1. 1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University

2. 2 2 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 2, Part A Descriptive Statistics: Tabular and Graphical Presentations n Summarizing Qualitative Data n Summarizing Quantitative Data n Qualitative Data – uses labels or names to identify categories of like items (nominal or ordinal). n Quantitative Data – uses numeric values that indicate how much or how many (interval or ratio).

3. 3 3 Slide © 2008 Thomson South-Western. All Rights Reserved Summarizing Qualitative Data n Frequency Distribution n Relative Frequency Distribution n Percent Frequency Distribution n Bar Graphs n Pie Charts You will be working with three types of raw data distributions: 1. Frequency distribution – raw data 2. Relative frequency distribution - fractions 3. Percent frequency distribution - percents

4. 4 4 Slide © 2008 Thomson South-Western. All Rights Reserved 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

5. 5 5 Slide © 2008 Thomson South-Western. All Rights Reserved 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

6. 6 6 Slide © 2008 Thomson South-Western. All Rights Reserved Frequency Distribution Poor Below Average Average Above Average Excellent 2 3 5 9 1 Total 20 RatingFrequency

7. 7 7 Slide © 2008 Thomson South-Western. All Rights Reserved 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

8. 8 8 Slide © 2008 Thomson South-Western. All Rights Reserved 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.

9. 9 9 Slide © 2008 Thomson South-Western. All Rights Reserved Relative Frequency and Percent Frequency Distributions Poor Below Average Average Above Average Excellent.10.15.25.45.05 Total 1.00 10 15 25 45 5 100 Relative RelativeFrequency Percent PercentFrequency Rating.10(100) = 10 1/20 =.05 fractionpercent

10. 10 Slide © 2008 Thomson South-Western. All Rights Reserved Bar Graph A bar graph is a graphical device for depicting A bar graph is a graphical device for depicting qualitative data summarized in a frequency, relative qualitative data summarized in a frequency, relative frequency, or percent frequency distribution. frequency, or percent frequency distribution. On one axis (usually the horizontal x axis), we specify On one axis (usually the horizontal x 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 y axis). vertical y 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. For qualitative data, the bars are separated to For qualitative data, the bars are separated to emphasize the fact that each class is a separate category. emphasize the fact that each class is a separate category.

11. 11 Slide © 2008 Thomson South-Western. All Rights Reserved Poor Below Average Below Average Above Average Above Average Excellent Frequency Rating Bar Graph 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 Marada Inn Quality Ratings Equal widths but not equal fractions or percents.

12. 12 Slide © 2008 Thomson South-Western. All Rights Reserved Bar Graphs n The number of classes in a frequency distribution is the same as the number of categories found in the data. n Classes with frequencies of 5% or less can be grouped into an aggregate class called “other.” n The sum of the frequencies in any frequency distribution always equals the number of observations. n The sum of the relative frequencies in any relative frequency distribution always equals 1.00. n The sum of the percentages in a percent frequency distribution always equals 100%.

13. 13 Slide © 2008 Thomson South-Western. All Rights Reserved 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 qualitative data. frequency distributions for 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. 14 Slide © 2008 Thomson South-Western. All Rights Reserved 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 Could be relative frequency

15. 15 Slide © 2008 Thomson South-Western. All Rights Reserved 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. 16 Slide © 2008 Thomson South-Western. All Rights Reserved Summarizing Quantitative Data n Frequency Distribution n Relative Frequency Distribution n Percent Frequency Distribution n Histogram n Cumulative Distributions n Ogive

17. 17 Slide © 2008 Thomson South-Western. All Rights Reserved 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.

18. 18 Slide © 2008 Thomson South-Western. All Rights Reserved Example: Hudson Auto Repair n Sample of Parts Cost($) for 50 Tune-ups

19. 19 Slide © 2008 Thomson South-Western. All Rights Reserved Frequency Distribution n Frequency Distribution – a tabular summary of data showing the number (frequency) of items in each of several non-overlapping classes. n Three steps to define the classes with quantitative data: 1. Determine the number of non-overlapping classes. 1. Determine the number of non-overlapping classes. 2. Determine the width of each class. 2. Determine the width of each class. 3. Determine the class limits. 3. Determine the class limits.

20. 20 Slide © 2008 Thomson South-Western. All Rights Reserved 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. Use enough classes to show the variation in Use enough classes to show the variation in the data. the data. Do not use so many classes that some contain Do not use so many classes that some contain only a few data items. only a few data items.

