ISO 9276-1:2025

International
Standard
ISO 9276-1
Third edition
Representation of results of particle
2025-10
size analysis —
Part 1:
Graphical representation
Représentation de données obtenues par analyse
granulométrique —
Partie 1: Représentation graphique
Reference number
© ISO 2025
All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may
be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting on
the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address below
or ISO’s member body in the country of the requester.
ISO copyright office
CP 401 • Ch. de Blandonnet 8
CH-1214 Vernier, Geneva
Phone: +41 22 749 01 11
Email: copyright@iso.org
Website: www.iso.org
Published in Switzerland
ii
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols . 5
4.1 General .5
4.2 Symbols .5
5 Particle size, measures and types . 6
5.1 General .6
5.2 Particle size, x .6
5.3 Measures and types of quantity .7
6 Graphical representation of PSDs. 7
6.1 Cumulative distribution, Q (x) .7
r
6.2 Representation in discrete classes as histogram, q̅ .9
r,i
6.3 Distribution density, q (x) .10
r
7 Graphical representation of cumulative distribution and distribution density on a
logarithmic abscissa .11
7.1 Cumulative distribution on a logarithmic abscissa .11
7.2 Distribution density on a logarithmic abscissa . 12
8 Graphical representation of non-normalized distributions .12
8.1 Graphical representation of concentration distribution . 12
8.2 Graphical representation of separated particle distributions . 13
9 Graphical representation of two-dimensional distribution .13
9.1 Graphical representation of two-dimensional PSD . 13
9.2 Graphical representation of shape and size distribution data.16
Annex A (informative) Example of graphical representation of particle size analysis results of
sieving . 19
Annex B (informative) Example of number concentration size distribution as result of aerosol
measurement .22
Annex C (informative) Example of graphical representation of two-dimensional PSD .24
Bibliography .27

iii
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out through
ISO technical committees. Each member body interested in a subject for which a technical committee
has been established has the right to be represented on that committee. International organizations,
governmental and non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely
with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are described
in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the different types
of ISO document should be noted. This document was drafted in accordance with the editorial rules of the
ISO/IEC Directives, Part 2 (see www.iso.org/directives).
ISO draws attention to the possibility that the implementation of this document may involve the use of (a)
patent(s). ISO takes no position concerning the evidence, validity or applicability of any claimed patent
rights in respect thereof. As of the date of publication of this document, ISO had not received notice of (a)
patent(s) which may be required to implement this document. However, implementers are cautioned that
this may not represent the latest information, which may be obtained from the patent database available at
www.iso.org/patents. ISO shall not be held responsible for identifying any or all such patent rights.
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and expressions
related to conformity assessment, as well as information about ISO's adherence to the World Trade
Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www.iso.org/iso/foreword.html.
This document was prepared by Technical Committee ISO/TC 24, Particle characterization including sieving,
Subcommittee SC 4, Particle characterization.
This third edition cancels and replaces the second edition (ISO 9276-1:1998), which has been technically
revised. It also incorporates the Technical Corrigendum ISO 9276-1:1998/Cor 1:2004.
The main changes are as follows:
— Clause 3 has been added;
— Formulae (9) and (10) have been added in 7.2;
— Clauses 8 and 9 have been added;
— Annex B and Annex C have been added.
A list of all parts in the ISO 9276 series can be found on the ISO website.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www.iso.org/members.html.

iv
Introduction
The characterization of a dispersed phase with respect to its particle size distribution (PSD) is a major
task, whenever the generation, emission, transport, application or consumption of particulate matter is
monitored or investigated. The spectrum of relevant materials is very broad and "particle" can refer to
grains, flakes, fibres, droplets, bubbles, micelles or pores. Within a particulate material, particles typically
vary with regard to their size. The distribution of this measurand is characterized by the relative quantity of
particles belonging to a specific size class.
There are a wide range of methods for the size analysis of particulate matter. For many reasons, the results
will, in general, not be the same. One of the main problems still encountered with most methods in use is
their unknown absolute accuracy. Apart from this, there are two principal origins of the differences among
measured PSDs.
Firstly, the measurand "particle size" is an ambiguous property being specified by the measurement
principle of the instrumentation and the way of signal analysis. Both specifications define the type of
particles which are intrinsically measured: agglomerates or their constituent particles; Pickering emulsion
droplets or the nanoparticles at their surface; complete core-shell particles or the core of the particles only.
Therefore, “size” can refer to a variety of physical and geometric properties.
Secondly, the distribution of size can be measured and expressed by different types (and measures) of
[1]
quantity.
A disperse system can be described by different types of PSD, which supplement each other in providing a
comprehensive view on the granulometric state of the particulate material. Moreover, each type of PSD can
be represented in various manners.
Beside numerical and mathematical representations, graphical ones are particularly popular and useful
because they allow for a quick comprehension of the main PSD features or a fast evaluation of differences
among product lots. There are various ways of plotting a PSD, which emphasize different details though
communicating the same information. This requires explanation, especially when a PSD is described by
discrete data.
A harmonized view on terms and a common understanding of how to graphically plot and interpret PSDs
support the communication between suppliers and clients and improve the comparability of measurement
data from different instruments.

