E-1.i

APPENDIX E-1. UNCERTAINTY AND VARIABILITY

TABLE OF CONTENTS

E-1.1 UNCERTAINTY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-1.1

E-1.2 VARIABILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-1.2

E-1.3 APPROACHES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-1.2
E-1.1

APPENDIX E-1. UNCERTAINTY AND VARIABILITY

To account for the uncertainty and variability, the LCC model was developed using Microsoft

Excel combined with Crystal Ball (a commercially available add-on). The model uses Monte Carlo

simulations to perform the analysis considering uncertainty and variability.

The analysis explicitly specifies both the uncertainty and variability in the model’s inputs

using probability distributions. The Monte Carlo simulation then takes thousands of random samples

from the probability distribution for each input within the model to calculate the outputs. The

distribution of the values calculated for the model outcome therefore reflects the probability of the

values that would occur. This technique provides solutions to a broad range of questions and helps

provide an insight into the likelihood of various possible outcomes. 

An importance analysis is also produced which shows the relative contribution of each of the

inputs to the total range of outputs. 

 A variety of graphic displays are used to illustrate the implications of the analysis results.

These include: a) a cumulative probability distribution and barcharts showing the percentage of US

households which would have a net life-cycle cost savings by owning a more energy-efficient water

heater or b) a chart that depicts importance to the various elements of the cost for each design option

considered.

One major assumption has been made throughout the analysis. One of two probability

distributions, triangular or normal, will be used where a specific form of uncertainty or variability

is totally or partially unknown. The triangular distribution is one of the simplest forms of probability

distributions. It uses three simple parameters, minimum, most-likely, and maximum, to describe the

probability distribution for a given set of data. It is commonly used in cases where the knowledge

about the factor of interest is limited. Normal distribution, on the other hand, is based on an

underlying assumption that the data follow a bell-shaped distribution. This is usually the case in

which a variable is influenced by many factors but none of them is dominant. When nothing but a

mean and variance about a random variable is known, the Department used a normal distribution to

describe the variable. 

E-1.1 UNCERTAINTY

When making observations of past events or speculating about the future, imperfect

knowledge is the rule rather than the exception. For example, the energy actually consumed by a

particular appliance type (such as the average U.S. water heater) has not been directly recorded very

often. Rather, energy consumption is usually estimated based upon available information. Even

direct laboratory measurements have some margin of error. When estimating numerical values

expected for quantities at some future date, the exact outcome is rarely known in advance.
E-1.2

E-1.2 VARIABILITY

Variability means that different applications or situations produce different numerical values

for a quantity. Specifying an exact value for a quantity may be difficult because the value depends

on something else. For example, the amount of hot water used per day by a household depends upon

the specific circumstances and behaviors of the occupants (e.g., number of persons, personal habits

about hot water use, etc.). Variability makes specifying an appropriate value more difficult. One’s

personal experience may not be representative of the entire population. On the other hand,

variability provides more information about the population under study. Surveys can be helpful here,

and analysis of surveys can relate the variable of interest (e.g., gallons of hot water use per day) to

other variables that are better known or easier to forecast (e.g., persons per household).

E-1.3 APPROACHES

Two approaches to consider uncertainty and variability are described here: 

Scenario analysis uses a single numerical value for a quantity in the calculation, then changes

the numerical values and repeats the calculation. A number of calculations are done, which provide

some indication of the extent to which the result depends upon the assumptions. For example, the

life-cycle cost of an appliance could be calculated for electricity prices of 2, 8, and 14 cents per kWh.

The advantages of scenario analysis are that the calculation is simple; a range of estimates

is used; and crossover points can be identified. When examining a given design option, a crossover

point occurs when the energy price rises to a point at which the savings in operating cost compensate

for the increased purchase expense of the design option. The disadvantage of scenario analysis is

that there is no information about the likelihood of each scenario.

In this analysis of water heaters, the only variables treated as scenarios are future energy

prices. These prices are taken from forecasts by DOE’s Energy Information Administration (EIA)1

and the Gas Research Institute (GRI).2

Probability analysis considers the probabilities within a range of values. For quantities with

variability (e.g., electricity prices in different households), surveys can be used to generate a

frequency distribution of numerical values (e.g., the number of households with electricity prices at

particular levels) to estimate the probability of each value. For quantities with uncertainty, statistical

or subjective measures such as a triangular probability distribution can be used to provide

probabilities (e.g., manufacturing cost to improve energy efficiency to some level may be estimated

to be $10 +/- $3). 

The advantage is that the probability approach provides the greatest information about the

outcome of the calculations, that is, the probability that the outcome will be in any particular range

is provided. The major disadvantage of the probability approach is that it requires more information,
E-1.3

namely information about the shapes and magnitudes of the variability and uncertainty of each

quantity.

Scenario and probability analysis provide some indication of the robustness of the policy given

the uncertainties and variability. A policy is robust when the impacts are acceptable over a wide

range of possible conditions.
E-1.4

REFERENCES

1. U.S. Department of Energy-Energy Information Administration, Annual Energy Outlook 1999

with Projections to 2020, December, 1998. Washington, DC. Report No. DOE/EIA0383(99). <http://www.eia.doe.gov/oiaf/aeo99/homepage.html>

2. Gas Research Institute, 1998 Edition of the GRI Baseline Projection, 1998. Arlington, VA.