Journal of Materials Science

, Volume 45, Issue 12, pp 3228–3241 | Cite as

Statistical quantification of the microstructural homogeneity of size and orientation distributions



Methodologies to quantify the microstructural homogeneity, or uniformity, have been developed based on the proposed statistical homogeneity theory. Two kinds of homogeneities are considered, for the size and orientation distributions, respectively. In the case of size distribution, the homogeneity is quantified using two parameters, H 0.1 and H 0.2, which are defined as the probabilities falling into the ranges of μ ± 0.1μ and μ ± 0.2μ, respectively, where μ is the mean size. Whereas in the case of orientation distribution, three parameters are used to quantify the homogeneity: H R, the mean resultant length that is a simple measure of the angular data concentration, and H 0.1 and H 0.2, which are the probabilities in particular angular ranges of the circular or spherical data. These homogeneity quantities are formularized using the common statistical models, and typical examples are demonstrated.


Probability Density Function Orientation Distribution Directional Data Microstructural Homogeneity Homogeneity Degree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols

a, b, c

Grain axes

COV or σ/μ

Coefficient of variation


Coefficient of variation of the mean near-neighbor distance

\( \left( {\overline{C} ,\,\overline{S} } \right) \)

Coordinates of mean resultant vector

D, D0.1 and D0.2

Dispersion quantities


Probability density function


Grain homogeneity parameter

H, H0.1 and H0.2

Homogeneity quantities


Directional homogeneity quantity


Dimensionless homogeneity parameter






Total number, or measurement number


Fitting parameter

\( \overline{R} \)

Mean resultant length


Correlation coefficient


Sample standard deviation


Size uniformity


Spatial uniformity

\( \overline{V} \)

Mean volume

\( \overline{x} \)

Sample mean

(x, y)

Mass center

\( (x_{\text{G}} ,y_{\text{G}} ) \)

Mass gravity center

\( (x_{\text{S}} ,y_{\text{S}} ) \)

Microstructural center

α and β

Fitting parameters




Fitting parameter


Population mean


Angle, or colatitude


Population mean resultant length


Population standard deviation

\( \sigma_{\text{ga}} \)

Standard deviation of the grain areas



The author thanks three reviewers for their in-depth critical comments and constructive suggestions to improve this article.


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Microscopy and Imaging CenterTexas A&M UniversityCollege StationUSA

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