Presenting Data in Tables & Charts Organizing Numerical Data Data with 20 or more observations should be organized The Ordered Array: arranges raw data in order from the smallest observation to the largest observation. Raw Data Arranged in an Ordered Array 5. Auto Cost ($) 18000 18000 21000 1000 24000 15000 30000 22000 12000 30000 17000 2300 5. Auto Cost ($) 1000 2000 2300 5000 6000 9000 10000 12000 12000 14000 15000 16000 17000 18000 27000 5000 18000 18000 46000 20000 26500 21000 The Ordered Array makes it easy to identify: • extreme values • typical values • range where the majority of values are concentrated Stem and Leaf Display: shows where raw data clusters over a range of observations. EXAMPLE: the following data represent the weekly salary checks earned by a sample of eight secretaries: $555 $490 $648 $832 $710 $590 $576 $623 First, put the values in ascending order and then use the 100s column as the stems, use the 10s column as the leaves, and either ignore the units column or round the units column and then use the 10s column as the leaves. $555 $490 $648 $832 $710 $590 $576 $623 4 5 6 7 8 | | | | | 9 579 24 1 3 To further illustrate, how we can organize data to present, analyze and interpret findings, we will study data from a previous QBA questionnaire: 1) USD students’ auto costs • 2) USD students’ maximum auto speeds Raw Data from student questionnaire (partial) 1. Age 19 18 24 20 19 20 19 19 19 21 2. Gender 0=M, 1=F 1 1 0 1 1 0 1 1 0 0 3. Live 4. Campus Study / 0=Off Week 1=On (hrs) 0 15 1 33 1 12 1 12.5 0 12.5 0 10 1 5 1 20 1 15 1 10 5. 9. Auto 6. 7. 8. No. units Cost Alch bev Sodas / Hrs. Paid this sem ($) / wk (#) wk (#) / wk (hrs) (#) 18000 4 5 13.5 14 18000 0 6.5 14 16 21000 10 3 4 23 1000 1 12.5 12 16.5 24000 2 0 20 17 15000 50 1 0 13 21 0 18 16 0 2 0 17 30000 2 13 0 16 22000 20 0 5 16 10. TV / video game / wk (hrs) 2 4 4 10 5 3 14 2 20 15 RAW 5. Auto Cost ($) 18000 18000 21000 1000 24000 15000 30000 ARRAY 5. Auto Cost ($) 1000 2000 2300 5000 6000 9000 10000 12000 12000 Stem & Leaf Auto Costs Stem unit: 10000 0 1 2 3 4 5 6 122569 0224567888 0124577 00 56 06 Stem & Leaf MPH Stem unit: 10 7 8 9 10 11 12 13 14 15 16 17 18 0 0 005 0000000558 0 0 0 0 0 0 00 05 5 0027 0 And just for fun, let’s look at GPA GPA GPA Stem unit: = 1 2 4556889 3 00001122333334445567889 4 0 How Else Can We Organize our Data? Numerical Data • Frequency Distribution • Relative Frequency Distribution • Percentage Frequency Distribution • Cumulative Frequency Distribution Frequency Distribution Freq. Dist. MPH (22 fast car (mph)) interval 70 80 90 100 110 120 130 140 150 160 170 180 Frequency Percentage 1 1 2 8 3 3 2 2 4 4 0 1 3.2% 3.2% 6.5% 25.8% 9.7% 9.7% 6.5% 6.5% 12.9% 12.9% 0.0% 3.2% Freq Distribution MPH (22. fast car (mph) interval Frequency Percentage 60 80 100 120 140 160 180 0 2 10 6 4 8 1 0.0% 6.5% 32.3% 19.4% 12.9% 25.8% 3.2% Histogram MPH fast car (mph) midpoints -70 90 Frequency Distribution Frequency Distribution for Numerical Data (5. Auto Cost ($)) 0 Frequency Percentage 10000 7 24.14% 20000 10 34.48% 30000 8 27.59% 40000 0 0.00% 50000 