing the Population with its Sampling Distribution

18 19 20 21 22 23 240

.1

.2

.3 P(X)

X

Sample Means Distribution

n = 2

A B C D (18) (20) (22) (24)

0

.1

.2

.3

PopulationN = 4

P(X)

X_

21 2.236 21 1.58X X

Central Limit TheoremAs sample size gets large enough…

the sampling distribution becomes almost normal regardless of shape of population

X

Confidence Interval for µ One can be 95% confident that the

true mean of an output variable falls somewhere between the following limits:

Descriptions of Shape Histograms; Percentiles Measures of symmetry

Skewness Weighted average cube of distance from

mean divided by the cube of the standard deviation

Symmetric distributions have 0 skew Positively skewed:

tail to right side is longer than that to the left Outcomes are biased towards larger values Mean > median > mode

Population Proportions p Proportion of population having a characteristic Sample proportion provides an estimate

number of successes

sample sizeS

Xp

n

Sampling Distribution of Sample Proportion

Mean:

Standard error:

p = population proportion

Sampling DistributionP(ps)

.3

.2

.1 0

0 . 2 .4 .6 8 1ps

Spp

1Sp

p p

n

Confidence Interval for p One can be 95% confident that the

true proportion for an output variable that has a certain characteristic falls somewhere between the following limits: