Annex 3

Glossary

ANNEX 3

GLOSSARY

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CO-CHAIRS, EDITORS AND EXPERTS
Co-chairs of the Expert Meeting on Cross-sectoral Methodologies f or
Uncertainty Estimation and Inventory Quality
Taka Hiraishi (Japan) and Buruhani Nyenzi (Tanzania)

R E VI E W E D I T O R
Richard Odingo (Kenya)

AUTHORS
Milos Tichy (Czech Republic) and Simon Bentley (Australia)

R E VI E WE R S
Roberto Acosta (UNFCCC Secretariat), Simon Eggleston (UK), Ian Galbally (Australia), Katarina Mareckova(Slovak Republic), Thomas Martinsen (IPCC/OECD), Jos Olivier (Netherlands), Jim Penman (UK), and Kristin
Rypdal (Norway)

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Contents
ANNEX 3 GLOSSARY
INDEX TO THE GLOSSARY ...................................................................................................................A3.4
A3.1 INTRODUCTION ............................................................................................................................A3.6
A3.1.1 Selection of entries..................................................................................................................A3.6
A3.1.2 Formulation of entries.............................................................................................................A3.6
A3.2 GLOSSARY .....................................................................................................................................A3.7
REFERENCES..........................................................................................................................................A3.21

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Index

G

A
Accuracy

A3.7

Good Practice

Activity data

A3.7

I

Arithmetic mean

A3.7

IndependenceAuto-correlation

A3.7

K

Auto-covariance

A3.8

Key source category

A3.12

Kurtosis

A3.12

B

A3.11

A3.12

Bias

A3.8

L

Bootstrap technique

A3.8

Latin hypercube samplingA3.12

Law of large numbers

A3.12

C
Central limit theorem

A3.8

Linear model

A3.12

Coefficient of variation

A3.8

Linear regression

A3.12

Comparability

A3.8

Lognormal distribution

A3.13

Completeness

A3.8

M

Confidence

A3.9

Mean

A3.13

Confidence interval

A3.9

MedianA3.13

Consistency

A3.9

Mode

A3.13

Correlation

A3.9

Model

A3.14

Correlation coefficient

A3.9

Moments (of random variable)

A3.14

Covariance

A3.10

Monte Carlo method

A3.14

Cumulative distribution function
See Distribution function
D
Decision tree

A3.10

Distribution function

A3.10

E

N
Non-linear model

A3.14

Normal distribution

A3.14

P
Parameters of population
PDF

A3.14

See Probability density function

Elasticity

A3.10

Emission Factor

A3.10

Error

A3.10

Estimation

A3.10

Estimator

A3.11

Population standard deviation
See Standard deviation

Expectation

A3.11

Precision

A3.15

Expected value

A3.11

Probability

A3.15

Expert Judgement

A3.11

Probability density function – PDF

A3.15

Extreme value

A3.11

Probability distribution

A3.16

A3.4

Percentile

A3.15

Population

A3.15

Population percentile

See Percentile

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Propagation of uncertainties

Glossary

A3.16

Q

Statistics

A3.18

Systematic and random errors

A3.18
A3.18

Quality Assurance (QA)

A3.16

Systematic error

Quality Control (QC)

A3.16

T

R

Time series

A3.18

Random error

A3.16

Transparency

A3.18

Random variable

A3.16

Trend

A3.18

Residual

A3.17

Triangular distribution

A3.19

S
SampleU
A3.17

Unbiased estimator

A3.19

Uncertainty

A3.19

See Percentile

Uncertainty analysis

A3.19

Sensitivity

A3.17

Uniform distribution

A3.19

Sensitivity analysis

A3.17

V

Sigma interval

A3.17

Validation

A3.20

Simple random sample

A3.17

Variability

A3.20

Skewness

A3.17

Variance

A3.20

Standard deviation

A3.18

Variance of sample mean

A3.20

Standard error of the mean

A3.18

Verification

A3.20

Statistic

A3.18

Sample mode
Sample percentile

See Mode

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ANNEX 3 GLOSSARY
A3.1

INTRODUCTION

The Glossary provides a convenient reference for inventory compilers and policy makers covering general
statistical terms and terms that have a particular meaning in the context of emission inventories.

A3 .1 .1

Selectio n o f entries

The main purposes for the selection of terms and the formulation of entries were to:•Distinguish between terms that have different meanings when used in the context of greenhouse gas
inventory compilation and when used in a technical, statistical, or mathematical sense – e.g. the term
‘consistency’;•

Provide unified notation for the basic terms (mostly statistical) considered fundamental to the practicalreporting of inventories;•

Define other terms that assist in the understanding and development of good practice guidelines for
uncertainties in national inventories.

