Statistics is assigns that allows us to make conclusions about some real world phenomena, so processes or systems using some data that are extracted from this real world. However, in this real world phenomena that can be very complex and to simplify their study, we replace them with in a sense simpler mathematical models. So statistics involves some mathematical modeling. Let us discuss now the very general framework of models in statistic. Let us assume that we study for example customers to of some shop. In this case, it is possible that we have data about these customers, and this data is presented in the form of a table. Columns of this table represent a some properties or features of our customers. Something like age, or income, or number of children and so on. Every row of this table represents one customer. For example, it is possible that we have a 17-year-old customer with income 1,500 and no children, and it is possible that we have another customer a 35-year-old with income 2,000 and one child and so on. So this is the general form of a presentation of the data. There is a terminology that is related to this table. Columns of this table are called variables, or features, or properties, or something like this. Usually it is features in Machine Learning and variables in statistics. So columns are called variables or features. Rows of this table are called observations, or objects or items. Rows are usually observations or objects. We can rewrite this table using formal mathematical notation. I can rewrite it in a form of a matrix. For example, I can denote these values by let there x with some indices for example x 1,1 x 1,2 and so on. x 1, d this is x 2,1 this is x 2,2 and so on x 2,d and so on. X n, 1 x n, 2, x n, d. So if I write this table in the form of this matrix, then it means that I have the features, the columns of this matrix and objects and observations and roles of this matrix. So this matrix is usually called object feature matrix or something like this. So you see that I use this upper indices to denote different variables, different columns of this matrix. So do not confuse this upper index wave power. It is not x 1 to the power d, but it is just x 1, d. So we have here d features and n observations. We can assume for a moment that all these variables are just real numbers. In fact as we will discuss soon, it is possible for some variables to be not numeric, but of some other kind. For example, variables like gender or country of origin are not numeric. But for a moment let us assume that all these values are just numbers. So x i, j is just a real number. In this case, we can consider these metrics either as a collection of rows and each row is a d-dimensional vector. So we can say that we will denote this by x_i and x_i is observation and it is d-dimensional vector. Or we can consider this table as a collection of its columns and each column is n-dimensional vector. So we can denote it by x raised upper index j, and this is n-dimensional vector. We assume that our data is obtained as a result of some random experiment. For example, it will be discussed some study that involves some people. Then it means that these people that are included in this study are selected from the general population in a somewhat random way. So we have some randomness in this table and we may assume that each row of this table is a realization of some random variable or system of random variables. Our model is rather simple, we have made assumption that there are several random variables or better say a system of random variables that correspond to variables of our data. So we have X_1, X_2 and so on X_d, a system of random variables such that our observations are independent samples from this system. So we can say that our observations X_1, X_2 and so on, X_n are independent and identically distributed with distribution of this system. This means independent and identically distributed. What does it mean on prospect that they are independent and identically distributed? It means that we assume that for example, income of a second person in our study does not depend on the income of the first person in our study. In the same wage income on the second person does not depend on the age of the first-person. So rows are independent of each other, but columns of course can't be dependent. For example, it is easy to assume that income in some way dependent on age that these two variables are correlated for example, because we expect that a person with larger age probably have larger income. So we assume that these variables are independent and they has the same distribution, that all rows of this table is obtained by sampling from the same population of all potential customers or all potential participants of our study. This is our basic model and the main goal of statistics is to observe this sample or observe this table and make some conclusions about these random variables, about the properties, about the correlations and some more complicated correlations between them. So to conclude our goal is to make conclusions on our random variables. This system of random variables is also called a population, it does not mean that we have to study an actual population of some people. It is just a general term that is used to refer to a system of random variables that generated our data that generated our sample. Now let us discuss how to study this data, how to describe them. To do so we have to discuss the so-called descriptive statistics.
