RMI Notes .docx

# Symmetrical mean median left skewed tail to the left

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Symmetrical Mean = median Left-skewed Tail to the left Negative Mean < median Right-skewed Tail to the right Positive Mean > median Using Dispersion When analyzing probability distributions, insurance and RM professionals use measures of dispersion to assess the credibility of the measures of central tendency used in analyzing loss exposures Dispersion The variation among values in a distribution Describes the extent to which the distribution is spread out rather than concentrated around the expected value

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The less dispersion around the distribution expected value, the greater the likelihood that actual results will fall within a given range of that expected value Less dispersion means less uncertainty about the expected outcomes Affects the shape of a distribution The more dispersed a distribution (larger SD), the flatter the distribution A less dispersed distribution forms a more peaked distribution 2 statistical measures of dispersion: Standard deviation A measure of dispersion between the values in a distribution and the expected value/mean of that distribution, calculated y taking the square root of the variance Steps for calculating SD of a probability distribution: Calculate the distribution expected value/mean Subtract the expected value from each distribution value to find the differences Square each of the resulting differences Multiply each square by the probability associated with the value Sum the resulting products Find the square root on the sum Steps for calculating SD of a set of individual outcomes not involving probabilities: Calculate the mean of the outcomes (the sum of the outcomes / # of outcomes) Subtract the mean from each of the outcomes Square each of the resulting differences Sum these squares Divide his sum by the number of outcomes - 1 (variance) Calculate the square root of the variance Coefficient of variation A measure of dispersion calculated by dividing a distribution’s SD by its mean In comparing 2 distributions: If both distributions have the same mean/expected value, then the distribution with the larger SD has the greater variability If the 2 distributions have different means/expected values, the CoV is often used to compare the 2 distributions to determine which has the greater variability relative to its mean/expected value Determine whether a particular loss control measure has made losses more/less predictable Used in comparing the variability of distributions that have different shapes, means, or SDs The distribution with the largest CoV has the greatest relative variability The higher the variability within a distribution, the more difficult it is to accurately forecast an
individual outcome Calculating alternative probabilities Alternative probability The probability that any one of two or more events will occur within a given period

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• Fall '12
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