1 1 Slide 統計學 Fall 2003 授課教師：統計系余清祥 日期： 2003 年 9 月 23 日 第二週：敘述性統計量

2 2 Slide Chapter 2 Descriptive Statistics: Tabular and Graphical Methods n Summarizing Qualitative Data n Summarizing Quantitative Data n Exploratory Data Analysis n Crosstabulations and Scatter Diagrams and Scatter Diagrams

3 3 Slide Summarizing Qualitative Data n Frequency Distribution n Relative Frequency n Percent Frequency Distribution n Bar Graph n Pie Chart

4 4 Slide Frequency Distribution n A frequency distribution is a tabular summary of data showing the frequency (or number) of items in each of several nonoverlapping classes. n The objective is to provide insights about the data that cannot be quickly obtained by looking only at the original data.

5 5 Slide 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 guests are shown below. Below Average AverageAbove Average Above Average Above AverageAbove Average Above Average Below Average Below Average Average Poor Poor Above Average Below Average Below Average Average Poor Poor Above Average ExcellentAbove Average Average Above AverageAverage Above Average Average

6 6 Slide n Frequency Distribution Rating Frequency Poor 2 Below Average 3 Average 5 Above Average 9 Excellent 1 Total 20 Example: Marada Inn

7 7 Slide Relative Frequency Distribution n The relative frequency of a class is the fraction or proportion of the total number of data items belonging to the class. n A relative frequency distribution is a tabular summary of a set of data showing the relative frequency for each class.

8 8 Slide Percent Frequency Distribution n The percent frequency of a class is the relative frequency multiplied by 100. n A percent frequency distribution is a tabular summary of a set of data showing the percent frequency for each class.

9 9 Slide Example: Marada Inn n Relative Frequency and Percent Frequency Distributions Relative Percent Relative Percent RatingFrequencyFrequency Poor.1010 Below Average.1515 Average.2525 Above Average.4545 Excellent.05 5 Total

10 Slide Bar Graph n A bar graph is a graphical device for depicting qualitative data that have been summarized in a frequency, relative frequency, or percent frequency distribution. n On the horizontal axis we specify the labels that are used for each of the classes. n A frequency, relative frequency, or percent frequency scale can be used for the vertical axis. n Using a bar of fixed width drawn above each class label, we extend the height appropriately. n The bars are separated to emphasize the fact that each class is a separate category.

11 Slide Example: Marada Inn n Bar Graph Poor Below Average Below Average Above Average Above Average Excellent Frequency Rating

12 Slide Pie Chart n The pie chart is a commonly used graphical device for presenting relative frequency distributions for qualitative data. n First draw a circle; then use the relative frequencies to subdivide the circle into sectors that correspond to 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 degrees of the circle.

13 Slide Example: Marada Inn n 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% Quality Ratings

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

15 Slide Summarizing Quantitative Data n Frequency Distribution n Relative Frequency and Percent Frequency Distributions n Dot Plot n Histogram n Cumulative Distributions n Ogive

16 Slide Example: Hudson Auto Repair 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, rounded to the nearest dollar, are listed below.

17 Slide Frequency Distribution n Guidelines for Selecting Number of Classes Use between 5 and 20 classes. Use between 5 and 20 classes. Data sets with a larger number of elements usually require a larger number of classes. Data sets with a larger number of elements usually require a larger number of classes. Smaller data sets usually require fewer classes. Smaller data sets usually require fewer classes.

18 Slide Frequency Distribution n Guidelines for Selecting Width of Classes Use classes of equal width. Use classes of equal width. Approximate Class Width = Approximate Class Width =

19 Slide Example: Hudson Auto Repair n Frequency Distribution If we choose six classes: Approximate Class Width = ( )/6 = 9.5  10 Approximate Class Width = ( )/6 = 9.5  10 Cost ($) Frequency Total 50 Total 50

20 Slide n Relative Frequency and Percent Frequency Distributions Relative Percent Relative Percent Cost ($) Frequency Frequency Cost ($) Frequency Frequency Total Total Example: Hudson Auto Repair

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

22 Slide 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.

23 Slide Example: Hudson Auto Repair n Dot Plot Cost ($)

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

25 Slide Example: Hudson Auto Repair n Histogram Parts Cost ($) Parts Cost ($) Frequency

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

27 Slide Example: Hudson Auto Repair n Cumulative Distributions Cumulative Cumulative Cumulative Relative Percent Cumulative Relative Percent Cost ($) Frequency Frequency Frequency < < < < < < < < < < < <

28 Slide 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.

29 Slide Example: Hudson Auto Repair n Ogive 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. 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 class, 69.5 is used for the class, and so on. Thus, 59.5 is used for the class, 69.5 is used for the class, and so on.

30 Slide Example: Hudson Auto Repair n Ogive with Cumulative Percent Frequencies Parts Parts Cost ($) Parts Parts Cost ($) Cumulative Percent Frequency

31 Slide Exploratory Data Analysis n The techniques of exploratory data analysis consist of simple arithmetic and easy-to-draw pictures that can be used to summarize data quickly. n One such technique is the stem-and-leaf display.

32 Slide Stem-and-Leaf Display n A stem-and-leaf display shows both the rank order and shape of the distribution of the data. n It is similar to a histogram on its side, but it has the advantage of showing the actual data values. n The first digits of each data item are arranged to the left of a vertical line. n To the right of the vertical line we record the last digit for each item in rank order. n Each line in the display is referred to as a stem. n Each digit on a stem is a leaf.