v
International Standard ISO 9276-1:2025(en)
Representation of results of particle size analysis —
Part 1:
Graphical representation
1 Scope
This document specifies guidelines and instructions for the graphical representation of particle size analysis
data in histograms, distribution densities and cumulative distributions. It also establishes a standard
nomenclature to obtain the histograms, distribution densities and cumulative distributions from measured
particle size data.
This document applies to the graphical representation of particle size distributions (PSDs) of solid particles,
droplets or gas bubbles covering all size ranges.
2 Normative references
There are no normative references in this document.
3 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
ISO and IEC maintain terminology databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at https:// www .electropedia .org/
3.1
particle
minute piece of matter with defined physical boundaries
Note 1 to entry: A physical boundary can also be described as an interface.
Note 2 to entry: A particle can move as a unit.
Note 3 to entry: This general particle definition applies to nano-objects.
[SOURCE: ISO 26824:2022 3.1.1]
3.2
particle size
x
d
linear dimension of a particle (3.1) determined by a specified measurement method and under specified
measurement conditions
Note 1 to entry: Different methods of analysis are based on the measurement of different physical properties.
Independent of the particle property actually measured, the particle size is reported as a linear dimension, e.g. as the
equivalent diameter.
Note 2 to entry: Possible types of particle size values are equivalent diameters from physical measurement or
equivalent circular diameters from image analysis as well as linear dimensions, e.g. sieve size of a particle, the smallest
aperture size through which a particle will pass if presented in the most favourable orientation.
Note 3 to entry: In this document and all other parts of the ISO 9276 series, the symbol x is used to denote the particle
size. However, it is recognized that the symbol d is also used to designate these values. Therefore the symbol x may be
replaced by d.
3.3
equivalent diameter
equivalent spherical diameter
x
j
d
j
diameter of a sphere that produces a response by a given particle-sizing method, that is equivalent to the
response produced by the particle being measured
Note 1 to entry: The physical property to which the equivalent diameter refers is indicated using a suitable subscript.
Note 2 to entry: For discrete-particle-counting in light-scattering instruments, a scattering intensity equivalent
diameter is used, in extinction measurement a light extinction equivalent diameter is used.
Note 3 to entry: Other material constants like density of the particle are used for the calculation of the equivalent
diameter like Stokes diameter or sedimentation equivalent diameter. The material constants, used for the calculation,
should be reported additionally. In some instants, refractive index and the spectral function should be reported as well.
Note 4 to entry: For inertial instruments, the aerodynamic diameter is used. Aerodynamic diameter is the diameter of
−3
a sphere of density 1 000 kg m that has the same settling velocity as the irregular particle.
[SOURCE: ISO 26824:2022 3.1.10, modified — Notes 2 and 3 to entry have been expanded.]
3.4
type of quantity
r
specification of the quantity of a distribution, a cumulative or a density measure
Note 1 to entry: The type is indicated by the general subscript, r, or by the appropriate value of r as follows:
number: r = 0
length: r = 1
area: r = 2
volume or mass: r = 3
Note 2 to entry: To include frequently used non-geometrical quantities, the specification of the quantity has been
extended to the following subscripts, replacing r:
— light extinction: subscript “ext”;
— light intensity: subscript “int”.
3.5
particle size distribution
PSD
distribution of particles as a function of particle size (3.2)
Note 1 to entry: Particle size distribution may be expressed as cumulative distribution (3.6) or a distribution density
(3.8) (distribution of the fraction of material in a size class, divided by the width of that class).
[SOURCE: ISO 26824:2022, 3.1.12]

