MIREES Academic year 20082009 STATISTICS Silvia Cagnone silvia

MIREES Academic year 2008/2009 STATISTICS Silvia Cagnone silvia. cagnone@unibo. it Department of Statistics University of Bologna

Readings q Mann P. , “Introductory Statistics”, 6 th edition, John Wiley & Sons, INC. , 2007. Chapters 1, 2, 3, 4. 4, 4. 5, 4. 6, 11. 3, 11. 4. 1, 13. 2. 1 -13. 2. 3, 13. 4, 13. 6 q Hyper. Stat Online Textbook (http: //davidmlane. com/hyperstat/) q Handouts and exercises download from http: //campus. cib. unibo. it/6033 Exam Project work and oral exam Office hours by e-mail silvia. cagnone@unibo. it

Aim of the course Basic concepts of the statistical method for the analysis and the interpretation of economic and social data. a. Univariate statistical analysis (one character) b. Bivariate statistical analysis (two characters) c. Brief introduction to the statistical inference Exercises in laboratory software Excel

INTRODUCTION TO STATISTICS

What is statistics? Two meanings: 1. Common usage Numerical facts (e. g the age of a student, the income of a family, the starting salary of a typical college graduate, etc. ). 2. Field or discipline of study Statistics is a group of methods used to collect, analyze, present, and interpret data and to make decisions

What is statistics? Statistics has two aspects: 1. theoretical or mathematical statistics deals development, derivation and proof of theorems, formulas, rules and laws; with the statistical 2. applied statistics involves the applications of those theorems, formulas, rules and laws to solve real-world problems.

Why statistics? n n n We use statistics when we need methods for extracting information from observed or collected data to obtain a deeper understanding from numbers about the situations they represent. Even professional statisticians have trouble understanding a data set (= a collection of numerical information) by merely looking at it. Statistics and data analysis provide methods that can help in the understanding of nearly every field of human experience.

Why statistics? Data set of 22. 385 individuals of Cambridge. Information on Sex, Height, Weight, Age and Smoking status. What we can say about this data set by merely looking at it? ?

Types of statistics § Descriptive statistics: consists of methods for organizing, displaying, and describing data by using tables, graphs, and summary measures. § Inferential Statistics: consists of methods that use sample results to help make decisions or predictions about a population. Population (target): the collection of all elements of interest Sample: the selection of a few elements from this population Probability acts as a link between descriptive and inferential statistics.

Descriptive statistics: example A sample of 30 employees from large companies was selected, and these employees were asked how stressful their jobs were. The responses of these employees are recorded next where Very represents Very stressful, Somewhat means Somewhat stressful, and None stands for Not stressful at all.

Inferential statistics: example 1. We may make some decisions about the political views of a college and university students based on political views of 1000 students selected from a few colleges and universities. 2. We may want to find the starting salary of a typical college graduate. To do so, we may select 2000 recent college graduates, find their starting salaries, and make a decision based on this information.

DESCRIPTIVE STATISTICS

Basic terms 1. ELEMENT or MEMBER or UNIT Specific subject or object (for example, a person, firm, item, state, or country) about which the information is collected 2. VARIABLE Characteristic under study that assumes different values for different elements. In contrast to a variable, the value of a CONSTANT is fixed. 3. OBSERVATION or MEASUREMENT The value of a variable for an element.

Basic terms: example Data set: 2001 Sales of Seven U. S. Companies Element/ Member/Unit Company Wal-Mart Stores IBM General Motors Dell Computer Procter & Gamble JC Penney Home Depot 2001 Sales Variable (millions of dollars) 217, 799 85, 866 Observation/ 177, 260 Measurement 31, 168 39, 262 32, 004 53, 553

Types of variables 1. QUANTITATIVE VARIABLE A variable that can be measured numerically. The data collected on a quantitative variable are called quantitative data. Examples: Incomes, heights, prices of homes, etc. n DISCRETE VARIABLE A variable whose values are countable. A discrete variable can assume only certain values with no intermediate values. (e. g. nr students in a class, nr components of a family, births in Forlì in 2007, etc. ) n CONTINUOUS VARIABLE A variable that can assume any numerical value over a certain interval or intervals. (e. g. age, weight, height, time to get to the school, etc. )

Types of variables 2. QUALITATIVE or CATEGORICAL VARIABLE A variable that cannot assume a numerical value but can be classified into two or more nonnumeric categories. The data collected on such a variable are called qualitative data. Examples: hair color, gender, etc.

Summation notation The summation operator is a mathematical notation used to denote the sum of values. For example, suppose a sample consists of 5 books and their prices are $25, $60, $37, $53 and $16. If we denote the variable price of a book by X, we have: x 1 (price of the first book) = $25; x 2 (price of the second book) = $60; … x 5 (price of the fifth book) = $16; Now, suppose we want to add the prices of all five books: x 1 + x 2 + x 3 + x 4 + x 5 = 25 + 60 + 37 + 53 + 16 = $191 or, briefly, xi = x 1 + x 2 + x 3 + x 4 + x 5 =25 + 60 + 37 + 53 + 16 = $191

Summation notation Example Annual salaries (in thousands of dollars) of four workers are 75, 42, 125, and 61. Find ∑x b) (∑x)² c) ∑x² a) Solution ∑x = x 1 + x 2 + x 3 + x 4 = 75 + 42 + 125 + 61 = 303 b) (∑x)² = (75 + 42 + 125 + 61)² =(303)² = 91. 809 c)∑x² = (75)² + (42)² + (125)² + (61)² = 5625 + 1764+ a) + 15, 625 + 721 = 26. 735

