1. Chapter 2, Part A Descriptive Statistics: Tabular and Graphical Presentations n Summarizing Categorical Data n Summarizing Quantitative Data Categorical data use labels or names Categorical data use labels or names to identify categories of like items. to identify categories of like items. Categorical data use labels or names Categorical data use labels or names to identify categories of like items. to identify categories of like items. Quantitative data are numerical values Quantitative data are numerical values that indicate how much or how many. that indicate how much or how many. Quantitative data are numerical values Quantitative data are numerical values that indicate how much or how many. that indicate how much or how many.

2. Summarizing Categorical Data n Frequency Distribution n Relative Frequency Distribution n Percent Frequency Distribution n Bar Chart n Pie Chart

3. A frequency distribution is a tabular summary of A frequency distribution is a tabular summary of data showing the frequency (or number) of items data showing the frequency (or number) of items in each of several non-overlapping classes. in each of several non-overlapping classes. A frequency distribution is a tabular summary of A frequency distribution is a tabular summary of data showing the frequency (or number) of items data showing the frequency (or number) of items in each of several non-overlapping classes. in each of several non-overlapping classes. The objective is to provide insights about the data The objective is to provide insights about the data that cannot be quickly obtained by looking only at that cannot be quickly obtained by looking only at the original data. the original data. The objective is to provide insights about the data The objective is to provide insights about the data that cannot be quickly obtained by looking only at that cannot be quickly obtained by looking only at the original data. the original data. Frequency Distribution

4. Guests staying at the Tanoa 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 guests are: Below Average Below Average Above Average Above Average Average Average Above Average Above Average Average Average Above Average Above Average Average Average Above Average Above Average Below Average Below Average Poor Poor Excellent Excellent Above Average Above Average Average Average Above Average Above Average Below Average Below Average Poor Poor Above Average Above Average Average Average Frequency Distribution n Example: Tanoa Hotel

5. Frequency Distribution Poor Below Average Average Above Average Excellent 2 3 5 9 1 1 Total 20 RatingFrequency n Example: Tanoa Hotel

6. The relative frequency of a class is the fraction or The relative frequency of a class is the fraction or proportion of the total number of data items proportion of the total number of data items belonging to the class. belonging to the class. The relative frequency of a class is the fraction or The relative frequency of a class is the fraction or proportion of the total number of data items proportion of the total number of data items belonging to the class. belonging to the class. A relative frequency distribution is a tabular A relative frequency distribution is a tabular summary of a set of data showing the relative summary of a set of data showing the relative frequency for each class. frequency for each class. A relative frequency distribution is a tabular A relative frequency distribution is a tabular summary of a set of data showing the relative summary of a set of data showing the relative frequency for each class. frequency for each class. Relative Frequency Distribution

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

8. Relative Frequency and Percent Frequency Distributions Poor Below Average Average Above Average Excellent.10.10.15.15.25.25.45.45.05.05 Total 1.00 10 10 15 15 25 25 45 45 5 5 100 100 Relative RelativeFrequency Percent PercentFrequency Rating.10(100) = 10 1/20 =.05 n Example: Tanoa Hotel

9. Bar Chart A bar chart is a graphical device for depicting A bar chart is a graphical device for depicting qualitative data. qualitative data. On one axis (usually the horizontal axis), we specify On one axis (usually the horizontal axis), we specify the labels that are used for each of the classes. the labels that are used for each of the classes. A frequency, relative frequency, or percent frequency A frequency, relative frequency, or percent frequency scale can be used for the other axis (usually the scale can be used for the other axis (usually the vertical axis). vertical axis). Using a bar of fixed width drawn above each class Using a bar of fixed width drawn above each class label, we extend the height appropriately. label, we extend the height appropriately. The bars are separated to emphasize the fact that each The bars are separated to emphasize the fact that each class is a separate category. class is a separate category.

10. Poor Below Average Below Average Above Average Above Average Excellent Frequency Rating Bar Chart 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 Tanoa Hotel Quality Ratings

11. Pie Chart The pie chart is a commonly used graphical device The pie chart is a commonly used graphical device for presenting relative frequency and percent for presenting relative frequency and percent frequency distributions for categorical data. frequency distributions for categorical data. n First draw a circle; then use the relative frequencies to subdivide the circle into sectors that correspond to to subdivide the circle into sectors that correspond to the relative frequency for each class. the relative frequency for each class. n Since there are 360 degrees in a circle, a class with a relative frequency of.25 would consume.25(360) = 90 relative frequency of.25 would consume.25(360) = 90 degrees of the circle. degrees of the circle.

