4 - Characterization of Data

Summary

This chapter covers myriad ways to statistically characterize data as a first step in their analysis. The concept of an estimator is introduced, and the common estimators of location (e.g, the sample mean and median), dispersion (e.g., the sample variance), shape (e.g., skewness), direction and association (e.g., correlation coefficient) are described, and related to their implementation in Matlab. The important limit theorems of statistics, the Laws of Large Numbers and the central limit theorems, are elucidated. Exploration data analysis tools are defined. These include the histogram, empirical cumulative distribution function, kernel density estimator, percent-percent plot and quantile-quantile plot, all of which are illustrated using data. Sampling distributions are defined, and the important central and non-central chi square, Student's t, F and correlation coefficient cases are elaborated. The distribution of order statistics is treated, leading to the distribution of the sample median and interquartile range. The joint distribution of the sample mean and sample variance is derived using Cochran's Therore, and their independence for Gaussian data is established.