We continue our multi-part discussion of quantitative metallography. We hope this information will be usable by many in their metallurgical investigations.  

Statistics
Other measurements are possible, but the ones described in previous blogs represent some of the simplest and most useful. Each can be repeated on a number of fields on the “plane of polish” so that a mean and standard deviation can be obtained. The number of fields measured influences the precision of the measurement. Manual measurements are tedious and time-consuming, so sampling statistics may be less than desired. Image analysis removes most of the barriers to inadequate sampling.  

A good measure of statistical precision is the 95% confidence interval (or confidence limit). This defines a range around the mean value where 95 times out of 100 a subsequently determined mean will fall. A mean volume fraction of 10 ±2% implies that in 95 of 100 measurements the mean value will be between 8 and 12%. The 95% confidence interval is determined by:

95% CI = ts/(n)^½        

where t is the student’s t factor (t is a function of the confidence level desired and the number of measurements and can be found in standard textbooks and in ASTM standards such as E 562, E 1245, and E 1383); s is the standard deviation; and n is the number of measurements.  

The relative accuracy, RA, of a measurement is determined by:

% RA = 95% CI / X x 100        

where X is the mean value. In general, a relative accuracy of 10% or less is considered to be satisfactory.  
Sampling
So far, we have discussed measurements on a single plane-of-polish on one specimen. Because we are usually dealing with large quantities of material (such as an entire “heat” of metal or alloy, a large heat-treatment lot of forgings or a large forging or casting), a single specimen may not be representative of the whole quantity. Ideally, random sampling of a large batch would be best, but practical considerations usually rule this out.  

In most cases, sampling is done at predetermined convenient locations, such as the extreme ends of a coil, bar, plate, etc. or at locations that will be subject to extensive marking. In some cases, excess metal is added to a forging or casting to provide test material as similar as possible to that of the component. Sampling is often a compromise and is rarely excessive due to cost considerations. However, inadequate sampling or sampling on non-representative material/locations may degrade the value of the measurements.