1. 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

2. 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

3. 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 _

4. 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 in years A B C D

5. .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

6. All Possible Samples of Size n=2 16 Samples Taken with Replacement 16 Sample Means Developing Sampling Distributions (continued)

7. Sampling Distribution of All Sample Means 18 19 20 21 22 23 24 0.1.2.3 P(X) X Sample Means Distribution 16 Sample Means _ Developing Sampling Distributions (continued)

8. 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 

9. Comparing the Population with its Sampling Distribution 18 19 20 21 22 23 24 0.1.2.3 P(X) X Sample Means Distribution n = 2 A B C D (18) (20) (22) (24) 0.1.2.3 Population N = 4 P(X) X _

10. Central Limit Theorem As sample size gets large enough… the sampling distribution becomes almost normal regardless of shape of population

11. Confidence Interval for µ One can be 95% confident that the true mean of an output variable falls somewhere between the following limits:

12. 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

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

14. Sampling Distribution of Sample Proportion Mean: Standard error: p = population proportion Sampling Distribution P(p s ).3.2.1 0 0. 2.4.6 8 1 psps

15. 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: