October 26, 2001Threshold from Standard Deviation1 Rich Christie University of Washington Distribution Design Working Group Webex Meeting October 26, 2001

Threshold from Standard Deviation2 Concept Set threshold R* as the mean (average) plus a multiple of standard deviation Days with reliability r i > R* are Major Event Days R* = μ + n·σ

October 26, 2001Threshold from Standard Deviation3 Normal (Gaussian) Distribution If daily reliability has a normal (Gaussian) probability distribution –Equivalent to frequency criteria –Area under pdf above R* (= p) is constant as mean and standard deviation vary –p converts to MED frequency f

October 26, 2001Threshold from Standard Deviation4 Normal (Gaussian) Distribution μ = 1 σ = 1 n = 1 R* = 2 μ = 1 σ = 2 n = 1 R* = 3 Areas [= p(x>R*)] the same

October 26, 2001Threshold from Standard Deviation5 Normal (Gaussian) Distribution With μ = 0, σ = 1

October 26, 2001Threshold from Standard Deviation6 Normal (Gaussian) Distribution p and f are independent of μ and σ But daily reliability does NOT have a normal distribution

October 26, 2001Threshold from Standard Deviation7 Log-Normal Distribution For Log-Normal Distribution –Most daily reliability data seems to be log- normal –Probability p and frequency f (MEDs/year) vary with mean μ and standard deviation σ, for same multiple n. –Effect due to skew (kurtosis) of distribution

October 26, 2001Threshold from Standard Deviation8 Log-Normal Distribution μ = 1 σ = 1 n = 1 R* = 2 μ = 1 σ = 2 n = 1 R* = 3 Areas [= p(x>R*)] differ

October 26, 2001Threshold from Standard Deviation9 Log-Normal Distribution MEDs/year decrease as mean μ decreases (Improving average reliability means fewer MEDs)

October 26, 2001Threshold from Standard Deviation10 Log-Normal Distribution MEDs/year increase as standard deviation σ decreases (Larger utilities have inherently lower standard deviation and thus would get higher MEDs/year.)