ORGANIZING DATA

Raw data When data are collected, the information obtained from each member of a population or a sample is recoded in the sequence in which it becomes available. This sequence of data is called raw data. RAW DATA Data recorded in the sequence in which they are collected. These data are also called ungrouped data, contain information on each member of population individually. because they a sample or

Raw data: Examples Ages of 50 students 21 18 25 22 25 19 20 19 28 23 24 19 31 21 18 25 22 19 20 37 29 19 23 22 27 34 19 18 22 23 26 25 23 21 21 27 22 19 20 25 37 25 23 19 21 33 23 26 21 24 M F M M M F F M M M F F M M F M F F M M M Gender of 50 students M F M M F F M F M M F

Organizing and Graphing Qualitative Data FREQUENCY DISTRIBUTION A frequency distribution for qualitative data lists all categories and the number of elements that belong to each of the categories. Example Variable Category Frequency

Frequency distribution How can we obtain a frequency distribution? Example A sample of 30 employees from large companies was selected, and these employees were asked how stressful their jobs were. The responses of these employees are recorded next where Very represents Very stressful, Somewhat means Somewhat stressful, and None stands for Not stressful at all. Raw data

Frequency distribution Stress on Job Very Somewhat None Tally ||||| ||||| | Frequency (ni) 10 14 6 Sum = ni = 30

Relative frequency and percentage distributions Relative frequency of a category: it is obtained by dividing the frequency of the category by the sum of all frequencies Percentage of a category: it is obtained by multiplying the relative frequency of the category by 100 Percentage = (Relative frequency) * 100 = fi * 100

Relative frequency and percentage distributions: example Stress on Job ( ) Frequency (ni) Very Somewhat None 10 14 6 ni = 30 = n Stress on Job ( ) Very Somewhat None Relative Frequency (fi) 10/30 =. 333 14/30 =. 467 6/30 =. 200 fi = 1. 00 Percentage. 333(100) = 33. 3. 467(100) = 46. 7. 200(100) = 20. 0 Sum = 100

Graphical presentation of qualitative data A graphic display can reveal at a glance the main characteristics of a data set. The bar graph and the pie chart are two types of graphs used to display qualitative data.

Bar graph A graph made of bars whose heights represent the frequencies of respective categories. Stress on Job Very Somewhat None Frequency (ni) 10 14 6 ni = 30 = N The categories are on the horizontal axis and all these categories are represented by intervals of the same width. We mark the frequencies on the vertical axis and their heights represent the frequency of the corresponding category. We leave a small gap between adjacent bars.

Bar graph for percentages The bar graphs for relative frequency and percentage distributions can be drawn simply by marking the relative frequencies or percentages, instead of the frequencies, on the vertical axis. Stress on Job Percentage Very Somewhat None 33. 3 46. 7 20. 0

Pie charts A circle divided into portions that represent the relative frequencies or percentages of a population or a sample belonging to different categories.

Organizing and Graphing Quantitative Data Often for quantitative data with a large number of different values, it is appropriated to prepare a frequency distribution based on classes Example: frequency distribution Weekly earning of 120 employees of a large company Variable Third class Lower limit of the sixth class Weekly Earnings (dollars) 400 -| 600 -| 800 -| 1000 -| 1200 -| 1400 -| 1600 Frequency column Number of Employees 9 22 39 15 9 Upper limit of the 6 sixth class Frequency of the third class

Relative frequency and percentage distribution Weekly Earnings (dollars) Number of Employees n Relative frequency f Percentage 400 -| 600 -| 800 -| 1000 -| 1200 -| 1400 -| 1600 14 22 49 20 9 6 14/120 = 0. 117 22/120 = 0. 183 49/120 = 0. 408 20/120 = 0. 167 9/120 = 0. 075 6/120 = 0. 050 0. 117*100 = 11. 7 0. 183*100 = 18. 3 0. 408*100 = 40. 8 0. 167*100 = 16. 7 0. 075*100 = 7. 5 0. 050*100 = 5. 0 n = 120 f=1 100

Cumulative frequency A cumulative frequency is the total number of values that fall below a certain value. To obtain the cumulative frequency of a class, we add the frequency of that class to the frequencies of all preceding classes. Weekly Earnings (dollars) Number of Employees n Cumulative frequency 400 -| 600 -| 800 -| 1000 -| 1200 -| 1400 -| 1600 14 22 49 20 9 6 14 14 + 22 = 36 14 +22 + 49 = 86 14 + 22 + 49 + 20 = 105 14 + 22 + 49 + 20 + 9 = 114 14 + 22 + 49 + 20 + 9 + 6 = 120 n = 120

Graphical presentation of quantitative data Quantitative data can be displayed mainly in a histogram. Actually, we can also draw a pie chart to display the percentage distribution for a quantitative data set. The procedure to construct a pie chart is similar to the one for qualitative data.

Histogram A graph in which classes are marked on the horizontal axis and the frequencies, relative frequencies, or percentages are marked on the vertical axis. The frequencies, relative frequencies, or percentages are represented by the heights of the bars. In a histogram, the bars are drawn adjacent to each other, to underline the continuity of the quantitative data.

Example histogram Weekly Earnings (dollars) Number of Employees Percentage 400 -| 600 -| 800 -| 1000 -| 1200 -| 1400 -| 1600 14 22 49 20 9 6 11. 7 18. 3 40. 8 16. 7 7. 5 5. 0