1. MM207-Statistics Unit 2 Seminar-Descriptive Statistics Dr Bridgette Stevens AIM:BStevensKaplan (add me to your Buddy list) 1

2. Chapter Outline 2.1 Frequency Distributions and Their Graphs 2.2 More Graphs and Displays 2.3 Measures of Central Tendency 2.4 Measures of Variation 2.5 Measures of Position 2 Larson/Farber 4th ed.

3. Section 2.1 Frequency Distributions and Their Graphs 3 Larson/Farber 4th ed.

4. Frequency Distribution A table that shows classes or intervals of data with a count of the number of entries in each class. The frequency, f, of a class is the number of data entries in the class. Larson/Farber 4th ed. 4 ClassFrequency, f 1 – 55 6 – 108 11 – 156 16 – 208 21 – 255 26 – 304 Lower class limits Upper class limits Class width 6 – 1 = 5

5. Constructing a Frequency Distribution Larson/Farber 4th ed. 5 1.Decide on the number of classes. Usually between 5 and 20; otherwise, it may be difficult to detect any patterns. 2.Find the class width. Determine the range of the data. Divide the range by the number of classes. Round up to the next convenient number.

6. Constructing a Frequency Distribution 3.Find the class limits. You can use the minimum data entry as the lower limit of the first class. Find the remaining lower limits (add the class width to the lower limit of the preceding class). Find the upper limit of the first class. Remember that classes cannot overlap. Find the remaining upper class limits. Larson/Farber 4th ed. 6

7. Example: Constructing a Frequency Distribution The following sample data set lists the number of minutes 50 Internet subscribers spent on the Internet during their most recent session. Construct a frequency distribution that has seven classes. 50 40 41 17 11 7 22 44 28 21 19 23 37 51 54 42 86 41 78 56 72 56 17 7 69 30 80 56 29 33 46 31 39 20 18 29 34 59 73 77 36 39 30 62 54 67 39 31 53 44 Larson/Farber 4th ed. 7

8. Solution: Constructing a Frequency Distribution 1.Number of classes = 7 (given) 2.Find the class width Larson/Farber 4th ed. 8 Round up to 12 50 40 41 17 11 7 22 44 28 21 19 23 37 51 54 42 86 41 78 56 72 56 17 7 69 30 80 56 29 33 46 31 39 20 18 29 34 59 73 77 36 39 30 62 54 67 39 31 53 44

9. Solution: Constructing a Frequency Distribution Larson/Farber 4th ed. 9 Lower limit Upper limit 7 Class width = 12 3.Use 7 (minimum value) as first lower limit. Add the class width of 12 to get the lower limit of the next class. 7 + 12 = 19 Find the remaining lower limits. 19 31 43 55 67 79

10. Solution: Constructing a Frequency Distribution The upper limit of the first class is 18 (one less than the lower limit of the second class). Add the class width of 12 to get the upper limit of the next class. 18 + 12 = 30 Find the remaining upper limits. Larson/Farber 4th ed. 10 Lower limit Upper limit 7 19 31 43 55 67 79 Class width = 12 30 42 54 66 78 90 18

11. Solution: Constructing a Frequency Distribution 4.Make a tally mark for each data entry in the row of the appropriate class. 5.Count the tally marks to find the total frequency f for each class. Larson/Farber 4th ed. 11 ClassTallyFrequency, f 7 – 18 IIII I 6 19 – 30 IIII 10 31 – 42 IIII IIII III 13 43 – 54 IIII III 8 55 – 66 IIII 5 67 – 78 IIII I 6 79 – 90 II 2 Σf = 50

12. Determining the Midpoint Midpoint of a class Larson/Farber 4th ed. 12 ClassMidpointFrequency, f 7 – 186 19 – 3010 31 – 4213 Class width = 12

13. Determining the Relative Frequency Relative Frequency of a class Portion or percentage of the data that falls in a particular class. Larson/Farber 4th ed. 13 ClassFrequency, fRelative Frequency 7 – 186 19 – 3010 31 – 4213