A3 .1 .2

Fo rmula tio n o f entries

The Glossary takes a pragmatic approach and provides one or more of the following types of definitions for eachentry. First, any definition developed specially for inventory applications is marked ‘Inventory definition.’ Insome cases, examples are used to illustrate specific meanings for inventory preparation. The second type is‘statistical definition,’ that are used to explain the statistical or common mathematical definition for a certain
term. Again, in some cases, examples are provided to clarify the application of these meanings for inventory use.
The final type of definitions are those that come from other sources – including pre-existing SBSTA or IPCC
definitions agreed by the Subsidiary Body for Scientific and Technological Advice (SBSTA) of the United
Nations Framework Convention on Climate Change (UNFCCC) (indicated by FCCC/SBSTA/1999/6 Add. 1), theRevised 1996 IPCC Guidelines for National Greenhouse Gas Inventories, and the International Standardization
Organization (ISO). The clauses, that fall under the following definitions: arithmetic mean, expectation,
population, probability, probability distribution, random variable, statistic and uncertainty include a definitionindicated by [7] are taken from the publication Guide to the Expression of Uncertainty in Measurement and havebeen reproduced with permission from the International Organization for Standardization (ISO). This ISO
publication can be obtained from any member body or directly from the Central Secretariat, ISO, Case Postale56, 1211 Geneva 20, Switzerland. Copyright remains with the ISO.
The definitions provided in this Glossary are not rigorous in a full mathematical or statistical sense. Most of thestatistical definitions given here lie within the context of ‘classical’ frequency-based statistical inference,
although it is acknowledged that this is not the only theory of statistical inference. As with any reference manual,compromises have been made between understandability, clarity, exactness, and brevity. To this end,
mathematical notation has been kept to a minimum.

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A3.2

Glossary

GLOSSARY

ACCURACY
Inventory definition: Accuracy is a relative measure of the exactness of an emission or removal estimate.
Estimates should be accurate in the sense that they are systematically neither over nor under trueemissions or removals, as far as can be judged, and that uncertainties are reduced as far as practicable.
Appropriate methodologies conforming to guidance on good practices should be used to promote accuracy
in inventories. (FCCC/SBSTA/1999/6 Add. 1)
Statistical definition: Accuracy is a general term which describes the degree to which an estimate of a quantity isunaffected by bias due to systematic error. It should be distinguished from precision as illustrated on Figure A3.1.
F i g u r e A 3 . 1 A c c u r a c y a n d P r e c i s i o n ( f r o m [3]* )ACTIVITY DATA
Inventory definition: Data on the magnitude of human activity resulting in emissions or removals taking
place during a given period of time. In the energy sector for example, the total amounts of fuel burned is annual
activity data for fuel combustion sources, and the total number of animals being raised, by species, is annual
activity data for methane emissions from enteric fermentation. (Revised 1996 IPCC Guidelines [9]*)ARITHMETIC MEAN
Statistical definition: The sum of the values divided by the number of values. [7]*AUTO-CORRELATION
Statistical definition: The correlation coefficient calculated for two data items in a time series.
Example: Observed animal numbers in two successive years are usually highly auto-correlated when the lifetimeof the animals significantly exceeds two years.

* See references (p A3.21).

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Glossary

AUTO-COVARIANCE
Statistical definition: The covariance calculated for two data items in a time series.

BIAS
Inventory definition: A systematic error of the observation method, whose value in most cases is unknown.
It can be introduced by using measuring equipment that is improperly calibrated, by selecting items from a wrong
population or by favouring certain elements of a population, etc.
Statistical definition: The difference between the expected value of a statistic and the parameter which it
estimates. See Unbiased estimator.
Example: Estimating the total fugitive emission from gas transport and distribution using only measurements ofleakage from high/medium pressure pipelines can lead to bias if the leakage in the lower pressure distribution
network (which is significantly more difficult to measure) is neglected.

BOOTSTRAP TECHNIQUE
Statistical definition: Bootstrap technique is a type of computationally intensive statistical methods which
typically use repeated resampling from a set of data to assess variability of parameter estimates.

CENTRAL LIMIT THEOREM
Statistical definition: A general name for the class of mathematical/statistical theorems which, verybroadly stated, says that the arithmetic mean of n independently distributed and random variables
approximates a normal distribution as n tends to infinity. This is true for underlying distributions of variables
likely to be encountered in practice, and certainly for any distributions likely to be encountered in the context ofgreenhouse gas inventories. For inventories, the theorem gives a guide to the interpretation of combined
variances of total emission (which is the sum of sectoral emissions). Also, under some conditions, the central
limit theorem can justify the approximation that the total emissions from a bottom-up inventory has a normal
distribution.

