Chapter 2

1. Chapter 2 Descriptive Statistics Part 1 • Summarizing Qualitative Data • Summarizing Quantitative Data Dr. Constance Lightner- Fayetteville State University

2. 2.1 Summarizing Qualitative Data • A frequency distribution is a tabular summary of data showing the frequency (or number) of items in each of several nonoverlapping classes. • The objective is to provide insights about the data that cannot be quickly obtained by looking only at the original data. Dr. Constance Lightner- Fayetteville State University

3. Example: Marada Inn Guests staying at Marada Inn were asked to rate the quality of their accommodations as being excellent, above average, average, below average, or poor. The ratings provided by a sample of 20 quests are shown below. Below Average Average Above Average Above Average Above Average Above Average Above Average Below Average Below Average Average Poor Poor Above Average Excellent Above Average Average Above Average Average Above Average Average Dr. Constance Lightner- Fayetteville State University

4. Example: Marada Inn • Frequency Distribution RatingFrequency Poor 2 Below Average 3 Average 5 Above Average 9 Excellent 1 Total 20 Dr. Constance Lightner- Fayetteville State University

5. Relative Frequency Distribution • The relative frequency of a class is the fraction or proportion of the total number of data items belonging to the class. • A relative frequency distribution is a tabular summary of a set of data showing the relative frequency for each class. Dr. Constance Lightner- Fayetteville State University

6. Percent Frequency Distribution • The percent frequency of a class is the relative frequency multiplied by 100. • Apercent frequency distribution is a tabular summary of a set of data showing the percent frequency for each class. Dr. Constance Lightner- Fayetteville State University

7. Example: Marada Inn Relative Frequency and Percent Frequency Distributions RelativePercent RatingFrequencyFrequency Poor .10 10 Below Average .15 15 Average .25 25 Above Average .45 45 Excellent .05 5 Total 1.00 100 Dr. Constance Lightner- Fayetteville State University

8. Bar Graph • A bar graph is a graphical device for depicting qualitative data. • On the horizontal axis we specify the labels that are used for each of the classes. • A frequency, relative frequency, or percent frequency scale can be used for the vertical axis. • Using a bar of fixed width drawn above each class label, we extend the height appropriately. • The bars are separated to emphasize the fact that each class is a separate category. Dr. Constance Lightner- Fayetteville State University

9. 9 8 7 6 Frequency 5 4 3 2 1 Rating Above Average Excellent Poor Below Average Average Example: Marada Inn: Bar Graph Dr. Constance Lightner- Fayetteville State University

10. Pie Chart • The pie chart is a commonly used graphical device for presenting relative frequency distributions for qualitative data. • First draw a circle; then use the relative frequencies to subdivide the circle into sectors that correspond to the relative frequency for each class. • Since there are 360 degrees in a circle, a class with a relative frequency of .25 would consume .25(360) = 90 degrees of the circle. Dr. Constance Lightner- Fayetteville State University

11. Exc. 5% Poor 10% Below Average 15% Above Average 45% Average 25% Quality Ratings Example: Marada Inn • Pie Chart Dr. Constance Lightner- Fayetteville State University

12. Steps for Frequency Distributions and Histograms using Quantitative Data 1.) Determine number of groups or classes 2.) Determine width of each class 3.) Create classes • Lower bounds • Upper bounds 4.) Determine frequency distribution (and/or relative frequency, percent frequency, cumulative frequency distribution) 5.) If required, draw a histogram based upon one of the distributions in step 4.)

13. Classes for Frequency Distributions and Histograms • For quantitative data, classes are interval ranges in which data can fall. • Classes must be defined so that: 1. All classes have equal width. (Class width is the difference between the lower limit of two successive classes). 2. There are no gaps between classes. 3. They are all non overlapping and consider all potential measurement values (i.e. all possible data values must fall in exactly one class). Dr. Constance Lightner- Fayetteville State University

14. Number of Classes and Class Width • In order to define classes, we must first decide how many classes to use, and the class width. • A general rule says that the number of classes should be the smallest whole number K that makes 2K greater than the total number of measurements n (from Bowerman, et. al.) • Class width = Largest measurement – Smallest measurement +0.05 K You can always round up to the nearest integer to simplify the creation of classes. NEVER ROUND DOWN. Dr. Constance Lightner- Fayetteville State University

15. Creating Classes • Once you determine the class width, you may begin creating the classes. • Creating Classes 1. Make the smallest measurement the lower limit for your first class. 2. Define the lower limit the remaining classes by adding the lower limit of the preceding class and the selected class width. 3. Define the upper limit for each class as the largest potential measurement less than the lower limit of the following class. Classes may be defined in any fashion, provided the above rules are met. On the Following slides we suggest one technique for defining classes. Dr. Constance Lightner- Fayetteville State University

