4.1. Basic Statistical Tools

A working knowledge of statistics is necessary to the understanding of the limitations of microprobe analysis and results. The sections below provide a very basic review of appropriate terms and statistical methods. The methodology presented is applicable to all analytical data that involve counting or multiple measurements. Such applications include X-ray fluorescence analysis, mass spectrometry, and inductively coupled or direct-coupled optical spectrometry. The National Institute of Standards and Technology (NIST) has a very thorough statistical handbook at: http://www.itl.nist.gov/div898/handbook/index.htm.

Mean

If x_i is an individual measurement and n measurements are made, then the mean value of the measurements is:

Standard Deviation

The standard deviation (s) from the mean of these measurements is defined as:

A population of measurements with normal or Gaussian distribution will have 68.3% of the population within ±1s , 95.4% within ±2s, 99.7% within ±3s, and 99.9% within ±4s (Figure St-1).

The variance of the measurements may be defined as s², and the coefficient of variation (also called relative error or relative standard deviation) is:

The relative error is often expressed as a percentage of the mean value.

Standard Error of the Mean

The error associated with the mean is less than that of an individual measurement. This is termed the standard error of the mean, and is defined as:

The production and counting of X-rays follow statistical patterns and have a Poisson distribution. At sufficiently high count rates this is identical to a normal distribution. Thus,

where C = number of X-rays counted. After determining the number of counts on a standard, the microprobe software reports a "sigma" ratio":

Ratios of up to 3.0 on a standard material are considered acceptable, although, as shown below, values greater than 1.0 indicate substantial heterogeneity. Such heterogeneity should not be present in a standard material and high sigma ratios probably indicate machine instability or operator problems.

Copyright 1997-2003, James H. Wittke

Last update: 01/18/2006 01:47 PM.