COEFFICIENT OF VARIATION
Statistical definition: The coefficient of variation, vx is the ratio of the population standard deviation, σx ,and mean, µx, where vx = σx/µx. It also frequently refers to the sample coefficient of variation, which is the ratioof the sample standard deviation and sample mean.1

COMPARABILITY
Inventory definition: Comparability means that estimates of emissions and removals reported by Parties
in inventories should be comparable among Parties. For this purpose, Parties should use the methodologies
and formats agreed by the Conference of the Parties (COP) for estimating and reporting inventories. The
allocation of different source/sink categories should follow the split of the Revised 1996 IPCC Guidelines for
National Greenhouse Gas Inventories, at the level of its summary and sectoral tables.

COMPLETENESS
Inventory definition: Completeness means that an inventory covers all sources and sinks as well as allgases included in the Revised 1996 IPCC Guidelines for National Greenhouse Gas Inventories in addition toother existing relevant source/sink categories which are specific to individual Parties (and therefore may

1 ‘Coefficient of variation’ is the term, which is frequently replaced by ‘error’ in a statement like ‘error is 5%’.

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Glossary

not be included in the IPCC Guidelines). Completeness also means full geographic coverage of sources andsinks of a Party2.

CONFIDENCE
Inventory definition: The term ‘confidence’ is used to represent trust in a measurement or estimate.
Having confidence in inventory estimates does not make those estimates more accurate or precise; however, it
will eventually help to establish a consensus regarding whether the data can be applied to solve a problem [6].*This usage of confidence differs substantially from the statistical usage in the term confidence interval.

CONFIDENCE INTERVAL
Statistical definition: A confidence interval is the range in which it is believed that the true value of aquantity lies. The level of belief is expressed by the probability, whose value is related to the size of theinterval. It is one of the ways in which uncertainty can be expressed (see Estimation).
In practice a confidence interval is defined by a probability value, say 95%, and confidence limits on either sideof the mean value x. In this case the confidence limits L1 and L2 would be calculated from the probability density
function such that there was a 95% chance of the true value of the quantity being estimated by x lying between L1and L2. Commonly L1 and L2 are the 2.5 percentile and 97.5 percentile respectively.
Example: ‘An emission is between 90 and 100 kt with a probability of 95%.’ Such a statement can be provided
when the confidence interval is calculated (the numerical values in this example are arbitrarily chosen).

CONSISTENCY
Inventory definition: Consistency means that an inventory should be internally consistent in all its
elements over a period of years. An inventory is consistent if the same methodologies are used for the baseand all subsequent years and if consistent data sets are used to estimate emissions or removals fromsources or sinks. Under certain circumstances referred to in paragraphs 10 and 11 of FCCC/SBSTA/1999/6
Add.1, an inventory using different methodologies for different years can be considered to be consistent if it has
been recalculated in a transparent manner taking into account any good practices.
Statistical definition: A statistical estimator for a parameter is said to be consistent, if the estimator tends towards
the parameter as the size of the sample used for the estimator increases – i.e. precision is improved by anincreasing number of observations.

CORRELATION
Statistical definition: Mutual dependence between two quantities. See Correlation coefficient.

CORRELATION COEFFICIENT
Statistical definition: A number lying between –1 and +1 which measures the mutual dependence between
two variables which are observed together. A value of +1 means that the variables have a perfect directstraight line relation; a value of –1 means that there is a perfect inverse straight line relation; and a value of 0means that there is no straight line relation. It is defined as the covariance of the two variables divided by the
product of their standard deviations.

2 According to the instruments of ratification, this is the acceptance, approval or accession of the Convention by a given

Party.
* See references (p A3.21).

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Glossary

COVARIANCE
Statistical definition: The covariance between two variables is a measure of the mutual dependence
between two variables.
The sample covariance of paired sample of random variables X and Y is calculated using the following formula:
1 n
2= ∑ ( x i − x )( y i − y ) where xi, yi, i = 1,…,n are items in the sample and x and y are sample means.
s xy
n i

CUMULATIVE DISTRIBUTION FUNCTION
See Distribution function.

DECISION TREEInventory definition: A decision tree is a flow chart describing the specific ordered steps which need to befollowed to develop an inventory or an inventory component in accordance with the principles of goodpractice.

DISTRIBUTION FUNCTION
Statistical definition: A distribution function or cumulative distribution function F(x) for a randomvariable X specifies the probability Pr(X ≤ x) that X is less than or equal to x.

ELASTICITY
Statistical definition: Elasticity (or normalised sensitivity) is a measure of how responsive one quantity is
to a change in another related quantity. The elasticity of a quantity Y that is affected by changes in anotherquantity X is defined as the percentage change in Y divided by the percentage change in X which caused thechange in Y.

