ABSTRACT
This chapter considers errors in the context of the data's mean and variance and other elementary statistical attributes. Applications of Taylor's theorem provide first order propagation rules for the mean and variance of a function in parameters with errors. The data's histogram approximates their mean- and variance-based statistical distribution that summarizes the data's integral and differential properties. Statistical error bars and inferences for the data assume the probability of finding a datum within an interval of the distribution is the area subtended by the distribution over the interval, where the distribution's total area or probability is 100%. Widely used distributions include the binomial distribution that describes the chances of obtaining n things taken x at a time. Another is the mean-centered bell-shaped Gaussian or normal distribution of data with random errors. By the central limits theorem, the means of roughly 25 or more data follow the normal distribution with standard error-based confidence limits that may be compared to infer datasets with disparate origins. For smaller datasets with approximately bell-shaped distributions, error bars and inference testing are established using the weaker Student t-distribution. For normally distributed data, the Chi-squared distribution provides confidence intervals on the variances that may be tested for disparate origins using Fisher's f-distribution, which also facilitates the analysis of variance [ANOVA] between two or more datasets. The Chi-squared distribution also can test the goodness of fit between the data and their presumed distribution. For distribution-free data, somewhat weaker non-parametric statistical tests may be applicable.
