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Introduction to Statistics

Basic definitions of statistics that are important to know its bigger picture.

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Introduction to Statistics

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  1. Elementary Statistics Muhammad Ali, A/P Statistics GPGC Mardan December 12, 2020 Muhammad Ali, A/P Statistics (GPGC Mardan) Elementary Statistics December 12, 2020 1/14

  2. Chapter 1: The Nature of Statistics CHAPTER OBJECTIVES What does the word statistics bring to mind? To most people, it sug- gests numerical facts or data, such as unemployment figures and farm prices. Two common definitions of the word statistics are as follows: When used as a plural verb it means that facts or data, either nu- merical or nonnumerical, organized and summarized so as to pro- vide useful and accessible information about a particular subject. In singular verb it means the science of organizing and summariz- ing numerical or nonnumerical information. Statisticians also analyze data for the purpose of making generaliza- tions and decisions. For example, a political analyst can use data from a portion of the voting population to predict the political preferences of the entire voting population. In this chapter, we introduce some basic terminology so that the various meanings of the word statistics will be- come clear to you. Muhammad Ali, A/P Statistics (GPGC Mardan) Elementary Statistics December 12, 2020 2/14

  3. Chapter 1: The Nature of Statistics 1.1 Statistics Basics You probably already know something about statistics. newspapers, surf the Web, watch the news on television, or follow sports, you see and hear the word statistics frequently. In this sec- tion we introduce the two major types of statistics: descriptive statistics and inferential statistics. We also introduce terminology that helps dif- ferentiate among various types of statistical studies. If you read DEFINITION 1.1: Descriptive Statistics Descriptive statistics consists of methods for organizing and summa- rizing information. Descriptive statistics includes the construction of graphs, charts, and tables and the calculation of various descriptive measures such as averages, measures of variation, and percentiles. We discuss descriptive statistics in detail in Chapters 2 and 3. Muhammad Ali, A/P Statistics (GPGC Mardan) Elementary Statistics December 12, 2020 3/14

  4. Chapter 1: The Nature of Statistics Definition 1.2: Population and Sample Population: The collection of all individuals or items under consider- ation in a statistical study. Sample: That part of the population from which information is obtained. “Fig. 1” depicts the relationship between a population and a sample from the population. Figure 1: Fig 1: Relation between population and sample Muhammad Ali, A/P Statistics (GPGC Mardan) Elementary Statistics December 12, 2020 4/14

  5. Chapter 1: The Nature of Statistics Population and Sample... Now that we have discussed the terms population and sample, we can define inferential statistics. Definition 1.3: Inferential Statistics Inferential statistics consists of methods for drawing and measuring the reliability of conclusions about a population based on information ob- tained from a sample of the population.Descriptive statistics and infer- ential statistics are interrelated. You must almost always use techniques of descriptive statistics to organize and summarize the information ob- tained from a sample before carrying out an inferential analysis. Muhammad Ali, A/P Statistics (GPGC Mardan) Elementary Statistics December 12, 2020 5/14

  6. Chapter 1: The Nature of Statistics Example 1.1: Classifying Statistical Studies The 2018 Pakistan General Election: Table 1 displays the voting results for the general elections held on 25, July 2018. Table 1: Final results of the Pakistan 2018 general elections Party PTI PML(N) PPP Votes 16,903,702 12,934,589 6,924,356 Percentage 31.82 24.35 13.03 Classification This study is descriptive. It is a summary of the votes cast by Pakistani voters in the 2018 general election. No inferences are made. Muhammad Ali, A/P Statistics (GPGC Mardan) Elementary Statistics December 12, 2020 6/14

  7. Chapter 1: The Nature of Statistics Example 1.2: Classifying Statistical Studies Suppose we are interested in the last Medical College Admission Test (MDCAT) marks of all the students appeared in the test. But it is not feasible to measure the test marks of all the students in Pakistan. So, now we will measure the marks of a smaller sample of students, for example 1000 students. The independent selection of 1000 students used a sample is an example of inferential statistics. Definition 1.4: The Development of Statistics Historically, descriptive statistics appeared before inferential statistics. Over the centuries, records of such things as births, deaths, marriages, and taxes led to the development of descriptive statistics. Inferential statistics is a newer arrival. Major developments began to occur with the research of Karl Pearson who published his findings in the early years of the twentieth century. Muhammad Ali, A/P Statistics (GPGC Mardan) Elementary Statistics December 12, 2020 7/14

  8. Chapter 1: The Nature of Statistics Definition 1.5: Parameter and Statistics A parameter is an unknown numerical summary of the population. A statistic is a known numerical summary of the sample which can be used to make inference about parameters. (Agresti Finlay, 1997). So, the inference about some specific unknown parameter is based on a statistic. We use known sample statistics in making inferences about unknown population parameters. The primary focus of most research studies is the parameters of the population, not statistics calculated for the particular sample selected. The sample and statistics describing it are important only in so far as they provide information about the unknown parameters. Muhammad Ali, A/P Statistics (GPGC Mardan) Elementary Statistics December 12, 2020 8/14