EMISSION FACTOR
Inventory definition: A coefficient that relates the activity data to the amount of chemical compound
which is the source of later emissioms. Emission factors are often based on a sample of measurement data,
averaged to develop a representative rate of emission for a given activity level under a given set of operating
conditions (Revised 1996 IPCC Guidelines [9]*).

ERROR
Statistical definition: In statistical usage, the term ‘error’ is a general term referring to the difference
between an observed (measured) value of a quantity and its ‘true’ (but usually unknown) value and does
not carry the pejorative sense of a mistake or blunder.

ESTIMATION
Statistical definition: Estimation is the assessment of the value of a quantity or its uncertainty through theassignment of numerical observation values in an estimation formula, or estimator. The results of an
estimation can be expressed as follows:* See references (p A3.21).

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Glossary•a point estimation which provide a number which can be used as an approximation to a parameter (such as
the sample standard deviation which estimates the population standard deviation), or•

an interval estimate specifying a confidence level.

Example: A statement like ‘The total emission is estimated to be 100 kt and its coefficient of variation is 5%’ isbased upon point estimates of the sample mean and standard deviation, whereas a statement such as ‘The total
emission lies between 90 and 110 kt with probability 95%’ expresses the results of estimation as a confidence
interval.

ESTIMATOR
Statistical definition: An estimator is a formula specifying how to calculate a sample estimate value of a
population parameter from the sampled data. For example, emission factors are often estimated as the
sample means of sets of measurements. There can be more than one estimator for a population parameter,
and each estimator in general has its own sampling properties with consistency and unbiasedness being
among the most important.

Examples of point estimators include the arithmetic mean x, which is a commonly used estimator for the expectedvalue (mean), and the sample variance s2, which is a commonly used estimator for the variance.

EXPECTATION
Statistical definition: 1. For a discrete random variable X taking the values xi with the probabilities pi, theexpectation is µ = E(X) = Σ pi xi; and 2. For a continuous random variable X having the probability density
function f(x), the expectation, if it exists, is µ = E(X) = ∫ x f(x) dx, the integral being extended over theinterval(s) of variation of X. [7]*EXPECTED VALUE
Statistical definition: See Mean.

EXPERT JUDGEMENT
Inventory definition: A carefully considered, well-documented qualitative or quantitative judgement madein the absence of unequivocal observational evidence by a person or persons who have a demonstrable
expertise in the given field.

EXTREME VALUE
Statistical definition: The extreme values of a sample are the maximum and minimum values of thesample. The statistical theory of extreme values is concerned with estimating the distributions of these
extreme values for large values of n.

GOOD PRACTICE
Inventory definition: Good Practice is a set of procedures intended to ensure that greenhouse gas
inventories are accurate in the sense that they are systematically neither over nor underestimates so far as
can be judged, and that uncertainties are reduced so far as possible.

Good Practice covers choice of estimation methods appropriate to national circumstances, quality assurance andquality control at the national level, quantification of uncertainties and data archiving and reporting to promotetransparency.
* See references (p A3.21).

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Glossary

INDEPENDENCE
Statistical definition: Two random variables are independent if there is a complete absence of association
between how their sample values vary. The most commonly used measure of the lack of independence
between two random variables is the correlation coefficient.

KEY SOURCE CATEGORY
Inventory definition: A key source category is one that is prioritised within the national inventory systembecause its estimate has a significant influence on a country’s total inventory of direct greenhouse gases interms of the absolute level of emissions, the trend in emissions, or both. (See Chapter 7, Methodological
Choice and Recalculation.)

KURTOSIS
Statistical definition: Kurtosis is a measure of the flatness of a PDF. It is a simple function of two moments.

Kurtosis is given by: γ =

µ4
µ 22

=

µ4
σ2

where µ2 and µ4 are the second and fourth population central moments. For

the normal distribution, kurtosis equals 3. The sample kurtosis has a corresponding definition, with sample
moments replacing population moments; it is very sensitive to ‘outlier’ points.

LATIN HYPERCUBE SAMPLING
Statistical definition: Latin hypercube sampling is a technique of selecting values for inputs to computer
realisation runs of a model by stratifying the range of each of the model inputs, and ensuring that input
values across the whole of each model input range are selected.

LAW OF LARGE NUMBERS
Statistical definition: A mathematical theorem which formalises the generally known wisdom that an
average becomes a better approximation to the mean as the number of observations is increased.

LINEAR MODEL
Statistical definition: A variable y is said to be linearly related to (or a linear function of) variables x1, x2,… if y can be expressed by the formula y= b0 + b1x1 + b2x2 + … where the b terms are constant numbers.
Whether a function is considered to be linear or not can vary depending upon the context in which it isapplied.

