8B-i

APPENDIX 8B. UNCERTAINTY AND VARIABILITY

 TABLE OF CONTENTS

8B.1 INTRODUCTION .................................................... 8B-1

8B.2 UNCERTAINTY ..................................................... 8B-1

8B.3 VARIABILITY ...................................................... 8B-1

8B.4 APPROACHES TO UNCERTAINTY AND VARIABILITY .................. 8B-2

8B.5 PROBABILITY ANALYSIS AND THE USE OF CRYSTAL BALL ............ 8B-3

LIST OF FIGURES

Figure 8B.5.1 Normal Probability Distribution ................................... 8B-3

Figure 8B.5.2 Triangular Probability Distribution ................................. 8B-3

Figure 8B.5.3 Uniform Probability Distribution ................................... 8B-3
8B-1

APPENDIX 8B. UNCERTAINTY AND VARIABILITY

8B.1 INTRODUCTION

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

the impact of a standard on life-cycle cost (LCC). 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 (i.e., 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 on 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. 

8B.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 piece of equipment (such as distribution transformer) is not directly recorded, but

rather estimated based on 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.

8B.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 a transformer is

run at a particular load depends on the specific circumstances and behaviors of the building

owners or utilities (e.g., square footage served, hours of operation). Variability makes specifying

an appropriate population value more difficult, since 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.,

building square footage).
8B-2

8B.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 performed that provide some indication of the extent to which the result depends

on the assumptions. For example, the LCC of a piece of equipment could be calculated for

energy rates of 2¢, 8¢, and 14¢ per kilowatt-hour (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 LCC is reduced, holding all other inputs constant. That is, the

crossover point is the energy rate at which an owner 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 tariffs in different regions), surveys can be used to generate a

frequency distribution of numerical values (e.g., the number of buildings subject to particular

tariffs) 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, i.e., it provides the probability that the outcome will be in a

particular range. 

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

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

wide range of possible conditions.

8B.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 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.
8B-3

Figure 8B.5.1 Normal

Probability

Distribution

Figure 8B.5.2 Triangular

Probability

Distribution

Figure 8B.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 is 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, 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 for normal, triangular, and

uniform are shown in Figures 8B.5.1, 8B.5.2, and 8B.5.3, respectively. 

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.