  9. Chapter 1: The Nature of Statistics Example 1.3: Parameter and Statistics Consider the research problem of finding out what percentage of 18-30 years old are using Facebook more than 2 hours daily. Parameter The proportion P of 18-30 years old using Facebook more than 2 hours daily. Statistics The proportion p of 18-30 years old using Facebook more than 2 hours daily calculated from the sample of 18-30 years old. Muhammad Ali, A/P Statistics (GPGC Mardan) Elementary Statistics December 12, 2020 9/14

  10. Chapter 1: The Nature of Statistics Definition 1.6: Variables A characteristic that varies from one person or thing to another is called a variable, i.e. a variable is any characteristic that varies from one indi- vidual member of the population to another. Examples of variables for humans are height, weight, number of siblings, sex, marital status, and eye color. The first three of these variables yield numerical information (yield numerical measurements) and are examples of quantitative (or numerical) variables, last three yield non-numerical information (yield non-numerical measurements) and are examples of qualitative (or cat- egorical) variables. Quantitative variables can be classified as either discrete or continuous. Muhammad Ali, A/P Statistics (GPGC Mardan) Elementary Statistics December 12, 2020 10/14

  11. Chapter 1: The Nature of Statistics Definition 1.6.1: Discrete Variables Variables, such as the numbers of children in family, the numbers of car accident on the certain road on different days, or the numbers of stu- dents taking basics of statistics course are the results of counting and thus these are discrete variables. As a definition, we can say that a variable is discrete if it has only a countable number of distinct possi- ble values. That is, a variable is discrete if it can assume only a finite number of values or as many values as there are integers. Definition 1.6.2: Continuous Variables Quantities such as length, weight, or temperature can in principle be measured arbitrarily accurately. Weight may be measured to the near- est gram, but it could be measured more accurately, say to the tenth of a gram.Such a variable, called continuous, is intrinsically different from a discrete variable. Muhammad Ali, A/P Statistics (GPGC Mardan) Elementary Statistics December 12, 2020 11/14

  12. Chapter 1: Nature of Statistics Definition 1.7: Measurement Scales of Variables The variables that we will generally encounter fall into four broad cate- gories: ratio scale, interval scale, ordinal scale, and nominal scale. It is important that we understand each. Ratio Scale For a variable X, taking two values, X1and X2, and the ratio the distance (X2-X1) are meaningful quantities. Also, there is a natu- ral ordering (ascending or descending) of the values along the scale. Therefore, comparisons such as X2≤X1or X2≥X1are meaningful. Most economic variables belong to this category. Thus, it is meaningful to ask how big this year’s GDP is compared with the previous year’s GDP. Personal income, measured in dollars, is a ratio variable; some- one earning 100,000 dollars is making twice as much as another person earning 50,000 dollars (before taxes are assessed). X1 X2and Muhammad Ali, A/P Statistics (GPGC Mardan) Elementary Statistics December 12, 2020 12/14

  13. Chapter 1: Nature of Statistics Interval Scale An interval scale variable satisfies the last two properties of the ratio scale variable but not the first. Thus, the distance between two time periods, say (2000–1995) is meaningful, but not the ratio of two time periods (2000/1995). At 11:00 a.m. PST on August 11, 2007, Portland, Oregon, reported a temperature of 60 degrees Fahrenheit while Talla- hassee, Florida, reached 90 degrees. Temperature is not measured on a ratio scale since it does not make sense to claim that Tallahassee was 50 percent warmer than Portland. This is mainly due to the fact that the Fahrenheit scale does not use 0 degrees as a natural base. Ordinal Scale A variable belongs to this category only if it satisfies the third property of the ratio scale (i.e., natural ordering).Examples are grading systems (A, B, C grades) or income class (upper, middle, lower). For these variables the ordering exists but the distances between the categories cannot be quantified. Muhammad Ali, A/P Statistics (GPGC Mardan) Elementary Statistics December 12, 2020 13/14

  14. Chapter 1: Nature of Statistics Students of economics will recall the indifference curves between two goods. Each higher indifference curve indicates a higher level of utility, but one cannot quantify by how much one indifference curve is higher than the others. Nominal Scale Variables in this category have none of the features of the ratio scale variables. Variables such as gender (male, female) and marital sta- tus (married, unmarried, divorced, separated) simply denote categories. Question: What is the reason why such variables cannot be expressed on the ratio, interval, or ordinal scales? As we shall see, econometric techniques you will learn in later semesters that may be suitable for ratio scale variables may not be suitable for nominal scale variables. There- fore, it is important to bear in mind the distinctions among the four types of measurement scales discussed above. Muhammad Ali, A/P Statistics (GPGC Mardan) Elementary Statistics December 12, 2020 14/14

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