11. Statistical Analysis of Experimental Data

Statistical methods are extremely important in engineering, because they provide a means for representing large amounts of data in a concise form that is easily interpreted and understood. Usually, the data are represented with a statistical distribution function that can be characterized by a measure of central tendency (the mean x ¯ ) and a measure of dispersion (the standard deviation S _x ). A normal or Gaussian probability distribution is by far the most commonly employed; however, in some cases, other distribution functions may have to be employed to adequately represent the data.

The most significant advantage resulting from the use of a probability distribution function in engineering applications is the ability to predict the occurrence of an event based on a relatively small sample. The effects of sampling error are accounted for by placing confidence limits on the predictions and establishing the associated confidence levels. Sampling error can be controlled if the sample size is adequate. Use of Studentʼs t distribution function, which characterizes sampling error, provides a basis for determining sample size consistent with specified levels of confidence. Studentʼs t distribution also permits a comparison to be made of two means to determine whether the observed difference is significant or whether it is due to random variation.

Statistical methods can also be employed to condition data and to eliminate an erroneous data point (one) from a series of measurements. This is a useful technique that improves the data base by providing strong evidence when something unanticipated is affecting an experiment.

Regression analysis can be used effectively to interpret data when the behavior of one quantity y depends upon variations in one or more independent quantities x ₁, x ₂, ..., x _n . Even though the functional relationship between quantities exhibiting variation remains unknown, it can be characterized statistically. Regression analysis provides a method to fit a straight line or a curve through a series of scattered data points on a graph. The adequacy of the regression analysis can be evaluated by determining a correlation coefficient. Methods for extending regression analysis to multivariate functions exist. In principle, these methods are identical to linear regression analysis; however, the analysis becomes much more complex. The increase in complexity is not a concern, because computer subroutines are available that solve the tedious equations and provide the results in a convenient format.

Many probability functions are used in statistical analyses to represent data and predict population properties. Once a probability function has been selected to represent a population, any series of measurements can be subjected to a chi-squared (χ ²) test to check the validity of the assumed function. Accurate predictions can be made only if the proper probability function has been selected.

Finally, statistical methods for accessing error propagation are discussed. These methods provide a means for determining error in a quantity of interest y based on measurements of related quantities x ₁, x ₂, ..., x _n and the functional relationship y = f(x ₁, x ₂, ..., x _n ).