2 6.90% 60000 1 3.45% 70000 1 3.45% Selecting the Number of Classes • There is no “correct” number of classes (K) to use in a frequency distribution. • However, the frequency distribution should have at least 5 classes, but no more than 20 Caution! • If you have too “FEW” classes (K), a large portion of your data, lies in one class. • However, if there are a number of empty classes, or too many classes with a frequency of 1 or 2, this may indicate too “MANY” classes (K). Approximate Number of Classes in Frequency Distribution # Observations Less than 50 50 – 200 200 – 500 500 – 1,000 1,000 – 5,000 5,000 – 50,000 More than 50,000 # Classes 5–7 7–9 9 - 10 10– 11 11- 13 13 – 17 17 - 20 What do you gain by organizing your data in a Frequency Distribution? Hint! From pages of raw data Answer • Reduce large numbers of data points to a workable number of classes and frequencies. • Study the frequency distribution and learn a great deal about the shape of the data set. Raw Data from student questionnaire (partial) 1. Age 19 18 24 20 19 20 19 19 19 21 2. Gender 0=M, 1=F 1 1 0 1 1 0 1 1 0 0 3. Live 4. Campus Study / 0=Off Week 1=On (hrs) 0 15 1 33 1 12 1 12.5 0 12.5 0 10 1 5 1 20 1 15 1 10 5. 9. Auto 6. 7. 8. No. units Cost Alch bev Sodas / Hrs. Paid this sem ($) / wk (#) wk (#) / wk (hrs) (#) 18000 4 5 13.5 14 18000 0 6.5 14 16 21000 10 3 4 23 1000 1 12.5 12 16.5 24000 2 0 20 17 15000 50 1 0 13 21 0 18 16 0 2 0 17 30000 2 13 0 16 22000 20 0 5 16 10. TV / video game / wk (hrs) 2 4 4 10 5 3 14 2 20 15 Frequency Distribution Freq. Dist. MPH (22 fast car (mph)) interval 70 80 90 100 110 120 130 140 150 160 170 180 Frequency Percentage 1 1 2 8 3 3 2 2 4 4 0 1 3.2% 3.2% 6.5% 25.8% 9.7% 9.7% 6.5% 6.5% 12.9% 12.9% 0.0% 3.2% Freq Distribution MPH (22. fast car (mph) interval Frequency Percentage 60 80 100 120 140 160 180 0 2 10 6 4 8 1 0.0% 6.5% 32.3% 19.4% 12.9% 25.8% 3.2% Histogram MPH fast car (mph) midpoints -70 90 Frequency Distribution Frequency Distribution for Numerical Data (5. Auto Cost ($)) 0 Frequency Percentage 10000 7 24.14% 20000 10 34.48% 30000 8 27.59% 40000 0 0.00% 50000 2 6.90% 60000 1 3.45% 70000 1 3.45% Constructing a Frequency Distribution • • • • • Gather the sample data Arrange data in an Ordered Array Select the number of classes to be used Determine class width: range/ # of classes Determine the class limits for each class so that the distribution is easy to interpret • Count the number of data values in each class (the raw frequencies) • Determine the Relative Frequencies Relative Frequency = Raw frequency count in each class -------------------------------------Total number of observations (n) Relative Frequency is essential for comparing the relationship between two data sets. To Convert Relative Frequency to Percent Frequency: Multiply Relative Frequency X 100 Example 15. A doctor's office staff has studied the waiting times for patients who arrive at the office with a request for emergency service. The following data were collected over a one-month period (the waiting times are in minutes). 