Presenting data 1 2 PRESENTING DATA The information that is collected must be presented effectively for statistical inference. Categorical and numerical data can be presented efficiently using charts and tables. 2.1 Presenting Categorical Variables You present a categorical variable by first sorting variable values according to the categories of the variable. Then you place the count, amount, or percentage (part of the whole) of each category into a summary table or into one of several types of charts. The Summary Table A two-column table in which category names are listed in the first column and the count, amount, or percentage of values are listed in a second column. Such as the following bank preference summary table indicates that people prefer in person at branch in their duties. Banking Preference? Percent The Bar Chart ATM 16% Automated or live telephone 2% Drive-through service at branch 17% In person at branch 41% Internet 24% A chart containing rectangles ( bars ) in which the length of each bar represents the count, amount, or percentage of responses of one category. Banking Preference Internet In person at branch Drive-through service at branch Automated or live telephone ATM 0% 5% 10% 15% 20% 25% 30% 35% 40% 45%
% in each category (bar graph) Cumulative % (line graph) Presenting data 2 The Pie Chart The pie chart is a circle broken up into slices that represent categories. The size of each slice of the pie varies according to the percentage in each category. The entire circle ( pie ) represents the total. Banking Preference 24% 16% 2% 17% ATM Automated or live telephone Drive-through service at branch In person at branch 41% Internet The Pareto Chart A special type of bar chart that presents the counts, amounts, or percentages of each category in descending order left to right, and also contains a superimposed plotted line that represents a running cumulative percentage. Pareto Chart For Banking Preference 100% 80% 60% 40% 20% 100% 80% 60% 40% 20% 0% In person at branch Internet Drivethrough service at branch ATM Automated or live telephone 0% When you have many categories, a Pareto chart enables you to focus on the most important categories by visually separating the vital few from the trivial many categories. For the banking preference data, the Pareto chart shows that two categories, person at branch and internet, account for more than one-half of all preferences.
Two-Way Cross-Classification Table Presenting data 3 A multicolumn table that presents the count or percentage of responses for two categorical variables. In a twoway table, the categories of one of the variables form the rows of the table, while the categories of the second variable form the columns. The outside of the table contains a special row and a special column that contain the totals. Cross-classification tables are also known as cross-tabulation tables. Counts of Particles Found Cross-Classified by Wafer Condition Row Percentages Table Wafer Condition Wafer Condition f o Good Bad Total f o Good Bad Total Particles Present Yes 14 36 50 No 320 80 400 Particles Present Yes 28 72 100 No 80 20 100 Total 334 116 450 Total 74.2 25.8 100 ColumnPercentages Table f o Particles Present Wafer Condition Good Bad Total Yes 4.2 31 11.1 No 95.8 69 88.9 Total 100 100 100 Overall Percentages Table Wafer Condition f o Particles Present Good Bad Total Yes 3.1 8 11.1 No 71.1 17.8 88.9 Total 74.2 25.8 100
Age of Surveyed College Students Age of Surveyed College Students Presenting data 4 2.2 Tables and Charts for Numerical Data To obtain a table, you first establish groups that represent separate ranges of values and then place each value into the appropriate group. To obtain a chart, you use the groups from the table. Ordered Array An ordered array is a sequence of data, in rank order, from the smallest value to the largest value. Shows range from minimum value to maximum value. Day Students 16 17 17 18 18 18 19 19 20 20 21 22 22 25 27 32 38 42 Night Students 18 18 19 19 20 21 Stem-and-Leaf Display 23 28 32 33 41 45 A simple way to see how the data are distributed and where concentrations of data exist The method is, separating the sorted data series into leading digits (the stems) and the trailing digits (the leaves). Day Students 16 17 17 18 18 18 19 19 20 20 21 22 22 25 27 32 38 42 Night Students 18 18 19 19 20 21 23 28 32 33 41 45
