The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 2 Displaying Distributions with Graphs 8 29 06 Lecture 2 1 1 Recall Statistics is the science of data Collecting Analyzing Decision making Fundamental concepts Population parameter sample statistic sample size Any questions 8 29 06 Lecture 2 1 2 Chapter 1 Looking at Data Distributioins 1 1 Displaying Distributions with Graphs 1 2 Displaying Distributions with Numbers 1 3 Density Curves and Normal Distributions 8 29 06 Lecture 2 1 3 NBA Draft 2005 Name Team Nationality Weight Height A Bogut Milwaukee Bucks Australia 245 7 0 M Williams Atlanta Hawks US 230 6 9 D Williams Utah Jazz US 210 6 3 C Paul New Orleans Hornets US 175 6 0 R Felton Charlotte Bobcats US 198 6 1 8 29 06 Lecture 2 1 4 Data contain Individuals the subjects described by the data Variables any characteristic of an individual A variable can take different values for different individuals 8 29 06 Lecture 2 1 5 NBA Draft 2005 Name Team Nationality Weight Height A Bogut Milwaukee Bucks Australia 245 7 0 M Williams Atlanta Hawks US 230 6 9 D Williams Utah Jazz US 210 6 3 C Paul New Orleans Hornets US 175 6 0 R Felton Charlotte Bobcats US 198 6 1 8 29 06 Lecture 2 1 6 Categorical Quantitative Variables A categorical variable places an individual into one of several groups or categories A quantitative variable takes numerical values for which arithmetic operations such as adding and averaging make sense 8 29 06 Lecture 2 1 7 NBA Draft 2005 Name Team Nationality Weight Height A Bogut Milwaukee Australia 245 7 0 M Williams Atlanta US 230 6 9 D Williams Utah US 210 6 3 C Paul New Orleans US 175 6 0 R Felton Charlotte US 198 6 1 Categorical variables Team Nationality Quantitative variables Weight Height 8 29 06 Lecture 2 1 8 NBA Draft 2005 Variables Team Nationality Categorical Weight Height Quantitative How many teams in the draft How many players drafted by each team How many players higher than 6 9 How many players between 200 and 250 pounds Equivalently what is the distribution for each variable 8 29 06 Lecture 2 1 9 Distributions of Variables The distribution of a variable indicates what values a variable takes and how often it takes these values For a categorical variable distribution categories count percent for each category For a quantitative variable distribution pattern of variation of its values 8 29 06 Lecture 2 1 10 Highest Level of Education for People Aged 25 34 Education Count millions Percent Less than high school 4 6 11 8 High school graduate 11 6 30 6 Some college 7 4 19 5 Associate degree 3 3 8 8 Bachelor s degree 8 6 22 7 Advanced degree 2 5 6 6 8 29 06 Lecture 2 1 11 Exploratory Data Analysis EDA Use statistical tools and ideas to help us examine data Goal to describe the main features of the data NEVER skip this EDA Displaying distributions with graphs Displaying distributions with numbers 8 29 06 Lecture 2 1 12 Basic Strategies for EDA Strategy I 1 One variable at a time 2 Relationships among the variables Strategy II 1 Graphical visualizations 2 Numerical summaries 8 29 06 Lecture 2 1 13 Graphic Techniques for Categorical Variables Bar Graph uses bars to represent the frequencies or relative frequencies such that the height of each bar equals the frequency or relative frequency of each category Frequencies counts Relative frequencies percent height indicates count or percent Pie Chart is a circle divided into a number of slices that represent the various categories such that the size of each slice is proportional to the percentage corresponding to that category area relative Note Pie chart requires to include all the categories that make up a whole 8 29 06 Lecture 2 1 14 Highest Level of Education for People Aged 25 34 Education Less than high school Count Percent millions 4 6 11 8 11 6 30 6 Some college 7 4 19 5 Associate degree 3 3 8 8 Bachelor s degree 8 6 22 7 Advanced degree 2 5 6 6 High school graduate 8 29 06 Lecture 2 1 15 Graphic Techniques for Quantitative Variables Stemplot Stem and Leaf Plot Histogram Time plot 8 29 06 Lecture 2 1 16 Stemplot Separate each observation into a stem consisting of all but the final rightmost digit and a leaf the final digit Stems may have as many digits as needed but each leaf contains only a single digit Write the stems in a vertical column with the smallest at the top and draw a vertical line at the right of this column Write each leaf in the row to the right of its stem in increasing order out from the stem 8 29 06 Lecture 2 1 17 of Home Runs per Season Babe Ruth New York Yankees 19201934 54 59 35 41 46 25 47 60 54 46 49 46 41 34 22 Mark McGwire St Louis Cardinals 1986 2001 3 49 32 33 39 22 42 9 9 39 52 58 70 65 32 29 Question Work out the stem plot of McGwire back to back stem plot of the two players 8 29 06 Lecture 2 1 18 Example Midterm Scores of STAT 101 The following data set contains the midterm exam scores of STAT 101 74 62 71 74 8 29 06 76 78 88 80 77 70 85 100 77 83 85 95 87 60 72 62 87 60 95 50 53 84 79 86 95 95 83 83 Lecture 2 1 82 85 97 86 79 79 78 93 79 84 87 73 84 36 19 Splitting Trimming Stems For a moderate number of obs Split each stem into two one with leaves 0 4 and the other with leaves 5 9 Increase of stems reduce of leaves Trimming If the observed values have too many digits you can trim them by rounding to a certain digit Disadvantage of stemplots Awkward for large data sets 8 29 06 Lecture 2 1 20 Example A study on litter size Data 170 observations 4 8 6 8 7 5 4 4 8 8 29 06 6 2 6 8 7 8 7 8 8 5 7 7 5 5 9 2 3 6 6 7 6 6 7 7 7 9 9 7 7 6 8 3 5 8 8 5 3 9 7 5 7 7 5 3 5 6 3 5 5 6 5 8 6 6 4 7 4 4 5 5 6 5 6 4 5 5 9 3 5 6 4 7 6 4 4 9 5 10 6 6 5 6 7 7 4 5 5 6 7 6 7 8 6 6 1 3 4 7 5 4 7 5 6 7 3 7 7 5 4 6 9 6 7 10 5 6 8 7 5 5 7 5 6 3 7 8 7 7 6 3 4 4 5 6 4 7 5 5 6 9 3 …