Threshold from Standard
Deviation
Rich Christie
University of Washington
Distribution Design Working Group
Webex Meeting
October 26, 2001
October 26, 2001
Threshold from Standard Deviation
1
Concept
• Set threshold R* as the mean (average) plus a
multiple of standard deviation
R* = μ + n·σ
• Days with reliability ri > R* are Major Event Days
October 26, 2001
Threshold from Standard Deviation
2
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, 2001
Threshold from Standard Deviation
3
Normal (Gaussian) Distribution
μ=1
σ=1
n=1
R* = 2
Areas [= p(x>R*)] the same
μ=1
σ=2
n=1
R* = 3
October 26, 2001
Threshold from Standard Deviation
4
Normal (Gaussian) Distribution
With μ = 0, σ = 1
Multiple n
1
2
3
4
5
6
October 26, 2001
Prob of MED p
0.159
0.0228
0.00135
3.1686E-05
2.87105E-07
9.90122E-10
Freq f, MED/yr
57.9
8.3
0.49
0.012
0.00010
0.00000036
Threshold from Standard Deviation
5
Normal (Gaussian) Distribution
p and f are independent of μ and σ
Mean
(Mu)
0
1
3
-118
Std dev Multiple Thresh(Sigma)
n
old R*
1
3
3
2
3
7
27
3
84
111.34
3
216.02
Prob of
MED p
0.00135
0.00135
0.00135
0.00135
Freq f,
MED/yr
0.49
0.49
0.49
0.49
• But daily reliability does NOT have a
normal distribution
October 26, 2001
Threshold from Standard Deviation
6
Log-Normal Distribution
• For Log-Normal Distribution
– Most daily reliability data seems to be lognormal
– 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, 2001
Threshold from Standard Deviation
7
Log-Normal Distribution
μ=1
σ=1
n=1
R* = 2
Areas [= p(x>R*)] differ
μ=1
σ=2
n=1
R* = 3
October 26, 2001
Threshold from Standard Deviation
8
Log-Normal Distribution
Std Dev
Mean Mu Sigma
1
1
0.9
1
0.8
1
0.7
1
Mult n
2
2
2
2
Threshold R*
3
2.9
2.8
2.7
Prob p
0.0413
0.0398
0.0378
0.0354
Freq f,
MED/yr
15.1
14.5
13.8
12.9
MEDs/year decrease as mean μ decreases
(Improving average reliability means fewer MEDs)
October 26, 2001
Threshold from Standard Deviation
9
Log-Normal Distribution
Std Dev
Mean Mu Sigma
1
1
1
0.9
1
0.8
1
0.7
Mult n
2
2
2
2
Threshold R*
3.0
2.8
2.6
2.4
Prob p
0.0413
0.0426
0.0436
0.0444
Freq f,
MED/yr
15.1
15.5
15.9
16.2
MEDs/year increase as standard deviation σ decreases
(Larger utilities have inherently lower standard deviation
and thus would get higher MEDs/year.)
October 26, 2001
Threshold from Standard Deviation
10