2 5 10 12 4 4 5 17 11 8 9 8 12 21 6 8 7 13 18 3 Use classes of 0 - 4, 5- 9, and so on. a. Show the frequency distribution. b. Show the relative frequency distribution. c. Show the cumulative frequency distribution. d. Show the relative cumulative frequency distribution. How Else Can We Organize our Data? Graphic Techniques to Describe Numerical Data 1) Histogram (continuous data) 2) Polygon 3) Ogive 4) Scattergram Histogram • Uni-modal • Bi-modal • Skewed: i) right or positively skewed ii) left or negatively skewed Histogram Auto Costs Histogram of Auto Cost 10 8 6 4 2 Midpoints ($) 65000 55000 45000 35000 25000 15000 0 -- Frequency 12 Histogram MPH Frequency Histogram 15 10 5 0 -- 70 90 110 Midpoints 130 150 170 Negative or Left Skewed Positive or Right Skewed Quiz Would incomes of employees in large firms tend to be positively or negatively skewed? Why? Quiz Do exam grades tend to be positively or negatively skewed? Why? A Scatter Diagram Graphs bivariate data to examine whether a relationship exists between two numerical variables. Is there a relationship between the price of their auto and the maximum MPH a USD student has driven? Scatter Diagram Speed vs. Cost 200 (MPH) 150 100 c 50 0 0 20000 40000 ($) 60000 80000 Is there a relationship between the number of alcoholic beverages consumed per week and the number of hours studied per week? Scattergram: Weekly Alchohol Amount vs Hours Studied 50 No. of Drinks 40 30 20 10 0 0 5 10 15 20 Hours 25 30 35 $Wk Entertainment vs #Alcoholic Bev/wk $Wk Entertain 200 150 100 50 0 0 10 20 30 Alcohol Bev/wk 40 50 60 GPA vs mph 4 GPA 3 2 1 0 50 70 90 110 130 mph 150 170 190 Alcoholic Beverages/wk MPH vs Alcoholic Beverages/wk 60 50 40 30 20 10 0 50 70 90 110 130 MPH 150 170 190 mph 0 70 0 80 4 90 21 95 8 100 4 100 15 100 1 100 7 100 0 100 2 100 6 105 12 105 3 108 50 120 0 120 2 120 0 130 10 130 2 135 MPH vs Alcoholic Beverages/wk Alcoholic Beverages/wk Alch bev / wk (#) 60 50 40 30 20 10 0 50 70 90 110 130 MPH 150 170 190 Scatter Diagram: Entertainment Vs. Cash On Hand Typical $ On-Hand 120 100 80 60 40 20 0 0 50 100 150 Entertainment $ per Week 200 Hours Paid Vs. Cash on Hand ($) Cash on Hand $ 120 100 80 60 40 20 0 0 10 20 30 Hours per week work 40 50 Tables & Charts for Categorical Data 1) Summary Table: similar to Frequency Distribution. 2) Contingency Table for Crosstabulation of Bivariate Categorical Data. 3) Bar Chart: graphical representation of frequency of occurrence. 4) Pie Chart: graphical emphasis of proportion 5) Pareto Diagram 6) Side-by-Side Bar Charts: for bivariate categorical data. Summary Table Presentation of Categorical Data Coke Classic Diet Coke Pepsi-Cola Diet Coke Coke Classic Coke Classic Dr. Pepper Diet Coke Pepsi-Cola Pepsi-Cola Coke Classic Dr. Pepper Sprite Coke Classic Diet Coke Coke Classic Coke Classic Sprite Coke Classic Diet Coke Coke Classic Diet Coke Coke Classic Sprite Pepsi-Cola Coke Classic Coke Classic Coke Classic Pepsi-Cola Coke Classic Sprite Dr. Pepper Pepsi-Cola Diet Coke Pepsi-Cola Coke Classic Coke Classic Coke Classic Pepsi-Cola Dr. Pepper Coke Classic Diet Coke Pepsi-Cola Pepsi-Cola Pepsi-Cola