Example: An emission E is usually expressed as the product of an emission factor F and an activity level A. In thecase where F is a fixed constant and E only varies when A does, E is linearly related to A. However, when both F
and A are considered to be variables (such as when applying the error propagation equation to estimate thevariance of E as a function of the variances and covariance of A and F) E is not a linear function of F and A.

LINEAR REGRESSION
Statistical definition: Linear regression provides a way of fitting a straight line to a set of observed data
points, taking into account the effects of observational variability.

Example: If emissions observations are plotted against corresponding activity levels, the slope of the line fitted
by a linear regression provides an estimate of the appropriate emission factor. The technique can also be used for
estimating a straight-line trend for a quantity which varies over time.

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Glossary

LOGNORMAL DISTRIBUTION
Statistical definition: The lognormal distribution is an asymmetric distribution, which starts from zero,
rises to a maximum and then tails off more slowly to infinity. It is related to the normal distribution: X has alognormal distribution if ln (X) has a normal distribution.

The PDF of the lognormal distribution is given by:f ( x) =1

σ l x 2π−(ln x − µ l ) 2

e

2σ l2

, for 0 ≤ x ≤ ∞The parameters required to specify the function are: µ l the mean of the natural log transform of the data; andσl 2 the variance of the natural log transform of the data. The data and information that the inventory compiler
can use to determine the input parameters are: mean = µ; variance = σ 2; and the relationships:µ2µ l = ln

(σ 2 + µ 2 )

and

σl =

σ 2

ln  2 + 1
µ




MEAN
Statistical definition: The mean, population mean, expectation or expected value is, broadly speaking, ameasure of a central value around which values sampled from a probability distribution tend to lie. The
sample mean or arithmetic average is an estimator for the mean. It is an unbiased and consistent estimatorof the population mean (expected value) and itself is a random variable with its own variance value. The
sample mean is the sum of values divided by the number of values:

x =

1 n
∑ x i (xi , i = 1,..,n are items of a sample)
n i

MEDIAN
Statistical definition: The median or population median is a value which divides the integral of a PDF into
two halves. For symmetric PDFs, it equals the mean. The median is the 50th population percentile.
The sample median is an estimator of the population median. It is the value that divides an ordered sample
into two equal halves. If there are 2n + 1 observations, the median is taken as the (n + 1)th member of the orderedsample. If there are 2n, it is taken as being halfway between the nth and (n + 1)th.

MODE
Statistical definition: Distributions can have one or more modes. In practice, we usually encounter distributions
with only one mode. In this case, the mode or population mode of a PDF is the measure of a central value
around which values sampled from a probability distribution tend to lie and is broadly speaking, the value
which has the highest probability of occurrence.
The sample mode is an estimator for the population mode calculated by subdividing the sample range into
equal subclasses, counting how many observations fall into each class and selecting the centre point of theclass (or classes) with the greatest number of observations.

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Glossary

MODEL
Statistical definition: A model is a quantitatively-based abstraction of a real-world situation which may
simplify or neglect certain features to better focus on its more important elements.

Example: the relationship that emissions equal an emission factor times an activity level is a simple model. The
term ‘model’ is also often used in the sense of a computer software realisation of a model abstraction whichcalculates a set of output values for a given set of input values – e.g. numerical global climate models.

MOMENTS (OF RANDOM VARIABLE)
Statistical definition: A population moment of a variable X about a given constant α is defined as the
expected value of the random variable (X – α)k , i.e. E(X – α)k. For the case where α equals the population
mean, µ, the moment E(X – µ)k is termed the kth central moment of X. They are important because statistical
calculations are usually based upon the moments of the PDF rather than the PDF itself. The most commonly
encountered moments are the mean and the variance.

The sample mean is the first moment around zero and the variance is the second central moment. Skewness andkurtosis are two frequently used functions of central moments which characterise the shape of the PDF.
The sample moments are estimators of population moments. The sample moment of kth order is thearithmetic mean of the kth power of the difference between the observed values and their average.

MONTE CARLO METHOD
Inventory definition: The principle of Monte Carlo analysis is to perform the inventory calculation manytimes by electronic computer, each time with the uncertain emission factors or model parameters andactivity data chosen randomly (by the computer) within the distribution on uncertainties specified initiallyby the user. Uncertainties in emission factors and/or activity data are often large and may not have normal
distributions. In this case the conventional statistical rules for combining uncertainties become very approximate.
Monte Carlo analysis can deal with this situation by generating an uncertainty distribution for the inventory
estimate that is consistent with the input uncertainty distributions on the emission factors, model parameters andactivity data.