21. 21 Slide © 2008 Thomson South-Western. All Rights Reserved 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 =

22. 22 Slide © 2008 Thomson South-Western. All Rights Reserved Frequency Distribution n Ultimately, the analyst uses judgment to determine the combination of the number of classes and class width that provides the best frequency distribution for summarizing the data. n In developing frequency distributions for qualitative data, we do not need to specify class limits because each data item naturally falls into a separate class. But with quantitative data, class limits are necessary to determine where each data value belongs. n The class limits define the smallest and largest data values that can be assigned to a class. n The class midpoint is the value halfway between the upper and lower class limits.

23. 23 Slide © 2008 Thomson South-Western. All Rights Reserved Frequency Distribution For Hudson Auto Repair, if we choose six classes: 50-59 60-69 70-79 80-89 90-99 100-109 2 13 16 7 7 5 Total 50 Parts Cost ($) Frequency Approximate Class Width = (109 - 52)/6 = 9.5  10

24. 24 Slide © 2008 Thomson South-Western. All Rights Reserved Relative Frequency and Percent Frequency Distributions n Relative frequency is the proportion of the observations belong to a class. n With n observations, Frequency of the class Relative frequency of a class = n The percent frequency of a class is the relative frequency multiplied by 100. The percent frequency of a class is the relative frequency multiplied by 100.

25. 25 Slide © 2008 Thomson South-Western. All Rights Reserved 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 ($).04.26.32.14.14.10 Total 1.00 Relative RelativeFrequency 4 26 32 14 14 10 100 Percent Frequency Frequency 2/502/50.04(100).04(100)

26. 26 Slide © 2008 Thomson South-Western. All Rights Reserved 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

27. 27 Slide © 2008 Thomson South-Western. All Rights Reserved 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.

28. 28 Slide © 2008 Thomson South-Western. All Rights Reserved 5060708090100110 50 60 70 80 90 100 110 Cost ($) Dot Plot (Interval or Ratio Data) Tune-up Parts Cost

29. 29 Slide © 2008 Thomson South-Western. All Rights Reserved Histogram (Interval or Ratio Data) 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 because all values between the lower limit of the because all values between the lower limit of the first class and the upper limit of the last class are first class and the upper limit of the last class are possible. possible.

30. 30 Slide © 2008 Thomson South-Western. All Rights Reserved 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

31. 31 Slide © 2008 Thomson South-Western. All Rights Reserved n Symmetric 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 Histogram (Interval or Ratio Data) Relative Frequency.05.10.15.20.25.30.35 0 0

32. 32 Slide © 2008 Thomson South-Western. All Rights Reserved Histogram (Interval or Ratio Data) n Moderately Skewed Left A longer tail to the left A longer tail to the left Example: exam scores Example: exam scores Relative Frequency.05.10.15.20.25.30.35 0 0

33. 33 Slide © 2008 Thomson South-Western. All Rights Reserved n Moderately Right Skewed A longer tail to the right A longer tail to the right Example: housing values Example: housing values Histogram (Interval or Ratio Data) Relative Frequency.05.10.15.20.25.30.35 0 0

34. 34 Slide © 2008 Thomson South-Western. All Rights Reserved Histogram (Interval or Ratio Data) n Highly Skewed Right (typical of business and economic data) A very long tail to the right A very long tail to the right Example: executive salaries Example: executive salaries Relative Frequency.05.10.15.20.25.30.35 0 0

35. 35 Slide © 2008 Thomson South-Western. All Rights Reserved 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.

36. 36 Slide © 2008 Thomson South-Western. All Rights Reserved Cumulative Distributions n Hudson Auto Repair <59 <69 <79 <89 <99 <109 Cost ($) Cumulative CumulativeFrequency RelativeFrequency CumulativePercent Frequency Frequency 2 15 31 38 45 50.04.30.62.76.90 1.00 4 30 62 76 90 100 2 + 13 15/5015/50.30(100).30(100)

37. 37 Slide © 2008 Thomson South-Western. All Rights Reserved Ogive n An ogive is a graph of a cumulative distribution. n The data values are shown on the horizontal x axis. n Shown on the vertical y 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 the mid-point of each class. n The plotted points are connected by straight lines. n The last entry in a cumulative frequency distribution always equals the total number of observations, or 1.00 (relative frequency), or 100% (percent frequency).

38. 38 Slide © 2008 Thomson South-Western. All Rights Reserved 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 Hudson Auto Repair

39. 39 Slide © 2008 Thomson South-Western. All Rights Reserved 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 Cumulative Percent Frequencies Tune-up Parts Cost

40. 40 Slide © 2008 Thomson South-Western. All Rights Reserved End of Chapter 2, Part A