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.2.1 Residential Dehumidifier Lifetime Parameters ......................................... 8-C-2

LIST OF FIGURES

Figure 8-C.2.1 Fraction of Portable Unit Dehumidifier Failing ......................................... 8-C-3

Figure 8-C.2.2 Cumulative Lifetime Length of Portable Unit Dehumidifiers ................... 8-C-3

Figure 8-C.2.3 Fraction of Whole-home Dehumidifier Failing.......................................... 8-C-4

Figure 8-C.2.4 Cumulative Lifetime Length of Whole-home Dehumidifiers.................... 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 types

of dehumidifiers (i.e. portable unit dehumidifiers, whole-home dehumidifiers) being considered 

for new energy efficiency standards. DOE characterized residential dehumidifier lifetimes using 

a Weibull probability distribution that ranged from the minimum to maximum lifetime estimates, 

as described in chapter 8, section 8.2.2. The Weibull distribution is recommended for evaluating

lifetime data, because it can be shaped to match low, 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 (alpha) = scale, and

β (beta) = 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 (1) specifying a Weibull distribution that 

has the calculated parameters (location, scale, and shape) in an assumption cell, then (2) 

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 portable unit 

dehumidifiers, DOE used lifetime estimates from the previous DOE rulemaking for 

dehumidifiers.4 DOE estimated that portable unit dehumidifier lifetimes do not vary by capacity 

size. DOE assumed whole-home dehumidifiers have the same life span as residential room air 

conditioner. DOE therefore applied the lifetime parameters derived for room air conditioners to 

whole-home dehumidifiers.5 DOE estimated that the maximum lifetime percentile for both 

product types was 99 percent.

Table 8-C.2.1 Residential Dehumidifier Lifetime Parameters

Product Application

Expert Opinion Values Weibull 

Parameters

Minimum

years

Average

years

Maximum

years

Maximum

percentile

%

Alpha 

(scale)

Beta 

(shape)

Portable unit 

Dehumidifiers -- 11.0 -- 99 11.00 4.20

Whole-home 

Dehumidifiers 3.0 10.5 20.00 99 20.30 2.50

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

distribution for each product type of residential dehumidifiers.

8-C-2
 Figure 8-C.2.1 Fraction of Portable Unit Dehumidifier Failing

Figure 8-C.2.2 Cumulative Lifetime Length of Portable Unit Dehumidifiers

0%

2%

4%

6%

8%

10%

12%

14%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Porbability of Failure (%)

Age of Equipment (Years)

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Cumulative Porbability (%)

Age of Equipment (Years)

8-C-3
Figure 8-C.2.3 Fraction of Whole-home Dehumidifier Failing

Figure 8-C.2.4 Cumulative Lifetime Length of Whole-home Dehumidifiers

0%

1%

1%

2%

2%

3%

3%

4%

4%

5%

5%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45

Porbability of Failure (%)

Age of Equipment (Years)

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45

Cumulative Porbability (%)

Age of Equipment (Years)

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.

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.

4 DOE–Energy Efficiency and Renewable Energy (EERE). Energy Conservation Program for 

Consumer Products: Technical Support Document: Energy Efficiency Program for Consumer 

Products and Commercial and Industrial Equipment, Residential Dishwashers, 

Dehumidifiers, and Cooking Products, and Commercial Clothes Washers. 2009. Washington, 

DC. (Last accessed October 10, 2013.) 

<http://www1.eere.energy.gov/buildings/appliance_standards/residential/cooking_products_f

inal_rule_tsd.html>

5 DOE–EERE. Energy Conservation Program for Consumer Products. Technical Support 

Document: Energy Efficiency Program for Consumer Products and Commercial and 

Industrial Equipment, Residential Clothes Dryers and Room Air Conditioners. 2011. 

Washington, D.C. (Last accessed October 25, 2013.) 

<http://www.regulations.gov/#!documentDetail;D=EERE-2007-BT-STD-0010-0053>

8-C-5