APPENDIX 8-C. LIFETIME DISTRIBUTIONS 

TABLE OF CONTENTS 

8-C.1 INTRODUCTION ................................................................................................. 8-C-1


8-C.2 DERIVATION OF WEIBULL DISTRIBUTION PARAMETERS ..................... 8-C-1


8-C.3 WATER HEATER LIFETIME DISTRIBUTIONS .............................................. 8-C-2


8-C.4 GAS-FIRED POOL HEATERS LIFETIME DISTRIBUTIONS.......................... 8-C-7


8-C.5 DIRECT HEATING EQUIPMENT LIFETIME DISTRIBUTIONS.................... 8-C-8


LIST OF TABLES 

Table 8-C.3.1 Water Heaters............................................................................................. 8-C-2


Table 8-C.4.1 Gas-Fired Pool Heaters.............................................................................. 8-C-7


Table 8-C.5.1 Direct Heating Equipment ......................................................................... 8-C-8


LIST OF FIGURES 

Figure 8-C.3.1 Fraction of the Gas-Fired Storage Water Heaters Failing ......................... 8-C-3


Figure 8-C.3.2 Cumulative Lifetime Length of Gas-Fired Storage Water Heaters ........... 8-C-3


Figure 8-C.3.3 Fraction of the Electric Water Heaters Failing .......................................... 8-C-4


Figure 8-C.3.4 Cumulative Lifetime Length of Electric Water Heaters ............................ 8-C-4


Figure 8-C.3.5 Fraction of the Oil-fired Storage Water Heaters Failing ........................... 8-C-5


Figure 8-C.3.6 Cumulative Lifetime Length of Oil-fired Storage Water Heaters ............. 8-C-5


Figure 8-C.3.7 Fraction of the Gas-Fired Instantaneous Water Heaters Failing................ 8-C-6


Figure 8-C.3.8 Cumulative Lifetime Length of Gas-Fired Instantaneous Water


Heaters ....................................................................................................... 8-C-6


Figure 8-C.4.2 Fraction of the Gas-Fired Pool Heater Failing........................................... 8-C-7


Figure 8-C.4.3 Cumulative Lifetime Length of Gas-Fired Pool Heaters........................... 8-C-8


Figure 8-C.5.2 Fraction of the Direct Heating Equipment Failing .................................... 8-C-9


Figure 8-C.5.3 Cumulative Lifetime Length of Direct Heating Equipment ...................... 8-C-9


8-C-i 

EE-2006-STD-0129 and EERE-2006-BT-STD-0129 COMMENT 32.2
APPENDIX 8-C. LIFETIME DISTRIBUTIONS 

8-C.1 INTRODUCTION 

For each product class, DOE characterized the product lifetime using a Weibull 

probability distribution that ranged from the minimum to maximum lifetime estimates described 

in Ch. 8. The Weibull distribution is recommended for applying lifetime data because it can be 

shaped to match low, most likely or average, and high values, yet still allow some probability of 

exceeding the high value. The Weibull distribution has a long tail, which the analyst can use to 

specify a distribution with some chance of exceeding the high value.1, 2 

8-C.2 DERIVATION OF WEIBULL DISTRIBUTION PARAMETERS 

DOE utilized an approach for calculating the Weibull distribution using product lifetime 

data to determine low, average, high value, and percentile of high value. A similar approach is 

3 described by Crystal Ball Technical Note, which uses most likely value instead of average.

Available data is used to assign low, average, and high values to a random variable with 

unknown distribution parameters. 

The Weibull distribution can be defined as: 

β ⎛

⎜

⎝

L ⎞

⎟

⎠

β −1 x −

⎛

⎜

⎝

⎞

⎟

⎠

β − x − L
 α f (x) = exp
 Eq. 1


α α

Where: 

L = location 

α = scale 

β = shape 

Therefore, the cumulative distribution is 

β ⎛

⎜

⎝

⎞

⎟

⎠

x−L − α F(x) = 1
− exp
 Eq. 2


Based on available data, Weibull distribution parameters are specified in the following 

manner: 

1.	 The output deviates must be greater than the expert opinion for low value, 

2.	 The average Xavg, must be equal to the average value from the available data, 

3.	 The high value, xb, must correspond to some particular percentile point (e. g., 95%, or 

90%) 

8-C-1 

EE-2006-STD-0129 and EERE-2006-BT-STD-0129 COMMENT 32.2
The values for the parameters in Eq.1 and Eq. 2 are determined using the approach 

3 outlined in Crystal Ball’s Technote.