14. Graphs of Frequency Distributions Frequency Histogram A bar graph that represents the frequency distribution. The horizontal scale is quantitative and measures the data values. The vertical scale measures the frequencies of the classes. Consecutive bars must touch. Larson/Farber 4th ed. 14 data values frequency

15. Example: Frequency Histogram Construct a frequency histogram for the Internet usage frequency distribution. Larson/Farber 4th ed. 15 Class Class boundariesMidpoint Frequency, f 7 – 18 6.5 – 18.512.56 19 – 3018.5 – 30.524.510 31 – 4230.5 – 42.536.513 43 – 5442.5 – 54.548.58 55 – 6654.5 – 66.560.55 67 – 7866.5 – 78.572.56 79 – 9078.5 – 90.584.52

16. Solution: Frequency Histogram (using Midpoints) Larson/Farber 4th ed. 16

17. Graphs of Frequency Distributions Frequency Polygon A line graph that emphasizes the continuous change in frequencies. Larson/Farber 4th ed. 17 data values frequency

18. Solution: Frequency Polygon Larson/Farber 4th ed. 18 You can see that the frequency of subscribers increases up to 36.5 minutes and then decreases. The graph should begin and end on the horizontal axis, so extend the left side to one class width before the first class midpoint and extend the right side to one class width after the last class midpoint.

19. Graphs of Frequency Distributions Relative Frequency Histogram Has the same shape and the same horizontal scale as the corresponding frequency histogram. The vertical scale measures the relative frequencies, not frequencies. Larson/Farber 4th ed. 19 data values relative frequency

20. Example: Relative Frequency Histogram Construct a relative frequency histogram for the Internet usage frequency distribution. Larson/Farber 4th ed. 20 Class Class boundaries Frequency, f Relative frequency 7 – 18 6.5 – 18.560.12 19 – 3018.5 – 30.5100.20 31 – 4230.5 – 42.5130.26 43 – 5442.5 – 54.580.16 55 – 6654.5 – 66.550.10 67 – 7866.5 – 78.560.12 79 – 9078.5 – 90.520.04

21. Solution: Relative Frequency Histogram 21 6.5 18.5 30.5 42.5 54.5 66.5 78.5 90.5 From this graph you can see that 20% of Internet subscribers spent between 18.5 minutes and 30.5 minutes online.

22. Graphs of Frequency Distributions Cumulative Frequency Graph or Ogive A line graph that displays the cumulative frequency of each class at its upper class boundary. The upper boundaries are marked on the horizontal axis. The cumulative frequencies are marked on the vertical axis. Larson/Farber 4th ed. 22 data values cumulative frequency

23. Solution: Ogive 23 6.5 18.5 30.5 42.5 54.5 66.5 78.5 90.5 From the ogive, you can see that about 40 subscribers spent 60 minutes or less online during their last session. The greatest increase in usage occurs between 30.5 minutes and 42.5 minutes.

24. Section 2.2 More Graphs and Displays Larson/Farber 4th ed. 24

25. Section 2.2 Objectives Graph quantitative data using stem-and-leaf plots and dot plots Graph qualitative data using pie charts and Pareto charts Graph paired data sets using scatter plots and time series charts Larson/Farber 4th ed. 25

26. Graphing Quantitative Data Sets Stem-and-leaf plot Each number is separated into a stem and a leaf. Similar to a histogram. Still contains original data values. Larson/Farber 4th ed. 26 Data: 21, 25, 25, 26, 27, 28, 30, 36, 36, 45 26 21 5 5 6 7 8 30 6 6 45

27. Example: Constructing a Stem-and- Leaf Plot The following are the numbers of text messages sent last month by the cellular phone users on one floor of a college dormitory. Display the data in a stem-and-leaf plot. Larson/Farber 4th ed. 27 155159 144 129 105 145 126 116 130 114 122 112 112 142 126 118118 108 122 121 109 140 126 119 113 117 118 109 109 119 139139 122 78 133 126 123 145 121 134 124 119 132 133 124 129 112 126 148 147

28. Solution: Constructing a Stem-and-Leaf Plot 28 Include a key to identify the values of the data. From the display, you can conclude that more than 50% of the cellular phone users sent between 110 and 130 text messages.