12. Below Average 15% Below Average 15% Average 25% Average 25% Above Average 45% Above Average 45% Poor 10% Poor 10% Excellent 5% Excellent 5% Tanoa HotelQuality Ratings Pie Chart

13. n Insights Gained from the Preceding Pie Chart Example: Tanoa Hotel One-half of the customers surveyed gave Marada One-half of the customers surveyed gave Marada a quality rating of “above average” or “excellent” a quality rating of “above average” or “excellent” (looking at the left side of the pie). This might (looking at the left side of the pie). This might please the manager. please the manager. For each customer who gave an “excellent” rating, For each customer who gave an “excellent” rating, there were two customers who gave a “poor” there were two customers who gave a “poor” rating (looking at the top of the pie). This should rating (looking at the top of the pie). This should displease the manager. displease the manager.

14. Summarizing Quantitative Data n Frequency Distribution n Relative Frequency and Percent Frequency Distributions Percent Frequency Distributions n Dot Plot n Histogram n Cumulative Distributions n Ogive

15. The manager of Nadi Auto would like to gain a better understanding of the cost of parts used in the engine tune-ups performed in the shop. She examines 50 customer invoices for tune-ups. The costs of parts, rounded to the nearest dollar, are listed on the next slide. n Example: Nadi Auto Repair Frequency Distribution

16. Sample of Parts Cost($) for 50 Tune-ups Frequency Distribution n Example: Nadi Auto Repair

17. Frequency Distribution 2. Determine the width of each class. 3. Determine the class limits. 1. Determine the number of non-overlapping classes. The three steps necessary to define the classes for a frequency distribution with quantitative data are:

18. Frequency Distribution n Guidelines for Determining the Number of Classes Use between 5 and 20 classes. Use between 5 and 20 classes. Data sets with a larger number of elements Data sets with a larger number of elements usually require a larger number of classes. usually require a larger number of classes. Smaller data sets usually require fewer classes. Smaller data sets usually require fewer classes. The goal is to use enough classes to show the variation in the data, but not so many classes that some contain only a few data items. The goal is to use enough classes to show the variation in the data, but not so many classes that some contain only a few data items.

19. Frequency Distribution n Guidelines for Determining the Width of Each Class Use classes of equal width. Use classes of equal width. Approximate Class Width = Approximate Class Width = Making the classes the same width reduces the chance of inappropriate interpretations. inappropriate interpretations. Making the classes the same width reduces the chance of inappropriate interpretations. inappropriate interpretations.

20. n Note on Number of Classes and Class Width In practice, the number of classes and the In practice, the number of classes and the appropriate class width are determined by trial appropriate class width are determined by trial and error. and error. Once a possible number of classes is chosen, the Once a possible number of classes is chosen, the appropriate class width is found. appropriate class width is found. The process can be repeated for a different The process can be repeated for a different number of classes. number of classes. Frequency Distribution Ultimately, the analyst uses judgment to Ultimately, the analyst uses judgment to determine the combination of the number of determine the combination of the number of classes and class width that provides the best classes and class width that provides the best frequency distribution for summarizing the data. frequency distribution for summarizing the data.

21. Frequency Distribution n Guidelines for Determining the Class Limits Class limits must be chosen so that each data Class limits must be chosen so that each data item belongs to one and only one class. item belongs to one and only one class. The lower class limit identifies the smallest The lower class limit identifies the smallest possible data value assigned to the class. possible data value assigned to the class. The upper class limit identifies the largest The upper class limit identifies the largest possible data value assigned to the class. possible data value assigned to the class. The appropriate values for the class limits The appropriate values for the class limits depend on the level of accuracy of the data. depend on the level of accuracy of the data. An open-end class requires only a lower class limit or an upper class limit. An open-end class requires only a lower class limit or an upper class limit.

22. Frequency Distribution If we choose six classes: 50-59 50-59 60-69 60-69 70-79 70-79 80-89 80-89 90-99 90-99 100-109 100-109 2 2 13 13 16 16 7 7 5 5 Total 50 Parts Cost ($) Frequency Approximate Class Width = (109 - 52)/6 = 9.5  10 n Example: Nadi Auto Repair

23. Relative Frequency and Percent Frequency Distributions 50-59 50-59 60-69 60-69 70-79 70-79 80-89 80-89 90-99 90-99 100-109 100-109 Parts Cost ($).04.04.26.26.32.32.14.14.10.10 Total 1.00 Relative RelativeFrequency 4 26 26 32 32 14 14 10 10 100 100 Percent Frequency Frequency 2/502/50.04(100).04(100) n Example: Nadi Auto Repair Percent frequency is the relative frequencymultiplied by 100. by 100.Percent frequency is the relative frequencymultiplied by 100. by 100.

24. Only 4% of the parts costs are in the $50-59 class. Only 4% of the parts costs are in the $50-59 class. The greatest percentage (32% or almost one-third) The greatest percentage (32% or almost one-third) of the parts costs are in the $70-79 class. of the parts costs are in the $70-79 class. 30% of the parts costs are under $70. 30% of the parts costs are under $70. 10% of the parts costs are $100 or more. 10% of the parts costs are $100 or more. Insights Gained from the % Frequency Distribution: Relative Frequency and Percent Frequency Distributions n Example: Nadi Auto Repair

25. Dot Plot n One of the simplest graphical summaries of data is a dot plot. n A horizontal axis shows the range of data values. n Then each data value is represented by a dot placed above the axis.