This solution can be checked using Crystal Ball by specifying a Weibull distribution with 

the calculated parameters (location, scale, and shape) in an assumption cell and generate a 

forecast that equals the assumption. Forecast histogram and statistics verify that the Weibull 

distribution matches the desired shape. 

8-C.3 WATER HEATER LIFETIME DISTRIBUTIONS 

Table 8-C.3.1 shows the average, minimum, maximum lifetime, and maximum percentile 

values used to determine the Weibull distribution parameters alpha and beta. The analysis 

utilized a maximum percentile of 99%. 

Table 8-C.3.1 Water Heaters 

Product Class 

Expert Opinion Values 

Weibull 

Parameters 

Minimum 

(years) 

Average 

(years) 

Maximum 

(years) 

Maximum 

percentile 

(%) 

Alpha 

(scale) 

Beta 

(shape) 

Gas-Fired/LPG Storage 7 12 20 99 5.6258 1.8233 

Electric Storage 6 14 20 99 8.9079 3.3778 

Oil-Fired Storage 5 9 13 99 4.5001 2.6548 

Gas-Fired Instantaneous 8 20 30 99 13.4203 3.0897 

Figure 8-C.3.2 to Figure 8-C.3.9 show the Weibull distribution as well as the cumulative 

Weibull distribution for each water heater product class. 

8-C-2 

EE-2006-STD-0129 and EERE-2006-BT-STD-0129 COMMENT 32.2
0%

2%

4%

6%

8%

10%

12%

14%

0 2 4 6 8 10 12 14 16 18 20 22 24

Age of Equipment (Years) 

Probability of Failure (%) 

Figure 8-C.3.2 Fraction of the Gas-Fired Storage Water Heaters 

Failing 

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 2 4 6 8 10 12 14 16 18 20 22 24

Age of Equipment (Years) 

Probability (%) 

Figure 8-C.3.3 Cumulative Lifetime Length of Gas-Fired Storage 

Water Heaters 

8-C-3 

EE-2006-STD-0129 and EERE-2006-BT-STD-0129 COMMENT 32.2
0%

2%

4%

6%

8%

10%

12%

14%

0 2 4 6 8 10 12 14 16 18 20 22 24

Age of Equipment (Years) 

Probability of Failure (%) 

Figure 8-C.3.4 Fraction of the Electric Water Heaters Failing


0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 2 4 6 8 10 12 14 16 18 20 22 24

Age of Equipment (Years) 

Probability (%) 

Figure 8-C.3.5 Cumulative Lifetime Length of Electric Water 

Heaters 

8-C-4 

EE-2006-STD-0129 and EERE-2006-BT-STD-0129 COMMENT 32.2
0%

5%

10%

15%

20%

25%

0 2 4 6 8 10 12 14 16

Age of Equipment (Years) 

Probability of Failure (%) 

Figure 8-C.3.6 Fraction of the Oil-fired Storage Water Heaters 

Failing 

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 2 4 6 8 10 12 14 16

Age of Equipment (Years) 

Probability (%) 

Figure 8-C.3.7 Cumulative Lifetime Length of Oil-fired Storage 

Water Heaters 

8-C-5 

EE-2006-STD-0129 and EERE-2006-BT-STD-0129 COMMENT 32.2
0%

2%

4%

6%

8%

10%

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

Age of Equipment (Years) 

Probability of Failure (%) 

Figure 8-C.3.8 Fraction of the Gas-Fired Instantaneous Water 

Heaters Failing 

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

Age of Equipment (Years) 

Probability (%) 

Figure 8-C.3.9 Cumulative Lifetime Length of Gas-Fired 

Instantaneous Water Heaters 

8-C-6 

EE-2006-STD-0129 and EERE-2006-BT-STD-0129 COMMENT 32.2
8-C.4 GAS-FIRED POOL HEATERS LIFETIME DISTRIBUTIONS 

Table 8-C.4.1 shows the average, minimum, maximum lifetime, and maximum percentile 

values used to determine the Weibull distribution parameters alpha and beta. The analysis 

utilized a maximum percentile of 99.99%. 