29. Graphing Quantitative Data Sets Dot plot Each data entry is plotted, using a point, above a horizontal axis 29 Data: 21, 25, 25, 26, 27, 28, 30, 36, 36, 45 26 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

30. Solution: Constructing a Dot Plot 30 From the dot plot, you can see that most values cluster between 105 and 148 and the value that occurs the most is 126. You can also see that 78 is an unusual data value. 155159 144 129 105 145 126 116 130 114 122 112 112 142 126 118118 108 122 121 109 140 126 119 113 117 118 109 109 119 139139 122 78 133 126 123 145 121 134 124 119 132 133 124 129 112 126 148 147

31. Graphing Qualitative Data Sets Pie Chart A circle is divided into sectors that represent categories. The area of each sector is proportional to the frequency of each category. Larson/Farber 4th ed. 31

32. Example: Constructing a Pie Chart The numbers of motor vehicle occupants killed in crashes in 2005 are shown in the table. Use a pie chart to organize the data. (Source: U.S. Department of Transportation, National Highway Traffic Safety Administration) Larson/Farber 4th ed. 32 Vehicle typeKilled Cars18,440 Trucks13,778 Motorcycles4,553 Other823

33. Solution: Constructing a Pie Chart Find the relative frequency (percent) of each category. 33 Vehicle typeFrequency, fRelative frequency Cars18,440 Trucks13,778 Motorcycles4,553 Other823 37,594

34. Solution: Constructing a Pie Chart 34 Vehicle type Relative frequency Central angle Cars0.49176º Trucks0.37133º Motorcycles0.1243º Other0.027º From the pie chart, you can see that most fatalities in motor vehicle crashes were those involving the occupants of cars.

35. Graphing Qualitative Data Sets Pareto Chart A vertical bar graph in which the height of each bar represents frequency or relative frequency. The bars are positioned in order of decreasing height, with the tallest bar positioned at the left. Larson/Farber 4th ed. 35 Categories Frequency

36. Constructing a Pareto Chart 36 Cause$ (million) Admin. error 7.8 Employee theft 15.6 Shoplifting14.7 Vendor fraud 2.9 From the graph, it is easy to see that the causes of inventory shrinkage that should be addressed first are employee theft and shoplifting.

37. Graphing Paired Data Sets Paired Data Sets Each entry in one data set corresponds to one entry in a second data set. Graph using a scatter plot. The ordered pairs are graphed as points in a coordinate plane. Used to show the relationship between two quantitative variables. 37 x y

38. Example: Interpreting a Scatter Plot As the petal length increases, what tends to happen to the petal width? 38 Each point in the scatter plot represents the petal length and petal width of one flower.

39. Graphing Paired Data Sets Time Series Data set is composed of quantitative entries taken at regular intervals over a period of time. e.g., The amount of precipitation measured each day for one month. Use a time series chart to graph. 39 time Quantitative data

40. Example: Constructing a Time Series Chart The table lists the number of cellular telephone subscribers (in millions) for the years 1995 through 2005. Construct a time series chart for the number of cellular subscribers. (Source: Cellular Telecommunication & Internet Association) Larson/Farber 4th ed. 40

41. Solution: Constructing a Time Series Chart 41 The graph shows that the number of subscribers has been increasing since 1995, with greater increases recently.

42. Section 2.3 Measures of Central Tendency Larson/Farber 4th ed. 42

43. Measures of Central Tendency Measure of central tendency A value that represents a typical, or central, entry of a data set. Most common measures of central tendency: Mean Median Mode Larson/Farber 4th ed. 43

44. Measure of Central Tendency: Mode Mode The data entry that occurs with the greatest frequency. If no entry is repeated the data set has no mode. If two entries occur with the same greatest frequency, each entry is a mode (bimodal). Larson/Farber 4th ed. 44

