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 DISTRIBUTION FOR PORTABLE AIR CONDITIONERS ............ 8D-2

LIST OF TABLES

Table 8D-3.1 Distribution Parameters for Portable ACs .................................................. 8D-2

LIST OF FIGURES

Figure 8D-3.1 Percent of Portable ACs Retiring each Year............................................... 8D-2
8D-1

APPENDIX 8D. LIFETIME DISTRIBUTIONS

8D-1 INTRODUCTION

In order to perform a life-cycle cost analysis, the U.S. Department of Energy (DOE)

characterized the lifetime of portable air conditioners (ACs) being considered for new energy 

efficiency standards. DOE characterized portable ACs lifetimes using a Weibull probability 

distribution encompassing lifetime estimates from minimum to maximum, as described in 

chapter 8, section 8.2.5. The Weibull distribution is recommended for evaluating lifetime data,

because it can be shaped to match low, most likely (or average), or 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 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 for portable ACs 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 

Crystal Ball software, 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:

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 (such as 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 DISTRIBUTION FOR PORTABLE AIR CONDITIONERS

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 portable ACs. 

DOE estimated that the maximum lifetime percentile for portable ACs was 99 percent.

Table 8D-3.1 Distribution Parameters for Portable ACs

Value Weibull Parameters

Minimum

years

Average

years

Maximum

years

Maximum

Percentile %

Alpha 

scale

Beta 

shape

3 10.47 20 99 10.66 2.64

Figure 8D-3.1 shows the Weibull distribution for the lifetime of portable ACs. The 

lifetime distribution is based on the retirement function, which indicates the number of portable 

ACs that fail and are retired in each year of their life. DOE used an average lifetime of 10.47

years in its analyses.

Figure 8D-3.1 Percent of Portable ACs Retiring each Year

0.0%

2.0%

4.0%

6.0%

8.0%

10.0%

12.0%

1 4 7 10 13 16 19 22 25 28

Age

Weibull Retirement Function for PACs 

(mean = 10.47 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 June 20, 2014.) 

<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>.