1. QMS 6351 Statistics and Research Methods Chapter 2 Descriptive Statistics: Tabular and Graphical Methods Prof. Vera Adamchik

2. Chapter 2 Outline Summarizing Qualitative Data Summarizing Quantitative Data

3. Raw Data When data are collected, the information obtained from each member of a population or a sample is recorded in the sequence in which it becomes available. This sequence of data recording is random and unranked. Such data, before they are grouped or ranked, are called raw data.

4. 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.

5. Example: Marada Inn 1.Below Average 2.Above Average 3.Average 4.Above Average 5.Above Average 6.Above Average 7.Above Average 8.Below Average 9.Below Average 10.Average11.Poor 12.Poor 13.Above Average 14.Excellent 15.Above Average 16.Average 17.Above Average 18.Average 19.Above Average 20.Average

6. Summarizing Qualitative Data Tabular Presentation Frequency Distribution Relative Frequency Distribution Percent Frequency Distribution Graphical Presentation Bar Graph Pie Chart

7. Frequency Distribution A frequency distribution for qualitative data is a tabular summary of a set of data showing all categories and the number of elements that belong to each of the categories. The objective is to provide insights about the data that cannot be quickly obtained by looking only at the original (raw) data.

8. Frequency Distribution Rating Frequency Poor 2 Below Average 3 Average 5 Above Average 9 Excellent 1 Total 20 Example: Marada Inn

9. 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 that class/Sum of all frequencies A relative frequency distribution is a tabular summary of a set of data showing the relative frequency for each class.

10. Percent Frequency Distribution The percent frequency of a class is the relative frequency multiplied by 100%. A percent frequency distribution is a tabular summary of a set of data showing the percent frequency for each class.

11. Example: Marada Inn Relative Frequency and Percent Frequency Distributions RatingRelative Freq. Percent Freq.,% Poor 2/20 =.10 10 Below Average 3/20 =.15 15 Average 5/20 =.25 25 Above Average 9/20 =.45 45 Excellent 1/20 =.05 5 Total 1.00 100

12. Bar Graph A bar graph is a graphical device for depicting qualitative data that have been summarized in a frequency, relative frequency, or percent frequency distribution.

13. Bar Graph On the horizontal axis we specify the labels 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.

14. Example: Marada Inn Bar Graph 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 Poor Below Average Below Average Above Average Above Average Excellent Frequency Rating

15. 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.

16. Example: Marada Inn Pie Chart Average 25% Average 25% Below Average 15% Below Average 15% Poor 10% Poor 10% Above Average 45% Above Average 45% Exc. 5% Exc. 5% Ratings

17. Example: Hudson Auto Repair The manager of Hudson 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, rounded to the nearest dollar, are listed below.

18. Summarizing Quantitative Data Tabular Presentation Frequency Distribution Relative Frequency Distribution Percent Frequency Distribution Cumulative Frequency Distribution Cumulative Relative Frequency Distribution Cumulative Percent Frequency Distribution Graphical Presentation Histogram Ogive

19. Frequency Distribution For quantitative data, an interval that includes all the values that fall within two numbers, the lower and upper limits, is called a class. The classes are non-overlapping. Frequencies (f)give the number of values that belong to different classes.

20. Guidelines Use between 5 and 20 classes. Larger (smaller) data sets usually require a larger (fewer) number of classes. Use classes of equal width. Approximate class width =

21. Example: Hudson Auto Repair Frequency Distribution If we choose six classes, approximate class width = (109 - 52)/6 = 9.5  10 Cost ($)Frequency 50-59 2 60-69 13 70-79 16 80-89 7 90-99 7 100-109 5 Total 50

22. Relative and Percent Frequency Relative Frequency of a Class = Frequency of that class/Sum of all frequencies Percent Frequency = (Relative Frequency)*100%

23. Relative Frequency and Percent Frequency Distributions Cost ($) Relative Freq. Percent Freq.,% 50-59 2/50 =.04 4 60-69 13/50 =.2626 70-79 16/50 =.3232 80-89 7/50 =.1414 90-99 7/50 =.1414 100-109 5/50 =.1010 Total 1.00 100 Example: Hudson Auto Repair

24. Cumulative Distribution The cumulative frequency (or cumulative relative frequency or cumulative percent frequency) distribution shows the number of items (or the proportion of items or the percentage of items) with values less than or equal to the upper limit of each class.

25. Example: Hudson Auto Repair Cumulative Distributions Cost, $ Cum.Freq.Cum.Rel.Freq.Cum.Perc.Freq. < 59 2.04 4 < 69 15.30 30 < 79 31.62 62 < 89 38.76 76 < 99 45.90 90 < 109 50 1.00 100

26. Histogram The variable of interest is placed on the horizontal axis and the frequency, relative frequency, or percent frequency is placed on the vertical axis. A rectangle is drawn above each class interval with its height corresponding to the interval’s frequency, relative frequency, or percent frequency. Unlike a bar graph, a histogram has no natural separation between rectangles of adjacent classes.

27. Example: Hudson Auto Repair Histogram 2 2 4 4 6 6 8 8 10 12 14 16 18 Frequency 50 60 70 80 90 100 110 Cost ($)

28. Ogive An ogive is a graph of a cumulative frequency (or cumulative relative frequency or cumulative percent frequency) distribution. The data values (class limits) are shown on the horizontal axis. An ogive is drawn by joining with straight lines the dots marked above the upper limits of classes at heights equal to the cumulative frequencies. Ogive starts at the lower limit of the first class and ends at the upper limit of the last class.

29. Example: Hudson Auto Repair Ogive 10 20 30 40 50 Cumulative Frequency 50 60 70 80 90 100 110 Cost ($)