Chapter 3 - Statistical Methods and Error Handling

Abstract

In this chapter, we review a number of basic statistical methods and procedures with an emphasis on data reduction, statistical estimation, error minimization, Gaussian statistics, and estimation methods (e.g., the maximum likelihood method). Starting with the mean, variance and standard deviation, the concept of probability is introduced and the results of computing “expected values” are discussed. The concept of estimation bias is outlined and the features of a number of probability density functions are described. The significance of the central limit theorem is stressed and its application to sample mean and variance is presented. The construction of confidence intervals is introduced and the need to pick the significance level for this confidence interval is emphasized. A method for selecting the sample size is introduced and applied to calculating satellite altimeter bias. The value of the maximum likelihood estimation method is presented and its applications are discussed. We next discuss linear estimation (or regression), demonstrating what variability is being minimized in this process. The general value of the method of least squares is clearly presented. The relationship between correlation and regression is introduced and the effects of random errors on correlations are discussed. We provide a brief discussion of hypothesis testing followed by a more in-depth discussion on the importance of determining the effective degrees of freedom. The concept of two types of error (bias and random) is introduced and techniques are discussed for identifying and treating such error values. Propagation of random error and the statistics of roundoff are both introduced and discussed. Various methods of interpolation are introduced and the merits of each are outlined. We define the covariance and correlation calculations and discuss how they apply to different analyses. The bootstrap and Jackknife nonparametric statistical methods are presented and discussed with examples for each.