Pepsi-Cola Coke Classic Dr. Pepper Pepsi-Cola Sprite Show: a. Freq distribution b. Relative Freq c. Percent Freq d. Bar graph e. Pareto diagram f. Pie chart. Soft drink Freq. Relative Freq Coke Classic 19 Diet Coke 8 Dr. Pepper 5 Pepsi-Cola 13 Sprite 5 .38 or 38% .16 or 16% .10 or 10% .26 or 26% .10 or 10% Total 1.00 or 100% 50 Contingency Table for Crosstabulation of Bivariate Categorical Data CROSSTABULATION OF QUALITY RATING AND MEAL PRICE FOR 300 LOS ANGELES RESTAURANTS Meal Price Quality Rating $10 – 19 $20 – 29 $30 – 39 $40 – 49 Total Good 42 40 2 0 84 Very Good 34 64 46 6 150 Excellent 2 14 28 22 66 78 118 76 28 300 Total ROW PERCENTAGES FOR EACH QUALITY RATING CATEGORY Meal Price Quality Rating $10 – 19 $20 – 29 $30 – 39 $40 – 49 Good 50.0 47.6 2.4 0.0 Very Good 22.7 42.7 30.6 4.0 Excellent 3.0 21.2 42.4 33.4 Total 100 100 100 Gender vs. Number of Alcoholic Drinks per Week Crosstabs Gender vs Alcohol/wk Count of Gender 0=M, 1=F Alcohol/wk Gender 0=M, 1=F 0 1 Grand Total 0 2 2 4 1 9 7 16 2 4 1 5 4 0 1 1 5 Grand Total 1 16 0 11 1 27 Contingency Table for Gender vs. Auto Costs Crosstabs of Gender vs Auto Cost Count of Gender 0=M, 1=F Auto Cost Gender 0=M, 1=F 0 1 Grand Total 1 5 2 7 2 4 6 10 3 4 2 6 5 2 0 2 6 1 0 1 7 Grand Total 0 16 1 11 1 27 Contingency Table of Gender vs. MPH Crosstabs Gender vs mph Count of Gender 0=M, 1=F mph Gender 0=M, 1=F 70-90 0 0 1 2 Grand Total 2 91-110 3 6 9 111-130 3 1 4 131-150 3 1 4 5 7 1 8 Grand Total 16 11 27 Contingency Table of Live on/off Campus by Gender On (1) Off (0) Campus Gender Male 0 Female 1 0 1 Totals 0 9 5 14 1 9 10 19 Totals 18 15 33 Gender 0=M, 1=F 0 1 0 0 0 0 1 1 1 1 0 0 0 1 0 1 0 0 0 0 0 1 1 0 1 0 1 Auto Cost Alch bev ($) / wk (#) 15000 50 18000 0 21000 10 9000 1 26500 6 2000 4 18000 6 20000 15 27000 36 24000 2 12000 15 16000 6 30000 2 1000 1 5000 12 18000 4 46000 7 6000 12 25000 0 60000 6 17000 20 66000 2 12000 0 45000 0 2300 3 10000 10 14000 2 mph 120 70 145 160 150 100 105 100 150 135 152 160 150 100 105 90 100 180 120 157 140 120 100 130 108 130 100 Auto CostAlcohol/wk 2 5 2 0 3 1 1 1 3 1 1 1 2 1 2 2 3 4 3 1 2 2 2 1 3 1 1 1 1 2 2 1 5 1 1 2 3 0 6 1 2 2 7 1 2 0 5 0 1 1 1 1 2 1 mph 3 1 4 5 5 2 2 2 5 4 5 5 5 2 2 1 2 5 3 5 4 3 2 3 2 4 2 The following data are for 30 observations on two qualitative variables, X and Y. The categories for X are A, B, and C; the categories for Y are 1 and 2. Categories for X are A,B, and C. The categories for Y are 1 and 2. Obs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 x A B B C B C B C A B A B C C C B C B C B C B C A B C C A B B y 1 1 1 2 1 2 1 2 1 1 1 1 2 2 2 2 1 1 1 1 2 1 2 1 1 2 2 1 1 2 a. Develop a crosstabulation for the data with x in the rows and y in the columns. b. Compute the row percentages. c. Compute the column percentages. d. What is the relationship, if any, between x and y? Side-by-side Bar Charts M al e Drop Page Fields Here C r os s t a b Ge nde r v s A ut o $ Count of Gender 0=M , 1=F Femal e A ut o Cost 1 7 6 Gender 0=M , 1=F 5 3 2 M al e 0 1 0 1 2 3 4 5 6 7 Pareto Diagram Separates the “vital few” from the “trivial many”.