Advanced Mathematics

My goals

By the end of this unit, I will be able to: 

• find measures of central tendency in two quantitative variables. 
• find measures of variability in two quantitative variables. 
• determine the linear regression line of a given series. 
• calculate a linear correlation coefficient 

Introduction

Descriptive statistics is a set of brief descriptive coefficients that summarises a given data set, which can either be a representation of the entire population or sample. Data may be qualitative such as sex, color and so on or quantitative represented by numerical quantity such as height, mass, time and so on. 

The measures used to describe the data are measures of central tendency and measures of variability or dispersion. Until now, we know how to determine the measures of central tendency in one variable. In this unit, we will use those measures in two quantitative variables known as double series. 

In statistics, double series include technique of analyzing data in two variables, when focus on the relationship between a dependent variable-y and an independent variable-x. The linear regression  method will be used in this unit. The estimation target is a function of the independent variable called the regression function which will be a function of a straight line. 

Descriptive statistics provide useful summary of security returns when performing empirical and analytical analysis, as they provide historical account of return behavior. Although past information is useful in any analysis, one should always consider the expectations of future events. Some variables are discrete, others are continuous. If the variable can take only certain values, for example, the number of apples on a tree, then the variable is discrete. If however, the variable can take any decimal value (in some range), for example, the heights of the children in a school, then the variables are continuous. In this unit, we will consider discrete variables. 

9.1. Covariance

Activity 9.1 

Complete the following table 

What can you get from the following expressions: 

In case of two variables, say x and y, there is another important result called covariance of x and y, denoted (x,y)

The covariance of variables x and y is a measure of how these two variables change together. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the smaller values, i.e. the variables tend to show similar behavior, the covariance is positive. In the opposite case, when the greater values of one variable mainly correspond to the smaller values of the other, i.e. the variables tend to show opposite behavior, the covariance is negative. If covariance is zero, the variables are said to be uncorrelated, meaning that there is no linear relationship between them.

Therefore, the sign of covariance shows the tendency in the linear relationship between the variables. The magnitude of covariance is not easy to interpret. 

Developing this formula, we have 

Example 9.1 

Find the covariance of x and y in the following data sets 

Solution

We have 

Thus, 

Example 9.2 

Find the covariance of the following distribution 

Solution

 Convert the double entry into a simple table and compute the arithmetic means