16. Example 2.1 • A basketball player practices free throws by taking 25 shots each day, and he records the number of shots missed each day in order to track his progress. The number of shots missed on days 1 through 30 are respectively: 17 15 16 18 14 15 13 12 10 11 11 10 9 10 9 9 9 10 8 10 6 8 9 8 7 9 8 7 5 8 Create a frequency distribution for this data. Dr. Constance Lightner- Fayetteville State University

17. Example 2.1: Creating Classes • For our example there are 30 measurements, n=30. • Determine the number of Classes • 24=16 and 25=32. Thus K=5 is the smallest whole number K that makes 2K greater than 30. So we will use 5 classes. • Determine Class Width • The class width = (18-5+.05)/5= 2.61. Round up to 3. • Create our classes Step 1: 5 is the smallest measurement, thus this will be the lower limit of our first class. Step 2: Class Lower limits for our 5 classes 5 5+3 = 8 8+3= 11 11+3 = 14 14+ 3= 17 Dr. Constance Lightner- Fayetteville State University

18. Example 2.1: Creating Classes • Step 3: Class Lower Class Upper Limit Limit 5 7 (Potential values are integers, so this is the largest possible measurement less than the lower limit of the next class- 8) 8 10 11 13 14 16 17 19 (If we had another class following this one it would have began with 20, thus 19 is the largest value less than 20) Dr. Constance Lightner- Fayetteville State University

19. Example 2.1: Frequency Distribution • Classes Frequency [5 - 7] 4 On Exactly 4 days he missed between 5-7 free throws [8 - 10] 16 On Exactly 16 days he missed between 8-10 free throws [11 – 13] 4 On Exactly 4 days he missed between 11-13 free throws [14 – 16] 4 On Exactly 4 days he missed between 14-16 free throws [17 – 19] 2 On Exactly 2 days he missed between 17-19 free throws 30 Total number of measurements Dr. Constance Lightner- Fayetteville State University

20. Relative Frequency Distribution • The relative frequency of a class is the fraction or proportion of the total number of data items belonging to the class. relative frequency of a class = frequency of the class the total # of measurements • A relative frequency distribution is a tabular summary of a set of data showing the relative frequency for each class. Dr. Constance Lightner- Fayetteville State University

21. Example 2.1: Frequency and Relative Frequency • Classes Frequency Relative Frequency [5 - 7] 4 4/30 [8 - 10] 16 16/30 [11 – 13] 4 4/30 [14 – 16] 4 4/30 [17 – 19] 2 2/30 30 1 • The sum of all frequencies always equals n. • The sum of all relative frequencies always equals 1. Dr. Constance Lightner- Fayetteville State University

22. Example 2.2 The manager of Hudson Auto would like to get a better picture of the distribution of costs for engine tune-up parts. A sample of 50 customer invoices has been taken and the costs of parts are listed below. Anderson Sweeney and Williams Dr. Constance Lightner- Fayetteville State University

23. Example 2.2: Creating Classes • Determine the number of Classes [recall n (# of measurements) =50] • 26 exceeds 50. Thus K=6 classes • Determine Class Width • Smallest measurement=52 Largest measurement=109 • Width= 109-52+.05 = 9.51 Round up to 10 6 Dr. Constance Lightner- Fayetteville State University

24. Example 2.2: Creating Classes • Create our 6 Classes • Step 1: 52 is the smallest measurement • Step 2: Lower Limits 52 62 72 82 92 102 Dr. Constance Lightner- Fayetteville State University

25. Example 2.2: Creating Classes Step 3: Create classes Lower Limit Upper Limit 52 61.9 Since measurements are recorded with 1 decimal place, 61.9 is the largest potential measurement smaller than 62 62 71.9 72 81.9 82 91.9 92 101.9 102 111.9 Since the next lower limit would have been 112. Dr. Constance Lightner- Fayetteville State University

26. Example 2.2: Frequency and Relative Frequency • Classes Frequency Relative Frequency [52 - 61.9] 2 2/50 [62 – 71.9] 15 15/50 [72 – 81.9] 16 16/50 [82 – 91.9] 6 6/50 [92 – 101.9] 7 7/50 [102 – 111.9] 4 4/50 50 1 Dr. Constance Lightner- Fayetteville State University

27. Histogram • A common graphical presentation of quantitative data is a histogram. • The variable of interest is placed on the horizontal axis. • A rectangle is drawn above each class interval with its height corresponding to the interval’s frequency, or relative frequency. • A histogram has no natural separation between rectangles of adjacent classes. Anderson Sweeney and Williams Dr. Constance Lightner- Fayetteville State University

28. Example 2.2: Histogram 18 16 14 12 Frequency 10 8 6 4 2 Parts Cost ($) 50 60 70 80 90 100 110 Dr. Constance Lightner- Fayetteville State University

29. The End Dr. Constance Lightner- Fayetteville State University