Measurements are performed to obtain statistical characteristics of the microstructure of a material. Analysis of the errors of estimation is an independent, well-developed discipline (Ref 11). The discussion here is limited only to the most simple, basic concepts, sufficient for preliminary analysis of experimental data. All the relationships listed subsequently are valid for the data complying with normal distribution. Two reasons for such a choice are:
O This type of distribution is very common for the results of measure-ments carried out on material microstructures. In some cases, normal behavior also is observed for logarithms of measured values. In computer-aided measurements, the change from X to log X is auto-mated.
Table 4 Guidelines for adequate magnification choice for digital measurements
Feature
Austenitic steels Size of field⬃200⫻ the mean grain section area
100–300 grains in a single field of view
Dispersed particles,⬃10%
area fraction
Graphite in cast iron Mean diameter or length of precipitates equal to 5–10% of the diagonal of the field of view
⬃100 precipitates visible in the field of view
Small dispersed particles
Carbides in tool steels Mean area of precipitates equal to at least 25 pixels
Ferritic-pearlitic steels A mean from magnifications optimal for both constituents of the feature of mean size
Preservation of the spatial distribution of features analyzed Analysis and Interpretation / 183
O The statistical properties of a normal distribution are well described and relatively simple to evaluate.
The most basic operation is an estimation of the mean value, x, of a given parameter, defined by:
x⫽兺i⫽1n i
n (Eq 5)
where xiis ith individual result, and n is the number of measurements.
Mean value characterizes the population studied, even if an element characterized by such a value does not exist ( this occurs, for example, if there are two groups of values, very low and very high; the mean value in this case falls between these two). This should be taken into account when features analyzed, such as grains, consist of two families of different size. However, the analysis of such cases is beyond the scope of this Chapter.
The most commonly used measure of the scatter of analyzed values is standard deviation, s, defined as a square root of the variance:
s⫽
冑
n⫺ 11 i兺⫽1n (xi⫺ x)2 (Eq 6)Standard deviation is often used for evaluation of the confidence level:
CL⫽ ta,n⫺1ў s (Eq 7)
where t␣,n–1is the value of the t-statistics (Student) for the significance level␣, and n is 1 degree of freedom. Exact values of t-statistics can be found in statistical tables, or are easily accessible in any software for statistical evaluation, including most popular spreadsheets.
Estimating the confidence level for␣ ⫽ 0.05, for example, means that when repeating the measurements, 95% (1 – ␣) of the results will be greater than x¯ – CL and smaller than x¯⫹ CL. Another useful conclusion is that if the number of measurements is large (>30), then approximately 99% of the results should not exceed the confidence level defined as follows:
CL0.99⫽ 3s (Eq 8)
To summarize, confidence level is a very convenient statistical charac-teristic for interpretation of results, and always is proportional to the standard deviation. Therefore, evaluation of the standard deviation is, in
most cases, sufficient for comparative analysis. In addition, Eq 6 refers to the so-called corrected standard deviation. This means that the expression takes into account the number of observations, and the values obtained can be compared even if the numbers of measurements are not identical.
(Nevertheless, using the same number of measurements is strongly recommended.)
Practically all microstructural elements (e.g., grains, pores, particles) have different sizes. This property can be characterized by means of distribution functions. For comparative purposes and based on the requirements of preliminary analysis, it is possible to apply much simpler methods, namely evaluation of the coefficient of variation, CV:
CV⫽s
x (Eq 9)
The interpretation of CV is simple: the greater the value of CV is, the more inhomogeneous the results are. Note that CV should be used for parameters complying with normal distribution only. An example of such a parameter is the amount of a given structural feature, such as porosity or graphite content in cast iron. Parameters describing size, such as grain size and particle diameter, frequently are described by means of log-normal distribution. Consequently, logarithms of the values have a log-normal distribution, which allows for application of the parameters defined above.
Example. Consider two pieces of cast iron (samples A and B) from which graphite content is to be determined. Both are prepared and tested in the same way (e.g., specimen preparation, magnification, image processing), and graphite content measured in the first 15 fields of view, yielding the basic statistical measures summarized below:
Both samples have the same mean graphite content, but the deviation in content (individual results) for sample B is four times higher than for sample A. Moreover, the scatter of results for sample B is so large that the 99% confidence level is approximately equal to half of the measured value. One interpretation of results is that the graphite content in sample B is inhomogeneous. Another is that the measurements for sample B were performed with different precision. If these interpretations cannot be
Sample Graphite contents Mean Deviation CV A 14.0, 14.3, 14.8, 13.7,
rejected a priori, it is wise to repeat the analysis or increase the number of measurements.
Advanced Statistical Analysis. The statistical parameters described above refer to a very limited subset of possible methods of interpretation.
Following are some other questions that can be answered by means of statistical analysis. Appropriate methods are described in detail in the literature, such as (Ref 10). The tasks of advanced statistical methods include:
O Comparison of mean values or deviations, which can help to decide whether or not the results obtained for two different materials are significantly different
O Comparison of distributions, which can be used to judge the extent of agreement with a given distribution function
O Correlation analysis, which is suitable to evaluate the relationship between two or more quantities (this analysis is often used to verify theoretical models for materials properties of materials)
O Verification of statistical hypotheses, which helps to eliminate some parameters that are not important from the point of view of a given analysis (e.g., checking if small changes in nonmetallic inclusion content have any effect on the grain size of steel)
Statistical interpretation can be complex. Fortunately, in the case of routine control, appropriate methods of statistical evaluation are included in the standardized procedures that describe the methodology of specimen preparation, testing procedure, and analysis of the results. Interpretation of nonstandard measurements could require an experienced statistician.