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

LIST OF TABLES

Table 8-C.1.1 Commercial Clothes Washers.................................................................... 8-C-2

LIST OF FIGURES

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

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

Multi-Family .............................................................................................. 8-C-3

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

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

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

8-C-i
APPENDIX 8-C. LIFETIME DISTRIBUTIONS

8-C.1 INTRODUCTION

The U.S. Department of Energy (DOE) characterized the lifetime of both product classes

of clothes washers being considered for new energy efficiency standards (front-loading and toploading). DOE characterized commercial clothes washer lifetimes using a Weibull probability 

distribution that ranged from the minimum to maximum lifetime estimates, as described in 

chapter 8, section 8.2.3. The Weibull distribution is recommended for evaluating lifetime data,

because it can be shaped to match low, most likely (or average), and high values. The probability 

of exceeding the high value is contained in the long tail of the Weibull distribution.

1,2

8-C.2 DERIVATION OF WEIBULL DISTRIBUTION PARAMETERS

Weibull distributions utilize available data to assign low, average, and high values to a 

random variable that has unknown distribution parameters. DOE applied Weibull distributions to 

product lifetime data to derive low, average, and high lifetime values, along with a percentile

containing a high value. A similar approach is described in a technical note to the software 

Crystal Ball, which uses a most likely value in place of an average value.3 The Weibull 

distribution can be defined as:

Where:

L = location,

α = scale, and

β = shape.

The cumulative distribution therefore is:

Based on available data, Weibull distribution parameters are specified as follows.

1. The output deviates must be greater than the expert opinion of 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

percent or 90 percent).

8-C-1
The values for the parameters in the equations were determined using the approach 

outlined in Crystal Ball’s technical note.

3

Crystal Ball can be used to check a solution by specifying a Weibull distribution that has 

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

forecast that equals that assumption. The forecast histogram and statistics will confirm whether

the Weibull distribution matches the desired shape.

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 8-C.1.1 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.

8-C-2
 Figure 8-C.2.1 Fraction of the Commercial Clothes Washers Failing, Multi- Family

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

 Multi-Family

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(%)

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 (%)

8-C-3
 Figure 8-C.2.3 Fraction of the Commercial Clothes Washers Failing, Laundromat

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

Laundromat

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

0 1 2 3 4 5 6 7 8 9 10 11

Age of Equipment (Years)

Probability of Failure (%)

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 1 2 3 4 5 6 7 8 9 10 11

Age of Equipment (Years)

Probability (%)

8-C-4
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. (Last accessed 

November, 2013.) <http://www.crystalball.com/support/simulation/cbl_gen_021A.html>

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