1 1 Slide © 2005 Thomson/South-Western Introduction to Statistics Chapter 2 Descriptive Statistics

2 2 Slide © 2005 Thomson/South-Western Slides Prepared by JOHN S. LOUCKS St. Edward’s University Slides Prepared by JOHN S. LOUCKS St. Edward’s University

3 3 Slide © 2005 Thomson/South-Western Chapter 2 Descriptive Statistics: Tabular and Graphical Presentations Part A n Summarizing Qualitative Data n Summarizing Quantitative Data

4 4 Slide © 2005 Thomson/South-Western Summarizing Qualitative Data n Frequency Distribution n Relative Frequency Distribution n Percent Frequency Distribution n Bar Graph n Pie Chart

5 5 Slide © 2005 Thomson/South-Western 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

6 6 Slide © 2005 Thomson/South-Western 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: 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

7 7 Slide © 2005 Thomson/South-Western Frequency Distribution Poor Below Average Average Above Average Excellent Total 20 RatingFrequency

8 8 Slide © 2005 Thomson/South-Western The relative frequency of a class is the fraction The relative frequency of a class is the fraction or 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 The relative frequency of a class is the fraction or 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

9 9 Slide © 2005 Thomson/South-Western 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.

10 Slide © 2005 Thomson/South-Western Relative Frequency and Percent Frequency Distributions Poor Below Average Average Above Average Excellent Total Relative RelativeFrequency Percent PercentFrequency Rating.10(100) = 10 1/20 =.05 Frequency

11 Slide © 2005 Thomson/South-Western Bar Graph A bar graph is a graphical device for depicting A bar graph 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.

12 Slide © 2005 Thomson/South-Western Poor Below Average Below Average Above Average Above Average Excellent Frequency Rating Bar Graph Marada Inn Quality Ratings

13 Slide © 2005 Thomson/South-Western Pie Chart The pie chart is a commonly used graphical device The pie chart is a commonly used graphical device for presenting relative frequency distributions for for presenting relative frequency distributions for qualitative data. qualitative data. n First draw a circle; then use the relative frequencies to subdivide the circle frequencies to subdivide the circle into sectors that correspond to the into sectors that correspond to the relative frequency for each class. relative frequency for each class. n Since there are 360 degrees in a circle, a class with a relative frequency of.25 would a class with a relative frequency of.25 would consume.25(360) = 90 degrees of the circle. consume.25(360) = 90 degrees of the circle.

14 Slide © 2005 Thomson/South-Western Below Average 15% Below Average 15% Average 25% Average 25% Above Average 45% Above Average 45% Poor 10% Poor 10% Excellent 5% Excellent 5% Marada InnQuality Ratings Marada Inn Quality Ratings Pie Chart

15 Slide © 2005 Thomson/South-Western n Insights Gained from the Preceding Pie Chart Example: Marada Inn 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.

16 Slide © 2005 Thomson/South-Western In Class Group Exercise NameFrequency Brown7 Davis6 Johnson10 Jones7 Smith12 Williams8 50 The Following are the voices gained by each representative of one class in a school

17 Slide © 2005 Thomson/South-Western 5 a NameFrequency Relative Frequency Percent Frequency Brown7.1414% Davis6.1212% Johnson % Jones7.1414% Smith % Williams8.1616% %

18 Slide © 2005 Thomson/South-Western 5 b

19 Slide © 2005 Thomson/South-Western 5c %Degrees Brown Davis Johnson Jones Smith Williams

20 Slide © 2005 Thomson/South-Western 5 d Most common: Smith, Johnson and Williams

21 Slide © 2005 Thomson/South-Western Summarizing Quantitative Data n Frequency Distribution n Relative Frequency and Percent Frequency Distributions n Dot Plot n Histogram n Cumulative Distributions n Ogive

22 Slide © 2005 Thomson/South-Western Example: Hudson Auto Repair The manager of Hudson Auto would like to have a better understanding of the cost of parts used in the engine tune-ups performed in the shop. He examines 50 customer invoices for tune-ups. The costs of parts, rounded to the nearest dollar, are listed on the next slide.