NON-LINEAR MODEL
Statistical definition: A model is non-linear if the relationship between its inputs and its outputs is nonlinear (see Linear model).

NORMAL DISTRIBUTION
Statistical definition: The normal (or Gaussian) distribution has the PDF given in the following equationand is defined by two parameters (the mean µ and the standard deviation σ):

f (x) =

1

σ 2π

−

e

( x − µ)2
2σ 2

, for - ∞ ≤ x ≤ ∞PARAMETERS OF POPULATION
Statistical definition: Parameters of the probability distribution that characterise the population. The
most commonly used population parameters are the moments – e.g. the mean and the standard deviation
for the normal distribution. A quantity used in describing the probability distribution of a random variable. [7]** See references (p A3.21).

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Glossary

PDF
See Probability density function.

PERCENTILE
Statistical definition: The kth percentile or population percentile is a value which separates the lowest kth
part of the integral of the PDF – i.e. an integral of a PDF tail from the kth percentile towards lower
probability densities.

The kth population percentile (0 ≤ k ≤ 100) of a population with a distribution function F(x) equals to z where z
satisfies F(z) = k/100
Sample kth percentile is an approximation for the population percentile which is derived from a sample. It
is the value below which k percent of the observations lie.

POPULATION
Statistical definition: The population is the totality of items under consideration. In the case of a randomvariable, the probability distribution is considered to define the population of that variable [7]*.

Example: all conceivable experiments or events of a given type.

PRECISION
Inventory definition: Precision is the inverse of uncertainty in the sense that the more precise something is,
the less uncertain it is.

PROBABILITY
Statistical definition: A probability is a real number in the scale 0 to 1 attached to a random event. ([7]*,C.2.1) There are different ways in which probability can be interpreted. One interpretation considers aprobability as having the nature of a relative frequency (i.e. the proportion of all outcomes corresponding to an
event), whilst another interpretation regards a probability as being a measure of degree of belief. The probability
that a random event E occurs is often denoted as Pr(E). Probabilities may also be expressed in percentage terms.
Probability theory is a branch of mathematics developed from axiomatic foundations, whose results underlay
statistical inference.

PROBABILITY DENSITY FUNCTION – PDF
Statistical definition: A probability density function (PDF) is a mathematical function which characterises
the probability behaviour of population. It is a function f(x) which specifies the relative likelihood of acontinuous random variable X taking a value near x, and is defined as the probability that X takes a value between
x and x+dx, divided by dx where dx is an infinitesimally small number. Most PDFs require one or moreparameters to specify them fully.

The probability that a continuous random variable X lies in between the values a and b is given by the interval of
the PDF, f(x), over the range between a and b.
a

Pr(a ≤ x < b) = ∫ f ( x)dxb

The PDF is the derivative (when it exists) of the distribution function:* See references (p A3.21).

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Annex 3f ( x) =GlossarydF ( x )dx

In practical situations, the PDF used is chosen from a relatively small number of standard PDFs and the main
statistical task is to estimate its parameters. Thus, for inventory applications, a knowledge of which PDF has beenused is a necessary item in the documentation of an uncertainty assessment.

PROBABILITY DISTRIBUTION
Statistical definition: A function giving the probability that a random variable takes any given value or
belongs to a given set of values. The probability on the whole set of values of the random variable equals 1.
[7]*

PROPAGATION OF UNCERTAINTIESStatistical definition: The rules for propagation of uncertainties specify how to algebraically combine thequantitative measures of uncertainty associated with the input values to the mathematical formulae used
in inventory compilation, so as to obtain corresponding measures of uncertainty for the output values. See
Chapter 6, Quantifying Uncertainties in Practice, and Annex 1, Conceptual Basis for Uncertainty
Analysis.

QUALITY ASSURANCE (QA)
Inventory definition: Quality Assurance (QA) activities include a planned system of review procedures
conducted by personnel not directly involved in the inventory compilation/development process to verify
that data quality objectives were met, ensure that the inventory represents the best possible estimate ofemissions and sinks given the current state of scientific knowledge and data available, and support theeffectiveness of the quality control (QC) programme.

QUALITY CONTROL (QC)
Inventory definition: Quality Control (QC) is a system of routine technical activities, to measure andcontrol the quality of the inventory as it is being developed. The QC system is designed to:(i)Provide routine and consistent checks to ensure data integrity, correctness, andcompleteness;(ii)

Identify and address errors and omissions;(iii)

Document and archive inventory material and record all QC activities.

QC activities include general methods such as accuracy checks on data acquisition and calculations andthe use of approved standardised procedures for emission calculations, measurements, estimating
uncertainties, archiving information and reporting. Higher tier QC activities include technical reviews ofsource categories, activity and emission factor data, and methods.

