“ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Statistics 101 Robert C. Patev NAD Regional Technical Specialist (978)

“ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Random Variables What’s a random variable? –A quantity that can assume a specific value, or lie within a specified range of values, with a specified probability –A function defined on a sample space that assigns a specific probability to each value or range of values

“ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Discrete functions –Probability mass function, pmf Continuous functions –Probability density function, pdf –Cumulative density function, CDF Statistics

“ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Example of pmf Statistics

“ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Example of pdf t, yearspdf Statistics

“ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Example of CDF t, yearspdfcdf Statistics

“ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Statistics –Commonly used terms Mean Standard deviation or variance Coefficient of variation Median Skewness Correlation Distributions Types Statistics

“ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Mean –The average of a set of values –Excel command – AVERAGE Expected value –the centroid of the probability distribution on a random variable Mean and Expected Value

“ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Variance –The average squared deviation of values from the mean or expected value Variance

“ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Standard Deviation Standard Deviation,  –Spread of data about the mean value –A measure of the uncertainty, or “width” of a distribution –Square root of the variance –EXCEL command – STDEV or STDEVP

“ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Coefficient of Variation (COV) COV, V x –Standard deviation divided by expected value –A dimensionless value, often expressed as a percentage

“ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Median –e.g., median income, median age –Rank value –For odd n, value with rank of (n+1)/2 –For even n, average of value with rank n/2 or (n/2) + 1 –Used to limit extreme values –50th percentile –Excel command – MEDIAN Median

“ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions A measure of the asymmetry of the probability distribution of a random variable Skewness x f(x) Right

“ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Correlation of Random Variables –Measures of the tendency for two random variables to assume values independently, or values with some relationship to each other –Range is from 1 to –1 1 is perfectly correlated positive -1 is perfectly correlated negative 0 is no correlation Correlation

“ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Random Variable A Random Variable B Negative Random Variable A Random Variable B Positive Correlation

“ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Examples of Distribution Types –Discrete Binomial Poisson –Continuous Normal Lognormal Exponential Weibull –Most Commonly Used »Uniform »Normal »Lognormal Statistics

“ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Standard Normal Standard Normal Variate –Yields a normal distribution with a mean of 0 and a standard deviation of 1 –N (0,1) or  –Useful in converting reliability index to probability of unsatisfactory performance –Excel command - NORMSDIST

“ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Standard Normal Distribution –Example Reliability index,  2.0 N(0,1) = P UP = 1 – = Standard Normal N (0,1)