Table 8-C.4.1 Gas-Fired Pool Heaters 

Product Class 

Expert Opinion Values 

Weibull 

Parameters 

Minimum 

(years) 

Average 

(years) 

Maximum 

(years) 

Maximum 

percentile 

(%) 

Alpha 

(scale) 

Beta 

(shape) 

Gas-Fired Pool Heaters 3 6 20 99.99 3.2703 1.3470 

Figure 8-C.4.1 and Figure 8-C.4.2 show the Weibull distribution as well as the 

cumulative Weibull distribution for each water heater product class. 

0%

5%

10%

15%

20%

25%

0 2 4 6 8 10 12 14 16 18 20

Age of Equipment (Years) 

Probability of Failure (%) 

Figure 8-C.4.1 Fraction of the Gas-Fired Pool Heater Failing


8-C-7 

EE-2006-STD-0129 and EERE-2006-BT-STD-0129 COMMENT 32.2
0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 2 4 6 8 10 12 14 16 18 20

Age of Equipment (Years) 

Probability (%) 

Figure 8-C.4.2	 Cumulative Lifetime Length of Gas-Fired Pool 

Heaters 

8-C.5 DIRECT HEATING EQUIPMENT LIFETIME DISTRIBUTIONS 

Table 8-C.5.1 shows the average, minimum, maximum lifetime, and maximum percentile 

values used to determine the Weibull distribution parameters alpha and beta. The analysis 

utilized a maximum percentile of 99%. 

Table 8-C.5.1	 Direct Heating Equipment 

Product Class 

Expert Opinion Values 

Weibull 

Parameters 

Minimum 

(years) 

Average 

(years) 

Maximum 

(years) 

Maximum 

percentile 

(%) 

Alpha 

(scale) 

Beta 

(shape) 

All DHE Equipment 10 15 20 99 5.6257 2.6548 

Figure 8-C.5.1 to Figure 8-C.5.2 show the Weibull distribution as well as the cumulative 

Weibull distribution for each water heater product class. 

8-C-8 

EE-2006-STD-0129 and EERE-2006-BT-STD-0129 COMMENT 32.2
0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

0 2 4 6 8 10 12 14 16 18 20 22

Age of Equipment (Years) 

Probability of Failure (%) 

Figure 8-C.5.1 Fraction of the Direct Heating Equipment Failing


0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 2 4 6 8 10 12 14 16 18 20 22 24

Age of Equipment (Years) 

Probability (%) 

Figure 8-C.5.2 Cumulative Lifetime Length of Direct Heating 

Equipment 

8-C-9 

EE-2006-STD-0129 and EERE-2006-BT-STD-0129 COMMENT 32.2
REFERENCES


1.	 Barnes, P. R., J. W. Van Dyke, B. W. McConnell, S. M. Cohn, and S. L. Purucker, The 

Feasibility of Replacing or Upgrading Utility Distribution Transformers During Routine 

Maintenance, 1995. Oak Ridge National Laboratory. Oak Ridge, TN. Report No. Report 

No. ORNL-6804/R1. <http://www.ornl.gov/~webworks/cpr/v823/rpt/78562.pdf> 

2.	 Karr, T., Making the Most of Life Test Data. Appliance Magazine, 2003. 

<http://www.appliancemagazine.com/print.php?article=197&zone=1&first=1> 

3.	 Crystal Ball, Technote: Derivation of Weibull Distribution Parameters When the 

Minimum, Most Likely Value, and a Percentile are Known, 2007. 

<http://www.crystalball.com/support/simulation/cbl_gen_021A.html> 

4.	 U.S. Department of Energy - Energy Information Administration, Residential Energy 

Consumption Survey: 2001 Public Use Data Files, 2001. 

<http://www.eia.doe.gov/emeu/recs/recs2001/publicuse2001.html> 

8-C-10 

EE-2006-STD-0129 and EERE-2006-BT-STD-0129 COMMENT 32.2