23 Slide © 2005 Thomson/South-Western Example: Hudson Auto Repair n Sample of Parts Cost for 50 Tune-ups

24 Slide © 2005 Thomson/South-Western 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 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

25 Slide © 2005 Thomson/South-Western 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 =

26 Slide © 2005 Thomson/South-Western Frequency Distribution For Hudson Auto Repair, if we choose six classes: Total 50 Parts Cost ($) Frequency Approximate Class Width = ( )/6 = 9.5  10

27 Slide © 2005 Thomson/South-Western Relative Frequency and Percent Frequency Distributions Parts Cost ($) Total 1.00 Relative RelativeFrequency Percent Frequency Frequency 2/50.04(100)

28 Slide © 2005 Thomson/South-Western 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. n Insights Gained from the Percent Frequency Distribution Relative Frequency and Percent Frequency Distributions

29 Slide © 2005 Thomson/South-Western 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.

30 Slide © 2005 Thomson/South-Western Cost ($) Dot Plot Tune-up Parts Cost

31 Slide © 2005 Thomson/South-Western 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.

32 Slide © 2005 Thomson/South-Western Histogram Total 50 Parts Cost ($) Frequency When plotting the Histogram, the points between When plotting the Histogram, the points between bars are the boundary limits The boundary limit is the mid value between the upper limit of the one class and the lower limit of the next class Boundary

33 Slide © 2005 Thomson/South-Western Histogram Parts Cost ($) Parts Cost ($) Frequency 49.5  Tune-up Parts Cost

34 Slide © 2005 Thomson/South-Western n Symmetric Left tail is the mirror image of the right tail Left tail is the mirror image of the right tail Histogram Relative Frequency

35 Slide © 2005 Thomson/South-Western Histogram n Moderately Skewed Left A longer tail to the left A longer tail to the left Relative Frequency

36 Slide © 2005 Thomson/South-Western n Moderately Right Skewed A Longer tail to the right A Longer tail to the right Histogram Relative Frequency

37 Slide © 2005 Thomson/South-Western Histogram n Highly Skewed Right A very long tail to the right A very long tail to the right Relative Frequency

38 Slide © 2005 Thomson/South-Western Cumulative frequency distribution  shows the Cumulative frequency distribution  shows the number of items with values less than or equal to number of items with values less than or equal to the upper limit of each class.. the 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 number of items with values less than or equal to the upper limit of each class.. 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 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.

39 Slide © 2005 Thomson/South-Western Cumulative Distributions n Hudson Auto Repair Less than 60 Less than 70 Less than 80 Less than 90 Less than 100 Less than 110 Cost ($) Cumulative CumulativeFrequency RelativeFrequency CumulativePercent Frequency Frequency /50.30(100)

40 Slide © 2005 Thomson/South-Western Cumulative Distributions n Hudson Auto Repair <59 <69 <79 <89 <99 <109 Cost ($) Cumulative CumulativeFrequency RelativeFrequency CumulativePercent Frequency Frequency /50.30(100)

41 Slide © 2005 Thomson/South-Western 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.

42 Slide © 2005 Thomson/South-Western 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 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. n Hudson Auto Repair

43 Slide © 2005 Thomson/South-Western Parts Parts Cost ($) Parts Parts Cost ($) Cumulative Percent Frequency (89.5, 76) Ogive with Cumulative Percent Frequencies Cumulative Percent Frequencies Tune-up Parts Cost

44 Slide © 2005 Thomson/South-Western Chapter 2 Descriptive Statistics: Tabular and Graphical Presentations Part B n Exploratory Data Analysis n Crosstabulations and Scatter Diagrams Scatter Diagrams x y

45 Slide © 2005 Thomson/South-Western Exploratory Data Analysis The techniques of exploratory data analysis consist of The techniques of exploratory data analysis consist of simple arithmetic and easy-to-draw pictures that can simple arithmetic and easy-to-draw pictures that can be used to summarize data quickly. be used to summarize data quickly. One such technique is the stem-and-leaf display. One such technique is the stem-and-leaf display.