RANDOM ERROR
See Systematic and random errors.

RANDOM VARIABLE
Statistical definition: A variable that may take any of the values of a specified set of values and with which* See references (p A3.21).

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Glossaryis associated a probability distribution. A random variable which may take only isolated values is said tobe ‘discrete.’ A random variable that may take any value within a finite or infinite interval is said to be‘continuous’. [7]*RESIDUAL
Statistical meaning: For an observed value whose behaviour is modelled by a statistical model, theresidual is the difference between the observed value and the value predicted by the model, e.g. by linear
regression. The residual is thus the component of an observation that cannot be explained by the model.

SAMPLE
Statistical meaning: A sample is a finite set of observations drawn from a population.

SENSITIVITY
Statistical definition: A sensitivity is a measurement of how responsive one quantity is to a change inanother related quantity. The sensitivity of a quantity Y that is affected by changes in another quantity X, isdefined as the change in Y divided by the change in X that caused the changes in Y.

SENSITIVITY ANALYSIS
Statistical definition: Sensitivity analysis is a study of a model algorithm to determine how sensitive (orstable) it is to variations of its input data or underlying assumptions. It is performed by varying input values
or model equations and observing how the model output varies correspondingly. The aim of such a sensitivity
analysis can include:•observing the range of output values corresponding to input variables lying within ‘reasonable’ ranges; and•calculating finite difference approximations for elasticities and sensitivities as required by somemethodologies for studying error propagation within a system.

SIGMA INTERVAL
Statistical definition: A c-sigma interval is a symmetric confidence interval centred on the mean andextending c times the standard deviation on either side.

SIMPLE RANDOM SAMPLE
Statistical definition: A sample of n items chosen from a population such that every possible sample has
the same probability of being chosen.

SKEWNESS
Statistical definition: Skewness is a measure of asymmetry of a PDF. It is a simple function of two
moments of the PDF, given by: γ =µ3µ 23 2=µ3

σ3where µ2, µ3, and σ, are central moments. Symmetric

distributions have γ = 0. The same name is frequently used for sample skewness, in which case both population
moments are replaced by sample moments.

* See references (p A3.21).

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Glossary

STANDARD DEVIATION
Statistical definition: The population standard deviation is the positive square root of the variance. It isestimated by the sample standard deviation that is the positive square root of the sample variance.

STANDARD ERROR OF THE MEAN
Statistical definition: A term often used to signify the sample standard deviation of the mean.

STATISTIC
Statistical definition: A statistic is a function of the sample random variables. [7]*STATISTICS
Statistical definition: Statistics can refer either in a general sense to the compilation of data, frequently
about human activities, or in a more specific sense to the branch of science concerned with the systematic
numerical treatment of data derived from aggregates of items.

SYSTEMATIC AND RANDOM ERRORS
Statistical definition: Systematic error is the difference between the true, but usually unknown, value of aquantity being measured, and the mean observed value as would be estimated by the sample mean of an
infinite set of observations. The random error of an individual measurement is the difference between an
individual measurement and the above limiting value of the sample mean.

SYSTEMATIC ERROR
Statistical definition: See Systematic and random errors.

TIME SERIES
Statistical definition: A time series is series of values which are affected by random processes and which
are observed at successive (but usually equidistant) time points.

TRANSPARENCY
Inventory definition: Transparency means that the assumptions and methodologies used for an inventory
should be clearly explained to facilitate replication and assessment of the inventory by users of thereported information. The transparency of inventories is fundamental to the success of the process for thecommunication and consideration of information.

TREND
Inventory definition: The trend of a quantity measures its relative trend over a time period, with a positive
trend value indicating growth in the quantity, and a negative value indicating a decrease. It is defined as
the ratio of the change in the quantity over the time period, divided by the initial value of the quantity,
and is usually expressed either as a percentage or a fraction.

* See references (p A3.21).

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Glossary

TRIANGULAR DISTRIBUTION
Statistical definition: An asymmetric triangular distribution function has a PDF
f(x) = 2(x – a) / {(b – a) (m – a)} when a ≤ x ≤ m and a < m ≤ b= 2(b – x) / {(b – a) (b – m)} when m ≤ x ≤ b and a ≤ m < b= 0 elsewhere,
where the parameters which specify the distribution are the minimum value a, the maximum value b, andthe most likely position (i.e. mode) m, subject to a ≤ m ≤ b.

UNBIASED ESTIMATOR
Statistical definition: An unbiased estimator is a statistic whose expected value equals the value of theparameter being estimated. Note that this term has a specific statistical meaning and that an estimate of aquantity calculated from an unbiased estimator may lack bias in the statistical sense, but may be biased in the
more general sense of the word if the sample has been affected by unknown systematic error. Thus, in statisticalusage, a biased estimator can be understood as a deficiency in the statistical evaluation of the collected data, andnot in the data themselves or in the method of their measurement or collection. For example, the arithmetic mean
(average) x is an unbiased estimator of the expected value (mean).