26. Dot Plot 5060708090100110 50 60 70 80 90 100 110 Cost ($) Tune-up Parts Cost n Example: Nadi Auto Repair

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

28. Histogram 2 2 4 4 6 6 8 8 10 12 14 16 18 Parts Cost ($) Parts Cost ($) Frequency 50  59 60  69 70  79 80  89 90  99 100-110 Tune-up Parts Cost n Example: Nadi Auto Repair

29. n Symmetric Histograms Showing Skewness Relative Frequency.05.10.15.20.25.30.35 0 0 Left tail is the mirror image of the right tail Left tail is the mirror image of the right tail Examples: heights and weights of people Examples: heights and weights of people

30. Histograms Showing Skewness n Moderately Skewed Left Relative Frequency.05.10.15.20.25.30.35 0 0 A longer tail to the left A longer tail to the left Example: exam scores Example: exam scores

31. n Moderately Right Skewed Histograms Showing Skewness Relative Frequency.05.10.15.20.25.30.35 0 0 A Longer tail to the right A Longer tail to the right Example: housing values Example: housing values

32. Histograms Showing Skewness n Highly Skewed Right Relative Frequency.05.10.15.20.25.30.35 0 0 A very long tail to the right A very long tail to the right Example: executive salaries Example: executive salaries

33. Cumulative frequency distribution  shows the Cumulative frequency distribution  shows the number of items with values less than or equal to the number of items with values less than or equal to the upper limit of each class.. upper limit of each class.. Cumulative frequency distribution  shows the Cumulative frequency distribution  shows the number of items with values less than or equal to the number of items with values less than or equal to the upper limit of each class.. upper limit of each class.. Cumulative relative frequency distribution – shows Cumulative relative frequency distribution – shows the proportion of items with values less than or the proportion of items with values less than or equal to the upper limit of each class. equal to the upper limit of each class. Cumulative relative frequency distribution – shows Cumulative relative frequency distribution – shows the proportion of items with values less than or the proportion of items with values less than or equal to the upper limit of each class. equal to the upper limit of each class. Cumulative Distributions Cumulative percent frequency distribution – shows Cumulative percent frequency distribution – shows the percentage of items with values less than or the percentage of items with values less than or equal to the upper limit of each class. equal to the upper limit of each class. Cumulative percent frequency distribution – shows Cumulative percent frequency distribution – shows the percentage of items with values less than or the percentage of items with values less than or equal to the upper limit of each class. equal to the upper limit of each class.

34. Cumulative Distributions The last entry in a cumulative frequency distribution The last entry in a cumulative frequency distribution always equals the total number of observations. always equals the total number of observations. The last entry in a cumulative relative frequency The last entry in a cumulative relative frequency distribution always equals 1.00. distribution always equals 1.00. The last entry in a cumulative percent frequency The last entry in a cumulative percent frequency distribution always equals 100. distribution always equals 100.

35. Cumulative Distributions n Nadi Auto Repair < 59 < 59 < 69 < 69 < 79 < 79 < 89 < 89 < 99 < 99 < 109 Cost ($) Cumulative CumulativeFrequency RelativeFrequency CumulativePercent Frequency Frequency 2 2 15 15 31 31 38 38 45 45 50 50.04.04.30.30.62.62.76.76.90.90 1.00 1.00 4 4 30 30 62 62 76 76 90 90 100 100 2 + 13 15/5015/50.30(100).30(100)

36. Ogive n An ogive is a graph of a cumulative distribution. n The data values are shown on the horizontal axis. n Shown on the vertical axis are the: cumulative frequencies, or cumulative frequencies, or cumulative relative frequencies, or cumulative relative frequencies, or cumulative percent frequencies cumulative percent frequencies n The frequency (one of the above) of each class is plotted as a point. n The plotted points are connected by straight lines.

37. Because the class limits for the parts-cost data are 50-59, 60-69, and so on, there appear to be one-unit gaps from 59 to 60, 69 to 70, and so on. Because the class limits for the parts-cost data are 50-59, 60-69, and so on, there appear to be one-unit gaps from 59 to 60, 69 to 70, and so on. Ogive These gaps are eliminated by plotting points halfway between the class limits. These gaps are eliminated by plotting points halfway between the class limits. Thus, 59.5 is used for the 50-59 class, 69.5 is used for the 60-69 class, and so on. Thus, 59.5 is used for the 50-59 class, 69.5 is used for the 60-69 class, and so on. n Nadi Auto Repair

38. Parts Parts Cost ($) Parts Parts Cost ($) 20 40 60 80 100 Cumulative Percent Frequency 50 60 70 80 90 100 110 (89.5, 76) Ogive with Cumulative Percent Frequencies Tune-up Parts Cost n Example: Nadi Auto Repair

40. End of Chapter 2, Part A