L-i

APPENDIX L. UNCERTAINTY AND VARIABILITY

TABLE OF CONTENTS

L.1 INTRODUCTION ................................................... L-1

L.2 UNCERTAINTY .................................................... L-1

L.3 VARIABILITY ..................................................... L-1

L.4 APPROACHES TO UNCERTAINTY AND VARIABILITY .................. L-2

L.5 PROBABILITY ANALYSIS AND THE USE OF CRYSTAL BALL ............ L-2

LIST OF FIGURES

Figure L.5.1 Normal Probability Distribution ................................... L-3

Figure L.5.2 Triangular Probability Distribution ................................. L-3

Figure L.5.3 Uniform Probability Distribution ................................... L-3
L-1

APPENDIX L. UNCERTAINTY AND VARIABILITY

L.1 INTRODUCTION

Analysis of an energy-efficiency standard involves calculations of impacts, for example,

the impact of a standard on consumer life-cycle cost (LCC). In order to perform the calculation,

the analyst must first: 1) specify the equation or model that will be used; 2) define the quantities in

the equation; and 3) provide numerical values for each quantity. In the simplest case, the equation

is unambiguous (contains all relevant quantities and no others), each quantity has a single

numerical value, and the calculation results in a single value. However, unambiguity and precision

are rarely the case. In almost all cases, the model and/or the numerical values for each quantity in

the model are not completely known (i.e., there is uncertainty) or the model and/or the numerical

values for each quantity in the model depend upon other conditions (i.e., there is variability).

Thorough analysis involves accounting for uncertainty and variability. While the simplest

analysis involves a single numerical value for each quantity in a calculation, arguments can arise

about what the appropriate value is for each quantity. Explicit analysis of uncertainty and

variability is intended to provide more complete information to the decision-making process. 

L.2 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. commercial air conditioner or heat pump) is

not directly recorded, but rather 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.

L.3 VARIABILITY

Variability means that different applications or situations produce different numerical

values when calculating a quantity. Specifying an exact value for a quantity may be difficult

because the value depends on something else. For example, the number of hours an air

conditioner is operated by a household depends upon the specific circumstances and behaviors of

the occupants (e.g., number of persons, personal habits about how comfortable the person wants

to be, etc.). Variability makes specifying an appropriate population value more difficult in as

much as any one value may not be representative of the entire population. Surveys can be helpful

here, and analysis of surveys can relate the variable of interest (e.g., hours of use) to other

variables that are better known or easier to forecast (e.g., persons per household).
L-2

L.4 APPROACHES TO UNCERTAINTY AND VARIABILITY

This section describes two approaches to uncertainty and variability: 

C scenario analysis, and 

C probability analysis.

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

changes one (or more) of 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

energy rates of 2, 8, and 14¢ per kWh. 

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

estimates is used; and crossover points can be identified. (An example of a crossover point is the

energy rate above which the life-cycle cost is reduced, holding all other inputs constant. That is,

the crossover point is the energy rate at which the consumer achieves savings in operating

expense that more than compensate for the increased purchase expense.) The disadvantage of

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

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

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

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

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

statistical or subjective measures 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 major disadvantage of the probability approach is that it requires more information,

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

quantity. The advantage of the probability approach is that it provides greater information about

the outcome of the calculations, that is, it provides the probability that the outcome will be in a

particular range. 

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.

L.5 PROBABILITY ANALYSIS AND THE USE OF CRYSTAL BALL

To quantify the uncertainty and variability that exist in inputs to the engineering, LCC, and

payback period (PBP) analyses, the Department used Microsoft Excel spreadsheets combined

with Crystal Ball, a commercially available add-in, to conduct probability analyses. The

probability analyses used Monte Carlo simulation and probability distributions.
L-3

Figure L.5.1 Normal

Probability

Distribution

Figure L.5.2 Triangular

Probability

Distribution

Figure L.5.3 Uniform

Probability

Distribution

Simulation refers to any analytical method meant to imitate a real-life system, especially

when other analyses are too mathematically complex or too difficult to reproduce. Without the

aid of simulation, a spreadsheet model will only reveal a single outcome, generally the most likely

or average scenario. Spreadsheet risk analysis uses both a spreadsheet model and simulation to

automatically analyze the effect of varying inputs on outputs of the modeled system. One type of

spreadsheet simulation is Monte Carlo simulation, which randomly generates values for uncertain

variables again and again to simulate a model. Monte Carlo simulation was named for Monte

Carlo, Monaco, where the primary attractions are casinos containing games of chance. Games of

chance such as roulette wheels, dice, and slot machines, exhibit random behavior. The random

behavior in games of chance is similar to how Monte Carlo simulation selects variable values at

random to simulate a model. When you roll a die, you know that either a 1, 2, 3, 4, 5, or 6 will

come up, but you do not know which for any particular roll. It's the same with the variables that

have a known range of values but an uncertain value for any particular time or event (e.g.,

equipment lifetime, discount rate, and installation cost). 

For each uncertain variable (one that has a range of possible values), possible values are

defined with a probability distribution. The type of distribution selected is based on the conditions

surrounding that variable. Probability distribution types include: 

During a simulation, multiple scenarios of a model are calculated by repeatedly sampling

values from the probability distributions for the uncertain variables and using those values for the

cell. Crystal Ball simulations can consist of as many trials (or scenarios) as desired—hundreds or

even thousands. During a single trial, Crystal Ball randomly selects a value from the defined

possibilities (the range and shape of the probability distribution) for each uncertain variable and

then recalculates the spreadsheet.