The median m is a value such that 50% of the observed values are smaller than or equal to it. In this section we generalize the idea to an arbitrary percentage. We could ask which value is such that 80% of the observations are smaller than or equal to it. Or, seeing things the other way around,… Continue reading CUMULATIVE FREQUENCIES AND PERCENTILES
Month: February 2023
Dispersion measures
Location measures do not tell us anything about dispersion of data. We may have two distributions sharing the same mean, median, and mode, yet they are quite different. Figure 4.8, repeated illustrates the importance of dispersion in discerning the difference between distributions sharing location measures. One possible way to characterize dispersion is by measuring the range X(n) − X(1) i.e.,… Continue reading Dispersion measures
Location measures: mean, median, and mode
We are all familiar with the idea of taking averages. Indeed, the most natural location measure is the mean. DEFINITION 4.5 (Mean for a sample and a population) The mean for a population of size n is defined as The mean for a sample of size n is The two definitions above may seem somewhat puzzling,… Continue reading Location measures: mean, median, and mode
SUMMARY MEASURES
A look at a frequency histogram tells us many things about the distribution of values of a variable of interest within a population or a sample. However, it would be quite useful to have a set of numbers capturing some essential features quantitatively; this is certainly necessary if we have to compare two histograms, since… Continue reading SUMMARY MEASURES
ORGANIZING AND REPRESENTING RAW DATA
We have introduced the basic concepts of frequencies and histograms in Section 1.2.1. Here we treat the same concepts in a slightly more systematic way, illustrating a few potential difficulties that may occur even with these very simple ideas. Imagine a car insurance agent who has collected the weekly number of accidents occurred during the last… Continue reading ORGANIZING AND REPRESENTING RAW DATA
WHAT IS STATISTICS?
A rather general answer to this question is that statistics is a group of methods to collect, analyze, present, and interpret data (and possibly to make decisions). We often consider statistics as a branch of mathematics, but this is the result of a more recent tendency. From a historical perspective, the term “statistics” stems from… Continue reading WHAT IS STATISTICS?
Introduction
Some fundamental concepts of descriptive statistics, like frequencies, relative frequencies, and histograms, have been introduced informally. Here we want to illustrate and expand those concepts in a slightly more systematic way. Our treatment will be rather brief since, within the framework descriptive statistics is essentially a tool for building some intuition paving. We introduce basic… Continue reading Introduction
Integrals in multiple dimensions
Definite integrals have been introduced in Section 2.13 as a way to compute the area below the curve corresponding to the graph of a function of one variable. If we consider a function (x, y) of two variables, there is no reason why we should not consider its surface plot and the volume below the surface, corresponding to a region D on… Continue reading Integrals in multiple dimensions
Partial derivatives: gradient and Hessian matrix
In Section 2.7 we defined the derivative of a function of a single variable as the limit of an increment ratio: If we have a function of several variables, we may readily extend the concept above by considering a point and perturbing one variable at a time. We obtain the concept of a partial derivative with respect to a single… Continue reading Partial derivatives: gradient and Hessian matrix
CALCULUS IN MULTIPLE DIMENSIONS
In this section we extend some concepts that we introduced in the previous concerning calculus for functions of one variable. What we really need for what follows is to get an intuitive idea of how some basic concepts are generalized when we consider a function of multiple variables, i.e., a function f(x1, x2, …, xn) = f(x) mapping a… Continue reading CALCULUS IN MULTIPLE DIMENSIONS