33 Slide Example: Hudson Auto Repair n Stem-and-Leaf Display

34 Slide Stretched Stem-and-Leaf Display n If we believe the original stem-and-leaf display has condensed the data too much, we can stretch the display by using two more stems for each leading digit(s). n Whenever a stem value is stated twice, the first value corresponds to leaf values of 0-4, and the second values corresponds to values of 5-9.

35 Slide Example: Hudson Auto Repair n Stretched Stem-and-Leaf Display

36 Slide Stem-and-Leaf Display n Leaf Units A single digit is used to define each leaf. A single digit is used to define each leaf. In the preceding example, the leaf unit was 1. In the preceding example, the leaf unit was 1. Leaf units may be 100, 10, 1, 0.1, and so on. Leaf units may be 100, 10, 1, 0.1, and so on. Where the leaf unit is not shown, it is assumed to equal 1. Where the leaf unit is not shown, it is assumed to equal 1.

37 Slide Example: Leaf Unit = 0.1 If we have data with values such as a stem-and-leaf display of these data will be Leaf Unit =

38 Slide Example: Leaf Unit = 10 If we have data with values such as a stem-and-leaf display of these data will be Leaf Unit =

39 Slide Crosstabulations and Scatter Diagrams n Thus far we have focused on methods that are used to summarize the data for one variable at a time. n Often a manager is interested in tabular and graphical methods that will help understand the relationship between two variables. n Crosstabulation and a scatter diagram are two methods for summarizing the data for two (or more) variables simultaneously.

40 Slide Crosstabulation n Crosstabulation is a tabular method for summarizing the data for two variables simultaneously. n Crosstabulation can be used when: One variable is qualitative and the other is quantitative One variable is qualitative and the other is quantitative Both variables are qualitative Both variables are qualitative Both variables are quantitative Both variables are quantitative n The left and top margin labels define the classes for the two variables.

41 Slide Example: Finger Lakes Homes n Crosstabulation The number of Finger Lakes homes sold for each style and price for the past two years is shown below. Price Home Style Price Home Style Range Colonial Ranch Split A-Frame Total Range Colonial Ranch Split A-Frame Total < $99, < $99, > $99, > $99, Total Total

42 Slide Example: Finger Lakes Homes n Insights Gained from the Preceding Crosstabulation The greatest number of homes in the sample (19) are a split-level style and priced at less than or equal to $99,000. The greatest number of homes in the sample (19) are a split-level style and priced at less than or equal to $99,000. Only three homes in the sample are an A-Frame style and priced at more than $99,000. Only three homes in the sample are an A-Frame style and priced at more than $99,000.

43 Slide Crosstabulation: Row or Column Percentages n Converting the entries in the table into row percentages or column percentages can provide additional insight about the relationship between the two variables.

44 Slide Example: Finger Lakes Homes n Row Percentages Price Home Style Price Home Style Range Colonial Ranch Split A-Frame Total Range Colonial Ranch Split A-Frame Total < $99, < $99, > $99, > $99, Note: row totals are actually due to rounding. Note: row totals are actually due to rounding.

45 Slide Example: Finger Lakes Homes n Column Percentages Price Home Style Price Home Style Range Colonial Ranch Split A-Frame Range Colonial Ranch Split A-Frame < $99, < $99, > $99, > $99, Total Total

46 Slide Scatter Diagram n A scatter diagram is a graphical presentation of the relationship between two quantitative variables. n One variable is shown on the horizontal axis and the other variable is shown on the vertical axis. n The general pattern of the plotted points suggests the overall relationship between the variables.

47 Slide Scatter Diagram n A Positive Relationship xy

48 Slide Scatter Diagram n A Negative Relationship xy

49 Slide Scatter Diagram n No Apparent Relationship xy

50 Slide Example: Panthers Football Team n Scatter Diagram The Panthers football team is interested in investigating the relationship, if any, between interceptions made and points scored. x = Number of y = Number of x = Number of y = Number of Interceptions Points Scored Interceptions Points Scored

51 Slide Example: Panthers Football Team n Scatter Diagram y x Number of Interceptions 1 23 Number of Points Scored

52 Slide Example: Panthers Football Team n The preceding scatter diagram indicates a positive relationship between the number of interceptions and the number of points scored. n Higher points scored are associated with a higher number of interceptions. n The relationship is not perfect; all plotted points in the scatter diagram are not on a straight line.

53 Slide Tabular and Graphical Procedures Data Qualitative Data Quantitative Data Tabular TabularMethods MethodsGraphical Methods MethodsGraphical FrequencyFrequency Distribution Distribution Rel. Freq. Dist.Rel. Freq. Dist. % Freq. Dist.% Freq. Dist. CrosstabulationCrosstabulation Bar GraphBar Graph Pie ChartPie Chart FrequencyFrequency Distribution Distribution Rel. Freq. Dist.Rel. Freq. Dist. Cum. Freq. Dist.Cum. Freq. Dist. Cum. Rel. Freq.Cum. Rel. Freq. Distribution Distribution Stem-and-LeafStem-and-Leaf Display Display CrosstabulationCrosstabulation Dot PlotDot Plot HistogramHistogram OgiveOgive ScatterScatter Diagram Diagram

54 Slide End of Chapter 2