46 Slide © 2005 Thomson/South-Western Stem-and-Leaf Display Each digit on a stem is a leaf. Each digit on a stem is a leaf. Each line in the display is referred to as a stem. Each line in the display is referred to as a stem. To the right of the vertical line we record the last To the right of the vertical line we record the last digit for each item in rank order. digit for each item in rank order. The first digits of each data item are arranged to the The first digits of each data item are arranged to the left of a vertical line. left of a vertical line. It is similar to a histogram on its side, but it has the It is similar to a histogram on its side, but it has the advantage of showing the actual data values. advantage of showing the actual data values. A stem-and-leaf display shows both the rank order A stem-and-leaf display shows both the rank order and shape of the distribution of the data. and shape of the distribution of the data.

47 Slide © 2005 Thomson/South-Western Example: Hudson Auto Repair The manager of Hudson Auto would like to have 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.

48 Slide © 2005 Thomson/South-Western Example: Hudson Auto Repair n Sample of Parts Cost for 50 Tune-ups

49 Slide © 2005 Thomson/South-Western Stem-and-Leaf Display a stem a leaf

50 Slide © 2005 Thomson/South-Western Crosstabulations and Scatter Diagrams Crosstabulation and a scatter diagram are two Crosstabulation and a scatter diagram are two methods for summarizing the data for two (or more) methods for summarizing the data for two (or more) variables simultaneously. variables simultaneously. Often a manager is interested in tabular and Often a manager is interested in tabular and graphical methods that will help understand the graphical methods that will help understand the relationship between two variables. relationship between two variables. Thus far we have focused on methods that are used Thus far we have focused on methods that are used to summarize the data for one variable at a time. to summarize the data for one variable at a time.

51 Slide © 2005 Thomson/South-Western Crosstabulation The left and top margin labels define the classes for The left and top margin labels define the classes for the two variables. the two variables. n Crosstabulation can be used when: one variable is qualitative and the other is one variable is qualitative and the other is quantitative, quantitative, both variables are qualitative, or both variables are qualitative, or both variables are quantitative. both variables are quantitative. A crosstabulation is a tabular summary of data for A crosstabulation is a tabular summary of data for two variables. two variables.

52 Slide © 2005 Thomson/South-Western Price Range A B C D Total < $99,000 > $99, Total Home Style Home Style Crosstabulation n Example: Finger Lakes Homes The number of Finger Lakes homes sold for each style and price for the past two years is shown below. quantitative variable variable qualitative

53 Slide © 2005 Thomson/South-Western PriceRange Total < $99,000 > $99, Total Home Style Home Style Crosstabulation Frequency distribution for the price variable Frequency distribution for the home style variable A B C D

54 Slide © 2005 Thomson/South-Western The general pattern of the plotted points suggests the The general pattern of the plotted points suggests the overall relationship between the variables. overall relationship between the variables. One variable is shown on the horizontal axis and the One variable is shown on the horizontal axis and the other variable is shown on the vertical axis. other variable is shown on the vertical axis. A scatter diagram is a graphical presentation of the A scatter diagram is a graphical presentation of the relationship between two quantitative variables. relationship between two quantitative variables. Scatter Diagram and Trend line A trend line is an approximation of the relationship. A trend line is an approximation of the relationship.

55 Slide © 2005 Thomson/South-Western Scatter Diagram n A Positive Relationship x y

56 Slide © 2005 Thomson/South-Western Scatter Diagram n A Negative Relationship x y

57 Slide © 2005 Thomson/South-Western Scatter Diagram n No Apparent Relationship x y

58 Slide © 2005 Thomson/South-Western Tabular and Graphical Procedures Qualitative Data Quantitative Data Tabular TabularMethods Methods Methods MethodsGraphical Methods MethodsGraphical Graphical Graphical FrequencyFrequency Distribution Distribution Rel. Freq. Dist.Rel. Freq. Dist. Percent Freq.Percent Freq. Distribution Distribution 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 DataData

59 Slide © 2005 Thomson/South-Western End of Chapter 2, Part B