45. Comparing the Mean, Median, and Mode All three measures describe a typical entry of a data set. Advantage of using the mean: The mean is a reliable measure because it takes into account every entry of a data set. Disadvantage of using the mean: Greatly affected by outliers (a data entry that is far removed from the other entries in the data set). Larson/Farber 4th ed. 45

46. Example: Comparing the Mean, Median, and Mode Find the mean, median, and mode of the sample ages of a class shown. Which measure of central tendency best describes a typical entry of this data set? Are there any outliers? Larson/Farber 4th ed. 46 Ages in a class 20 21 22 23 24 65

47. Solution: Comparing the Mean, Median, and Mode Larson/Farber 4th ed. 47 Mean: Median: 20 years (the entry occurring with the greatest frequency) Ages in a class 20 21 22 23 24 65 Mode:

48. Solution: Comparing the Mean, Median, and Mode Larson/Farber 4th ed. 48 Mean ≈ 23.8 years Median = 21.5 years Mode = 20 years The mean takes every entry into account, but is influenced by the outlier of 65. The median also takes every entry into account, and it is not affected by the outlier. In this case the mode exists, but it doesn't appear to represent a typical entry.

49. Solution: Comparing the Mean, Median, and Mode Larson/Farber 4th ed. 49 Sometimes a graphical comparison can help you decide which measure of central tendency best represents a data set. In this case, it appears that the median best describes the data set.

50. Weighted Mean The mean of a data set whose entries have varying weights. where w is the weight of each entry x Larson/Farber 4th ed. 50

51. Example: Finding a Weighted Mean You are taking a class in which your grade is determined from five sources: 50% from your test mean, 15% from your midterm, 20% from your final exam, 10% from your computer lab work, and 5% from your homework. Your scores are 86 (test mean), 96 (midterm), 82 (final exam), 98 (computer lab), and 100 (homework). What is the weighted mean of your scores? If the minimum average for an A is 90, did you get an A? Larson/Farber 4th ed. 51

52. Solution: Finding a Weighted Mean Larson/Farber 4th ed. 52 SourceScore, xWeight, wx∙w Test Mean860.5086(0.50)= 43.0 Midterm960.1596(0.15) = 14.4 Final Exam820.2082(0.20) = 16.4 Computer Lab980.1098(0.10) = 9.8 Homework1000.05100(0.05) = 5.0 Σw = 1Σ(x∙w) = 88.6 Your weighted mean for the course is 88.6. You did not get an A.

53. Mean of Grouped Data Mean of a Frequency Distribution Approximated by where x and f are the midpoints and frequencies of a class, respectively Larson/Farber 4th ed. 53

54. Finding the Mean of a Frequency Distribution In Words In Symbols Larson/Farber 4th ed. 54 1.Find the midpoint of each class. 2.Find the sum of the products of the midpoints and the frequencies. 3.Find the sum of the frequencies. 4.Find the mean of the frequency distribution.

55. Example: Find the Mean of a Frequency Distribution Use the frequency distribution to approximate the mean number of minutes that a sample of Internet subscribers spent online during their most recent session. Larson/Farber 4th ed. 55 ClassMidpointFrequency, f 7 – 1812.56 19 – 3024.510 31 – 4236.513 43 – 5448.58 55 – 6660.55 67 – 7872.56 79 – 9084.52

56. Solution: Find the Mean of a Frequency Distribution Larson/Farber 4th ed. 56 ClassMidpoint, xFrequency, f(x∙f) 7 – 1812.5612.5∙6 = 75.0 19 – 3024.51024.5∙10 = 245.0 31 – 4236.51336.5∙13 = 474.5 43 – 5448.5848.5∙8 = 388.0 55 – 6660.5560.5∙5 = 302.5 67 – 7872.5672.5∙6 = 435.0 79 – 9084.5284.5∙2 = 169.0 n = 50Σ(x∙f) = 2089.0

57. The Shape of Distributions Larson/Farber 4th ed. 57 Symmetric Distribution A vertical line can be drawn through the middle of a graph of the distribution and the resulting halves are approximately mirror images.