Presenting data 5 The Frequency and Percentage Distribution A table of grouped numerical data that contains the names of each group in the first column, the counts (frequencies) of each group in the second column, and the percentages of each group in the third column. This table can also appear as a two-column table that shows either the frequencies or the percentages. EXAMPLE The following Fan Cost Index data shown provides the cost in a recent year of attending an NBA professional basketball league game, including four tickets, two beers, four soft drinks, four hot dogs, two game programs, two caps, and the parking fee for one car for each of the 29 teams. Team Fan Cost Index Team Fan Cost Index Atlanta 244.48 Minnesota 231.38 Boston 358.72 New Jersey 328.90 Charlotte 196.90 New Orleans 182.30 Chicago 335.00 New York 394.52 Cleveland 317.90 Orlando 229.82 Dallas 339.23 Philadelphia 269.48 Denver 271.16 Phoenix 302.04 Detroit 282.00 Portland 251.86 Golden State 206.52 Sacramento 318.30 Houston 270.94 San Antonio 303.79 Indiana 250.57 Seattle 229.50 LA Clippers 317.00 Toronto 320.47 LA Lakers 453.95 Utah 235.75 Memphis 228.28 Washington 194.56 The frequency and percentage distribution for the NBA Fan Cost Index is as follows: Fan Cost Index ($) Frequency Percentage 150 to under 200 3 10.34% 200 to under 250 7 24.14% 250 to under 300 6 20.69% 300 to under 350 10 34.48% 350 to under 400 2 6.90% 400 to under 450 0 0.00% 450 to under 500 1 3.45% 29 100.00% Frequency and percentage distributions enable you to quickly determine differences among the many groups of values. In this example, you can quickly see that most of the fan cost indexes are between $200 and $350, and that very few fan cost indexes are either below $200 or above $350. Example: A manufacturer of insulation randomly selects 20 winter days and records the daily high temperature 24, 35, 17, 21, 24, 37, 26, 46, 58, 30, 32, 13, 12, 38, 41, 43, 44, 27, 53, 27 Solution: Sort raw data in ascending order: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Find range: 58-12 = 46 Select number of classes: 5 (usually between 5 and 15) Compute class interval (width): 10 (46/5 then round up)
Determine class boundaries (limits): Class 1: 10 to less than 20 Class 2: 20 to less than 30 Class 3: 30 to less than 40 Class 4: 40 to less than 50 Class 5: 50 to less than 60 Compute class midpoints: 15, 25, 35, 45, 55 Count observations & assign to classes Class Frequency Relative Frequency Percentage Cumulative Frequency Presenting data 6 Cumulative Percentage 10 but less than 20 3.15 15 3 15 20 but less than 30 6.30 30 9 45 30 but less than 40 5.25 25 14 70 40 but less than 50 4.20 20 18 90 50 but less than 60 2.10 10 20 100 Total 20 1.00 100 Histogram A bar chart in which the frequencies or percentages in each group of numerical data are represented as individual bars on the vertical Y axis and the variable is plotted on the horizontal X axis. In a histogram, in contrast to a bar chart of categorical data, no gaps exist between adjacent bars. EXAMPLE The following histogram presents the fan cost index data of the preceding example. The values below the bars (175, 225, 275, 325, 375, 425, 475) are midpoints, the approximate middle value for each group of data. As with the frequency and percentage distributions, you can quickly see that very few fan cost indexes are either below $200 or above $350.
Presenting data 7 The Time-Series Plot A chart in which each point represents the value of a numerical variable at a specific time. By convention, the X axis (the horizontal axis) always represents units of time, and the Y axis (the vertical axis) always represents units of the variable. Time-series plots can reveal patterns over time, patterns that you might not see when looking at a long list of numerical values. EXAMPLE The following data shows the mean hotel room rate in dollars for 1996 through 2006: Year Rate($) Year Rate($) 1996 70.63 2001 88.27 1997 75.31 2002 83.54 1998 78.62 2003 82.52 1999 81.33 2004 86.23 2000 85.89 2005 90.88 2006 97.78 In this example, the plot reveals that mean hotel rates were generally rising between 1996 and 2006, but declined in the years immediately after 2001.
Presenting data 8 The Scatter Plot A chart that plots the values of two numerical variables for each observation. In a scatter plot, the X axis (the horizontal axis) always represents units of one variable, and the Y axis (the vertical axis) always represents units of the second variable. Job Labor hours Cubic feet Job Labor hours Cubic feet M-1 24.00 545 M-19 25.00 557 M-2 13.50 400 M-20 45.00 1,028 M-3 26.25 562 M-21 29.00 793 M-4 25.00 540 M-22 21.00 523 M-5 9.00 220 M-23 22.00 564 M-6 20.00 344 M-24 16.50 312 M-7 22.00 569 M-25 37.00 757 M-8 11.25 340 M-26 32.00 600 M-9 M-10 M-11 M-12 M-13 M-14 M-15 M-16 M-17 M-18