APPENDIX 8C. LIFETIME DISTRIBUTIONS 

TABLE OF CONTENTS 

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

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

LIST OF TABLES 

Table 8C.2.1 Commercial Clothes Washers......................................................................8C-2 

LIST OF FIGURE 

Figure 8C.2.1 Fraction of the Commercial Clothes Washers Failing, Multi-Family .........8C-3 

Figure 8C.2.2 Cumulative Lifetime Length of Commercial Clothes Washers, MultiFamily ..........................................................................................................8C-3 

Figure 8C.2.3 Fraction of the Commercial Clothes Washers Failing, Laundromat ...........8C-4 

Figure 8C.2.4 Cumulative Lifetime Length of Commercial Clothes Washers, 

Laundromat..................................................................................................8C-4 

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APPENDIX 8C. LIFETIME DISTRIUBTIONS 

8C.1 INTRODUCTION 

DOE characterized the product lifetime using a Weibull probability distribution that 

ranged from the minimum to maximum lifetime estimates described in chapter 8, section 8.2.3. 

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 

8C.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 

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

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: 

Eq. 1 

where 

L = location 

Alpha = scale 

Beta = shape 

Therefore, the cumulative distribution is 

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%) 

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The values for the parameters in Eq.1 and Eq. 2 are determined using the approach 

outlined in Crystal Ball’s Technote.3 

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. 

Table 8C.2.1 shows the average, minimum, maximum lifetime, and maximum percentile 

values used to determine the Weibull distribution parameters alpha and beta. For commercial 

clothes washers, DOE developed two lifetime estimates based on product application—one for 

multi-family applications and another for Laundromats. DOE estimated that product lifetimes 

did not vary based on whether the washer was top-loading or front-loading. DOE estimated that 

the maximum lifetime percentile for both product applications was 99 percent. 

Table 8C.2.1 Commercial Clothes Washers 

Product Application 

Expert Opinion Values 

Weibull 

Parameters 

Minimum 

years 

Average 

years 

Maximum 

years 

Maximum 

percentile 

% 

Alpha 

(scale) 

Beta 

(shape) 

Multi-Family 7 11.3 15.5 99 4.782 2.655 

Laundromat 5 7.1 9.3 99 2.391 2.655 

Figures 8C.2.1 to 8C.2.4 show the Weibull distribution as well as the cumulative Weibull 

distribution for each commercial clothes washer product application. 

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0%

5%

10%

15%

20%

25%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Age of Equipment (Years) 

Probability of Failure(%) 

Figure 8C.2.1 Fraction of the Commercial Clothes Washers Failing, 

Multi-Family 

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Age of Equipment (Years) 

Probability (%) 

Figure 8C.2.2 Cumulative Lifetime Length of Commercial Clothes 

Washers, Multi-Family 

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0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

0 1 2 3 4 5 6 7 8 9 1 0 1 1

Age of Equipment (Years) 

Probability of Failure (%) 

Figure 8C.2.3 Fraction of the Commercial Clothes Washers Failing, 

Laundromat 

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 1 2 3 4 5 6 7 8 9 1 0 1 1

Age of Equipment (Years) 

Probability (%) 

Figure 8C.2.4 Cumulative Lifetime Length of Commercial Clothes 

Washers, Laundromat 

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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. 

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

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

Most Likely Value, and a Percentile are Known. 2007 [cited; Available from: 

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

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