UNCERTAINTY
Statistical definition: An uncertainty is a parameter, associated with the result of measurement thatcharacterises the dispersion of the values that could be reasonably attributed to the measured quantity.
[7]* (e.g. the sample variance or coefficient of variation)

Inventory definition: A general and imprecise term which refers to the lack of certainty (in inventory
components) resulting from any causal factor such as unidentified sources and sinks, lack of transparency etc.

UNCERTAINTY ANALYSIS
Statistical definition: An uncertainty analysis of a model aims to provide quantitative measures of theuncertainty of output values caused by uncertainties in the model itself and in its input values, and to
examine the relative importance of these factors.

UNIFORM DISTRIBUTION
Statistical definition: A random variable with a uniform or rectangular distribution is confined to lie
within a range over which all values are equally probable. If the upper and lower limits of the range are aand b respectively, the PDF is a flat function from a to b (the two parameters defining the PDF).

The PDF of a uniform distribution is given by:
 1
f ( x) =  b − a 0

for a ≤ x ≤ belsewhere

where

µ =

a+b
2

is the mean and* See references (p A3.21).

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σ2 =Glossary(b − a) 2
12

is the variance.

VALIDATION
Inventory definition: Validation is the establishment of sound approach and foundation. In the context ofemission inventories, validation involves checking to ensure that the inventory has been compiled correctly inline with reporting instructions and guidelines. It checks the internal consistency of the inventory. The legal use
of validation is to give an official confirmation or approval of an act or product. [6]*VARIABILITY
Statistical definition: This refers to observed differences attributable to true heterogeneity or diversity ina population. Variability derives from processes which are either inherently random or whose nature and effects
are influential but unknown. Variability is not usually reducible by further measurement or study, but can becharacterised by quantities such as the sample variance. [6]*VARIANCE
Statistical definition: The variance or population variance is a parameter of a PDF, which expresses thevariability of the population. It is the second central moment of a random variable. The sample variance isdefined as a measure of dispersion, which is the sum of the squared deviations of observations from their
1 n
2average, divided by one less than the number of observations. [7]* s 2 =∑ ( x i − x)n −1 i

VARIANCE OF SAMPLE MEAN
Statistical definition: The mean of a sample taken from a population is itself a random variable with its
own characteristic behaviour and its own variance. For such sample means, the appropriate estimate of
the variance is not the sample variance, which estimates the variability associated with a single simple
value, but a lower value, equal to the sample variance divided by the sample size.

VERIFICATION
Inventory definition: Verification refers to the collection of activities and procedures that can be followed
during the planning and development, or after completion of an inventory that can help to establish its
reliability for the intended applications of that inventory. Typically, methods external to the inventory areused to check the truth of the inventory, including comparisons with estimates made by other bodies or withemission and uptake measurements determined from atmospheric concentrations or concentration gradients ofthese gases. [6]** See references (p A3.21).

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Glossary

REFERENCES[1] Brandt S. (1970). Computational Methods in Statistics and Data Analysis. North-Holland, Amsterdam,
Netherlands.
[2] Wonnacott R. and Wonnacott T. (1990). Introductory Statistics for Business and Economics. John Wiley,
New York, USA.
[3] Zijp W.L. (1987). Treatment of measurement of Uncertainties. ECN-194, Petten, Netherlands.
[4] Martin B.R. (1971). Statistics for Physicists. Academic Press, London, UK.
[5] Kendall M. G, Stuart A. (1966). The Advanced Theory of Statistics. Charles Grifiin & Co.Ltd, London, UK.
[6] Intergovernmental Panel on Climate Change (IPCC) (1998). Meeting Report: Managing Uncertainty in
National Greenhouse Gas Inventories, Report of the meeting held in Paris 13-15, October 1998,
IPCC/OECD/IEA Programme on National Greenhouse Gas Inventories.
[7] International Organization for Standardization (ISO) (1993). Guide to the expression of uncertainty inmeasurement. ISBN 92-67-10188-9, ISO, Geneva, Switzerland.
[8] Kotz, S. and Johnson, N.L. (1988). Encyclopaedia of Statistical Sciences. 9 volumes, John Wiley & Sons,
New York, USA.
[9] Intergovernmental Panel on Climate Change (IPCC) (1997). Revised 1996 IPCC Guidelines for National
Greenhouse Gas Inventories: Volume 1 Reporting Instructions. J.T. Houghton et al., IPCC/OECD/IEA,
Paris, France.

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