3.6
cumulative distribution
cumulative distribution function
Q (x)
r
distribution of the fraction of material smaller than or equal (undersize) given particle sizes
Note 1 to entry: If the cumulative distribution, Q (x), is calculated from histogram (3.10) data, only individual points
r
Q = Q (x ) are obtained. Each individual point of the distribution, Q (x ), defines the relative quantity of particles
r,i r i r i
smaller than or equal to x . The continuous curve is calculated by suitable interpolation algorithms. The normalized
i
cumulative distribution extends between 0 and 1, i.e. 0 and 100 %.
i i
QQ==ΔΔqx with 1 ≤≤vi ≤n.
r,iv∑∑r, r,v v
v==11v
where
r is the (subscript) type of quantity;
i is the (subscript) number of the size class with upper limit x ;
i
ν is the (subscript) number of the size class with upper limit x ;
ν
n is the total number of size classes;
Q is the relative quantity of particles in size class with upper limit x .
r,ν ν
Note 2 to entry: When plotted on a graph paper with a logarithmic abscissa, the cumulative values, Q , i.e. the ordinates
r,i
of a cumulative distribution, do not change. However, the course of the cumulative distribution curve changes but the
relative quantities smaller than a certain particle size remain the same. Therefore, the following formula holds:
Q (x) = Q (In x)
r r
Note 3 to entry: The cumulative oversize distribution is given by 1−Q (x).
r
3.7
cumulative oversize distribution
cumulative oversize distribution function
1 - Q (x)
r
distribution of the fraction of material larger (oversize) than given particle sizes
Note 1 to entry: Each individual point of the distribution defines the relative quantity of particles larger than the
particle size. Therefore, the abscissa value of each point is x , the lower size of the particle size class, i.
i-1
3.8
distribution density
distribution density function
q (x)
r
distribution of the fraction of material in a size class, divided by the width of that class
Note 1 to entry: Under the presupposition that the cumulative distribution, Q (x) (3.6), is differentiable, the continuous
r
distribution density, q (x), is obtained from
r
dQx()
r
qx()=
r
dx
Conversely, the cumulative distribution, Q (x), is obtained from the distribution density, q (x), by integration:
r r
x
i
Qx = qx dx
() ()
ri r
∫
x
min
Note 2 to entry: The more common term “density distribution” can be misunderstood in the context of sedimentation
methods, so an alternative has been adopted.
Note 3 to entry: Differential distribution is also called in statistics “density of a probability or frequency”.

3.9
distribution density on a logarithmic abscissa
distribution density function on a logarithmic abscissa
q* (x)
r
distribution density (3.8), transformed for a logarithmic abscissa
Note 1 to entry: The density values of a histogram (3.10), q* = q̅* (x , x ), can be recalculated using the following
r,i r i-1 i
formula which indicates that the corresponding areas underneath the distribution density curve remain constant. In
particular, the total area is equal to 1 or 100 %, independent of any transformation of the abscissa.
q̅*(ξ ξ ) · Δξ = q̅(x , x ) · Δx     where ξ is any function of x
r i-1, i i r i-1 i i
Thus, the following transformation can be carried out to obtain the distribution density on a logarithmic abscissa:
qxΔΔQ
qx(),xxΔ
ri,,i r ii
* ri−1 ii
q = = = =⋅qx
ri, ri, i
lnxx−ln x x
   
ii−1 i i
ln ln
   
x x
   
i−1 i−1
Note 2 to entry: Replacing the natural logarithm by the logarithm with base 10 results in a distribution density, which
differs by the factor ln 10 = 2,303,
*lg
qq=⋅x ⋅ ln10
ri, i
ri,
3.10
histogram
q̅
r,i
graphical representation of a distribution density q (x), comprising a successive series of rectangular
r
columns with the height of mean distribution density, qx() , and the width of Δx , the area of each
ri, i
represents the relative quantity ΔQ (x)
r,i
Note 1 to entry: The relative quantity ΔQ (x) relates to the mean distribution density by the following formula:
r,i
ΔQ
ΔQx(),x
ri,
ri−1 i
ΔΔQQ= xx,,=qx xx ⋅=Δ  or  qq x ,xx = =
() () ()
ri,,ri−−11ir ii ir ir i−1 i
Δx Δx
i i
Note 2 to entry: The sum of all the relative quantities, ΔQ forms the area beneath the histogram q (x), normalized to
r,i r
100 % or 1 (condition of normalization). Therefore, following formula holds:
n n
ΔΔQq=⋅ x ==1 100%
ri,,ri i
∑∑
i==11i
3.11
concentration distribution density
concentration distribution density function
distribution of the concentration of material in a size class, divided by the width of that class
Note 1 to entry: In aerosol measurement e.g. the distribution density of the particle number concentration is
represented as a function of the particle size.
Note 2 to entry: The concentration distribution density can be calculated from the distribution density function of the
particle size by multiplication with the overall sizes measured total concentration.
[SOURCE: ISO 26824:2022 3.1.17]

