8-B-i

APPENDIX 8-B. UNCERTAINTY AND VARIABILITY IN LCC ANALYSIS FOR 

BOILERS

TABLE OF CONTENTS

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

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

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

8-B.4 APPROACHES TO UNCERTAINTY AND VARIABILITY .................................. 8-B-1

8-B.5 PROBABILITY ANALYSIS AND THE USE OF CRYSTAL BALL...................... 8-B-2

LIST OF FIGURES

Figure 8-B.5.1 Normal, Triangular, Uniform, Weibull, and Custom Probability 

Distributions............................................................................................... 8-B-3
8-B-1

APPENDIX 8-B. UNCERTAINTY AND VARIABILITY IN LCC ANALYSIS

8-B.1 INTRODUCTION

Analysis of energy conservation standards 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: 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.

8-B.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. water heater, direct heating equipment, or 

pool heater) 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.

8-B.3 VARIABILITY

Variability results when 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, water heater energy consumption 

depends upon the specific circumstances and behaviors of the occupants (e.g., number of 

persons, length and temperature of showers, etc.). Variability makes specifying an appropriate 

population value more difficult inasmuch 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).

8-B.4 APPROACHES TO UNCERTAINTY AND VARIABILITY

This section describes two approaches to uncertainty and variability: 
8-B-2

• scenario analysis, and 

• 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.

8-B.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, DOE 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.

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 
8-B-3

NORMAL TRIANGULAR UNIFORM

WEIBULL CUSTOM

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 a 1, 2, 3, 4, 5, or 6 will come up, but you do not know which for any particular roll. 

The same applies to 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:

Figure 8-B.5.1 Normal, Triangular, Uniform, Weibull, and Custom 

Probability Distributions

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.