LECTURE NOTES Introduction to Statistics 1

LECTURE NOTES Introduction to Statistics 1

Francis Joseph H. Campen~a November 5, 2012

Contents

1 Overview of Statistics

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1.1 Introduction & Some Definition of Terms . . . . . . . . . . . . 3

1.2 Data and Variables . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Scales of Measurement . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Mathematical Notations . . . . . . . . . . . . . . . . . . . . . 8

2 Describing Data with Graphs

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2.1 Frequency Distribution . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Graphical Representations of Data . . . . . . . . . . . . . . . 15

3 Numerical Measures

19

3.1 Measures of Center . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Measures of Variability . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Measures of Relative Position . . . . . . . . . . . . . . . . . . 23

3.4 Measure of Skewness . . . . . . . . . . . . . . . . . . . . . . . 25

3.5 Box and Whiskers Plot . . . . . . . . . . . . . . . . . . . . . . 27

4 Basic Probability Concepts

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4.1 Sample Space and Events . . . . . . . . . . . . . . . . . . . . 32

4.2 Operations with Events . . . . . . . . . . . . . . . . . . . . . . 34

4.3 Counting Techniques . . . . . . . . . . . . . . . . . . . . . . . 35

4.3.1 Tree Diagram . . . . . . . . . . . . . . . . . . . . . . . 35

4.3.2 Fundamental Principle of Counting . . . . . . . . . . . 38

4.3.3 Permutation . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3.4 Combination . . . . . . . . . . . . . . . . . . . . . . . 44

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CONTENTS

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4.4 Probability of an Event . . . . . . . . . . . . . . . . . . . . . . 45 4.5 Some Probability Laws . . . . . . . . . . . . . . . . . . . . . . 48

5 Probability Distributions

59

5.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2 Mean and Variance of Discrete Random Variables . . . . . . . 63

5.3 Properties of the Mean and Variance of a Random Variable . . 65

6 Discrete Probability Distributions

67

6.1 Discrete Uniform Probability Distributions . . . . . . . . . . . 67

6.2 Binomial Probability Distributions . . . . . . . . . . . . . . . 68

6.3 Hypergeometric Probability Distributions . . . . . . . . . . . . 71

6.4 Negative Binomial Probability Distributions . . . . . . . . . . 72

6.5 Geometric Probability Distributions . . . . . . . . . . . . . . . 72

6.6 Poisson Probability Distributions . . . . . . . . . . . . . . . . 73

7 Continuous Probability Distributions

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7.1 Normal Probability Distributions . . . . . . . . . . . . . . . . 78

8 Estimation of Parameters

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8.1 Sampling and Sampling Distribution . . . . . . . . . . . . . . 84

8.2 Sampling Procedures . . . . . . . . . . . . . . . . . . . . . . . 87

8.3 Estimating the Population Mean . . . . . . . . . . . . . . . . 91

9 Statistical Tables and Formulas

95

Bibliography

99

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Chapter

Overview of Statistics

Statistics is everywhere in the media, inside the school, at the office, on the bus. We live our life making choices. And most of the time we make decisions and choose which path to go to based on incomplete information. Some can say that people live their lives comfortably with some level of uncertainty. This is what makes statisticians unique. It is their ability to quantify uncertainties and make them precise. This is a reason why statisticians can make categorical statements with confidence and assurance about their level of uncertainty.

In this chapter we will give an overview of what statistics is all about by learning some of the terms and basic concepts that are used in this field. Most often people have difficulty studying or learning statistics because of all the jargons that are needed in order to understand statistical concepts. Basic concepts are presented here to introduce the elementary ideas and theories in statistics. Most of the theories and mathematical concepts are not presented here but merely used as tools in understanding statistics. It is assumed that these theoretical foundations have been proven and thus no a proof of these statements are not necessary.

1.1 Introduction & Some Definition of Terms

The word statistics comes from the German word "statistik" which means political science dealing with state affairs. We may wonder why the word statistics would actually be concerned with government affairs. This may be

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CHAPTER 1. OVERVIEW OF STATISTICS

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answered in these historical facts.

? As early as 3800 BC there were records of population in Babylonia

? In Biblical times, census were undertaken by Moses in 1491 BC and by David in 1017 BC. The most famous Roman Census was recorded in the Bible on Luke 2 : 1 - 4, N? ow in those days a decree went forth from Caesar Augustus for all the inhabited earth to be registered. This registration took place when Quirinius was governor of Syria and all people went traveling to be registered, each one to his own city.?'

? The Athenians and other classical Greeks took census for adult male citizens in wartime and the general populace during the shortage of food supply.

As we can see, statistics is generally all about data and our interpretation about the data. We now formally define what we mean by statistics. Definition Statistics is a science that deals with the methods of collecting, organizing, summarizing, analysis, and interpretation of data in such a way that valid conclusions can be drawn from them.

Based from our definition we can identify two major areas in statistics, descriptive statistics, and inferential statistics.

Definition Descriptive statistics consists of those methods concerned with collecting and describing a set of data to yield meaningful information. Inferential statistics comprises those methods concerned with the analysis of a data, which is a subset or a part of the population leading to predictions or inferences about the population.

APPLICATIONS AND USE OF STATISTICS

The uses and/or application of statistics are unlimited. Sometimes we just don't get to know that what we are doing or merely observing is already an application of statistics. Here are some of the examples of how and where statistics is used:

1. In education, statistics is used to describe test results. How many passed or how many failed the test, the performance of a certain batch

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