8D-i

APPENDIX 8D. LIFETIME DISTRIBUTIONS

TABLE OF CONTENTS

8D.1 INTRODUCTION .................................................................................................. 8D-1

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

8D.3 LIFETIME DISTRIBUTIONS FOR DISHWASHERS......................................... 8D-2

LIST OF TABLES

Table 8D.3.1 Distribution Parameters for Dishwashers ................................................... 8D-2

LIST OF FIGURES

Figure 8D.3.1 Likelihood that Dishwasher Will Fail........................................................ 8D-2
8D-1

APPENDIX 8D. LIFETIME DISTRIBUTIONS

8D.1 INTRODUCTION

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

of dishwashers being considered for new energy efficiency standards (standard-sized and 

comact). DOE characterized dishwasher 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

8D.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).
8D-2

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.

8D.3 LIFETIME DISTRIBUTIONS FOR DISHWASHERS

Table 8D.3.1 shows the average, minimum, and maximum lifetimes plus maximum 

percentile values used to determine the Weibull distribution parameters α and β for residential 

dishwashers. DOE estimated that the maximum lifetime percentile for both standard-sized and

compact was 99 percent.

Table 8D.3.1 Distribution Parameters for Dishwashers

Values Weibull Parameters

Minimum Average

years

Maximum

years

Maximum

years Percentile

Alpha 

(%) (scale)

Beta 

(shape)

5 15.4 50 99 16.29 2.31

Figure 8D.3.1 show the Weibull distribution for the lifetime of both standard-sized and 

compact dishwashers.

Figure 8D.3.1 Likelihood that Dishwasher Will Fail

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

7.0%

8.0%

0 5 10 15 20 25 30 35 40 45 50

Weibull Retirement Function for 

Dishwashers (mean = 15.43 years)
8D-3

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. Appliance Magazine. October 2003. (Last 

accessed march 23, 2010.) 

<www.appliancemagazine.com/ae/editorial.php?article=197&zone=215&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 December 4, 2008.) 

Available from: www.crystalball.com/support/simulation/cbl_gen_021A.html.