4 Symbols
4.1 General
In this document, the symbol x is used to denote the particle size, for instance the diameter of a sphere.
However, it is recognized that the symbol d is also widely used to designate the parameter of particle size.
Therefore, in the context of the ISO 9276 series, the symbol x may be replaced by d, if the measured values
can be represented by an equivalent diameter.
Symbols for particle size other than x or d should not be used.
Subscripts used to confer the sense of different types of particle size or identity are divided by a comma sign
in this and all other parts of the ISO 9276 series.
4.2 Symbols
AR aspect ratio, minimum Feret-diameter divided by maximum Feret-diameter
C total particle concentration in type of quantity r
r
d particle size, diameter of sphere (see 3.2)
i (subscript) number of the size class with upper limit x :  Δx = x - x
i i i i-1
n total number of size classes
r index for type of quantity (see 3.4)
q (x) distribution density by number
q (x) distribution density by length
q (x) distribution density by surface or projected area
q (x) distribution density by volume or mass
q (x) distribution density (general) (see 3.8)
r
q* (x) distribution density in a representation with a logarithmic abscissa (see 3.9)
r
*lg
q (x) distribution density in a representation with a logarithmic abscissa with base 10 (see 3.9)
r
q̅average distribution density of the class i:      q̅ = q̅ (x x )
r,i r,i, r i-1, i
q̅histogram (general) (see 3.10)
r,i
Q (x) cumulative distribution by number
Q (x) cumulative distribution by length
Q (x) cumulative distribution by surface or projected area
Q (x) cumulative distribution by volume or mass
Q (x) cumulative distribution (general) (see 3.6)
r
Q cumulative distribution of the class i:         Q  = Q (x )
r,i r,i r i
ΔQ increment of cumulative distribution with class width Δx :  ΔQ = ΔQ (x x ) = Q (x ) - Q (x )
r,i i r,i r i-1, i r i r i-1
x particle size, diameter of a sphere (see 3.2)
x size below which there are no particles
min
x size above which there are no particles
max
x upper size of a particle size class
i
x lower size of a particle size class
i-1
Δx = x - x , width of the particle size class
i i i-1
x̅  arithmetic mean size of the particle size class
x̅ geometric mean size of the particle size class
g
x minimum Feret-diameter
Fmin
x maximum Feret-diameter
Fmax
ν (integer, see subscript i)
ξ = ξ (x) transformed coordinate
5 Particle size, measures and types
5.1 General
In a graphical representation of particle size analysis data and in most graphical representations, the
independent variable particle size x is plotted on the abscissa (see Figure 1). While the dependent variable,
which in the graphical representation of particle size analysis data is a parameter that characterizes the
quantity in the specified type of quantity Q or q, is plotted on the ordinate.
5.2 Particle size, x
For the notation of particle size, see 4.1.
There is no sole definition of particle size. Different methods of analysis are based on the measurement
of different physical properties. Independent of the particle property actually measured, the particle size
is reported as a linear dimension. For instance, the particle size of an irregularly shaped particle can be
defined as the diameter of a sphere having a similar physical property, related to the measurement of the
irregularly shaped particle; this is known as the equivalent diameter (see 3.3). The physical property to
which the equivalent diameter refers may be indicated using a suitable subscript, for example:
— x  equivalent surface area diameter;
S
— x  equivalent volume diameter;
V
— x  (projection) area equivalent diameter or equivalent circle diameter from image analysis.
A
Other definitions are based on linear dimensions like the Feret diameter, the distance between two parallel
tangents on opposite sides of the image of a particle or, e.g. the sieve size of a particle, the smallest aperture size
through which a particle will pass if presented in the most favourable attitude or the geodesic length of fibres.
For deriving the equivalent diameter from the physical property, measured primarily, e.g. the settling
velocity, additional material constants must be used, e.g. particle density. It is common practice to use the
material constant of the real, irregular particle in this case.

5.3 Measures and type
...