58. The Shape of Distributions Larson/Farber 4th ed. 58 Uniform Distribution (rectangular) All entries or classes in the distribution have equal or approximately equal frequencies. Symmetric.

59. The Shape of Distributions Larson/Farber 4th ed. 59 Skewed Left Distribution (negatively skewed) The “tail” of the graph elongates more to the left. The mean is to the left of the median.

60. The Shape of Distributions Larson/Farber 4th ed. 60 Skewed Right Distribution (positively skewed) The “tail” of the graph elongates more to the right. The mean is to the right of the median.

61. Section 2.3 Summary Determined the mean, median, and mode of a population and of a sample Determined the weighted mean of a data set and the mean of a frequency distribution Described the shape of a distribution as symmetric, uniform, or skewed and compared the mean and median for each Larson/Farber 4th ed. 61

62. Section 2.4 Measures of Variation Larson/Farber 4th ed. 62

63. Deviation, Variance, and Standard Deviation Population Variance Population Standard Deviation Larson/Farber 4th ed. 63 Sum of squares, SS x

64. Example: Using Technology to Find the Standard Deviation Sample office rental rates (in dollars per square foot per year) for Miami’s central business district are shown in the table. Use a calculator or a computer to find the mean rental rate and the sample standard deviation. (Adapted from: Cushman & Wakefield Inc.) Larson/Farber 4th ed. 64 Office Rental Rates 35.0033.5037.00 23.7526.5031.25 36.5040.0032.00 39.2537.5034.75 37.7537.2536.75 27.0035.7526.00 37.0029.0040.50 24.5033.0038.00

65. Solution: Using Technology to Find the Standard Deviation Larson/Farber 4th ed. 65 Sample Mean Sample Standard Deviation

66. Interpreting Standard Deviation Standard deviation is a measure of the typical amount an entry deviates from the mean. The more the entries are spread out, the greater the standard deviation. 66

67. Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule) For data with a (symmetric) bell-shaped distribution, the standard deviation has the following characteristics: 67 About 68% of the data lie within one standard deviation of the mean. About 95% of the data lie within two standard deviations of the mean. About 99.7% of the data lie within three standard deviations of the mean.

68. Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule) Larson/Farber 4th ed. 68 68% within 1 SD 34% 99.7% within 3 SD 2.35% 95% within 2 SD 13.5%

69. Example: Using the Empirical Rule In a survey conducted by the National Center for Health Statistics, the sample mean height of women in the United States (ages 20-29) was 64 inches, with a sample standard deviation of 2.71 inches. Estimate the percent of the women whose heights are between 64 inches and 69.42 inches. Larson/Farber 4th ed. 69

70. Solution: Using the Empirical Rule Larson/Farber 4th ed. 70 55.8 7 58.5 8 61.2 9 6466.7 1 69.4 2 72.1 3 34% 13.5% Because the distribution is bell-shaped, you can use the Empirical Rule. 34% + 13.5% = 47.5% of women are between 64 and 69.42 inches tall.

71. Chebychev’s Theorem The portion of any data set lying within k standard deviations (k > 1) of the mean is at least: Larson/Farber 4th ed. 71 k = 2: In any data set, at least of the data lie within 2 standard deviations of the mean. k = 3: In any data set, at least of the data lie within 3 standard deviations of the mean.

72. Box-and-Whisker Plot Box-and-whisker plot Exploratory data analysis tool. Highlights important features of a data set. Requires (five-number summary): Minimum entry First quartile Q 1 Median Q 2 Third quartile Q 3 Maximum entry Larson/Farber 4th ed. 72

73. Drawing a Box-and-Whisker Plot 1.Find the five-number summary of the data set. 2.Construct a horizontal scale that spans the range of the data. 3.Plot the five numbers above the horizontal scale. 4.Draw a box above the horizontal scale from Q 1 to Q 3 and draw a vertical line in the box at Q 2. 5.Draw whiskers from the box to the minimum and maximum entries. Larson/Farber 4th ed. 73 Whiske r Maximu m entry Minimu m entry Box Median, Q 2 Q3Q3 Q1Q1