2. Chapter 4 Displaying and Summarizing Quantitative Data CHAPTER OBJECTIVES At the conclusion of this chapter you should be able to: n 1)Construct graphs that appropriately describe quantitative data n 2)Calculate and interpret numerical summaries of quantitative data. n 3)Combine numerical methods with graphical methods to analyze a data set. n 4)Apply graphical methods of summarizing data to choose appropriate numerical summaries. n 5)Apply software and/or calculators to automate graphical and numerical summary procedures.

3. Displaying Quantitative Data Histograms Stem and Leaf Displays

4. Relative Frequency Histogram of Exam Grades 0.05.10.15.20.25.30 405060708090 Grade Relative frequency 100

5. Frequency Histogram

6. Histograms A histogram shows three general types of information: n It provides visual indication of where the approximate center of the data is. n We can gain an understanding of the degree of spread, or variation, in the data. n We can observe the shape of the distribution.

7. All 200 m Races 20.2 secs or less

8. Histograms Showing Different Centers

9. Histograms Showing Different Centers (football head coach salaries)

10. Histograms - Same Center, Different Spread (football head coach salaries)

11. Excel Example: 2012-13 NFL Salaries

12. Statcrunch Example: 2012-13 NFL Salaries

13. Grades on a statistics exam Data: 75 66 77 66 64 73 91 65 59 86 61 86 61 58 70 77 80 58 94 78 62 79 83 54 52 45 82 48 67 55

14. Frequency Distribution of Grades Class Limits Frequency 40 up to 50 50 up to 60 60 up to 70 70 up to 80 80 up to 90 90 up to 100 Total 2 6 8 7 5 2 30

15. Relative Frequency Distribution of Grades Class Limits Relative Frequency 40 up to 50 50 up to 60 60 up to 70 70 up to 80 80 up to 90 90 up to 100 2/30 =.067 6/30 =.200 8/30 =.267 7/30 =.233 5/30 =.167 2/30 =.067

16. Relative Frequency Histogram of Grades 0.05.10.15.20.25.30 405060708090 Grade Relative frequency 100

17. Based on the histo- gram, about what percent of the values are between 47.5 and 52.5? 1. 50% 2. 5% 3. 17% 4. 30% Countdown 10

18. Stem and leaf displays n Have the following general appearance stemleaf 18 9 21 2 8 9 9 32 3 8 9 40 1 56 7 64

19. Stem and Leaf Displays n Partition each no. in data into a “stem” and “leaf” n Constructing stem and leaf display 1) deter. stem and leaf partition (5-20 stems) 2) write stems in column with smallest stem at top; include all stems in range of data 3) only 1 digit in leaves; drop digits or round off 4) record leaf for each no. in corresponding stem row; ordering the leaves in each row helps

20. Example: employee ages at a small company 18 21 22 19 32 33 40 41 56 57 64 28 29 29 38 39; stem: 10’s digit; leaf: 1’s digit n 18: stem=1; leaf=8; 18 = 1 | 8 stemleaf 18 9 21 2 8 9 9 32 3 8 9 40 1 56 7 64

21. Suppose a 95 yr. old is hired stemleaf 18 9 21 2 8 9 9 32 3 8 9 40 1 56 7 64 7 8 95

22. Number of TD passes by NFL teams: 2012-2013 season ( stems are 10’s digit) stemleaf 4343 03 247 26677789 201222233444 113467889 08

23. Pulse Rates n = 138

24. Advantages/Disadvantages of Stem-and-Leaf Displays n Advantages 1) each measurement displayed 2) ascending order in each stem row 3) relatively simple (data set not too large) n Disadvantages display becomes unwieldy for large data sets

25. Population of 185 US cities with between 100,000 and 500,000 n Multiply stems by 100,000

26. Back-to-back stem-and-leaf displays. TD passes by NFL teams: 1999-2000, 2012-13 multiply stems by 10 1999-20002012-13 2403 637 2324 665526677789 43322221100201222233444 9998887666167889 4211134 08

27. Below is a stem-and-leaf display for the pulse rates of 24 women at a health clinic. How many pulses are between 67 and 77? Stems are 10’s digits 1. 4 2. 6 3. 8 4. 10 5. 12 Countdown 10

28. Interpreting Graphical Displays: Shape n A distribution is symmetric if the right and left sides of the histogram are approximately mirror images of each other. Symmetric distribution Complex, multimodal distribution  Not all distributions have a simple overall shape, especially when there are few observations. Skewed distribution  A distribution is skewed to the right if the right side of the histogram (side with larger values) extends much farther out than the left side. It is skewed to the left if the left side of the histogram extends much farther out than the right side.

29. Heights of Students in Recent Stats Class

30. Shape (cont.)Female heart attack patients in New York state Age: left-skewedCost: right-skewed

31. AlaskaFlorida Shape (cont.): Outliers An important kind of deviation is an outlier. Outliers are observations that lie outside the overall pattern of a distribution. Always look for outliers and try to explain them. The overall pattern is fairly symmetrical except for 2 states clearly not belonging to the main trend. Alaska and Florida have unusual representation of the elderly in their population. A large gap in the distribution is typically a sign of an outlier.

32. Center: typical value of frozen personal pizza? ~$2.65

33. Spread: fuel efficiency 4, 8 cylinders 4 cylinders: more spread8 cylinders: less spread

34. Other Graphical Methods for Economic Data n Time plots plot observations in time order, with time on the horizontal axis and the vari- able on the vertical axis ** Time series measurements are taken at regular intervals (monthly unemployment, quarterly GDP, weather records, electricity demand, etc.)

35. Unemployment Rate, by Educational Attainment

36. Water Use During Super Bowl

37. Winning Times 100 M Dash

38. Numerical Summaries of Quantitative Data Numerical and More Graphical Methods to Describe Univariate Data

39. 2 characteristics of a data set to measure n center measures where the “middle” of the data is located n variability measures how “spread out” the data is

40. The median: a measure of center Given a set of n measurements arranged in order of magnitude, Median=middle valuen odd mean of 2 middle values,n even n Ex. 2, 4, 6, 8, 10; n=5; median=6 n Ex. 2, 4, 6, 8; n=4; median=(4+6)/2=5

41. Student Pulse Rates (n=62) 38, 59, 60, 60, 62, 62, 63, 63, 64, 64, 65, 67, 68, 70, 70, 70, 70, 70, 70, 70, 71, 71, 72, 72, 73, 74, 74, 75, 75, 75, 75, 76, 77, 77, 77, 77, 78, 78, 79, 79, 80, 80, 80, 84, 84, 85, 85, 87, 90, 90, 91, 92, 93, 94, 94, 95, 96, 96, 96, 98, 98, 103 Median = (75+76)/2 = 75.5

42. Medians are used often n Year 2011 baseball salaries Median $1,450,000 (max=$32,000,000 Alex Rodriguez; min=$414,000) n Median fan age: MLB 45; NFL 43; NBA 41; NHL 39 n Median existing home sales price: May 2011 $166,500; May 2010 $174,600 n Median household income (2008 dollars) 2009 $50,221; 2008 $52,029

43. The median splits the histogram into 2 halves of equal area

44. Examples n Example: n = 7 17.5 2.8 3.2 13.9 14.1 25.3 45.8 n Example n = 7 (ordered): n 2.8 3.2 13.9 14.1 17.5 25.3 45.8 n Example: n = 8 17.5 2.8 3.2 13.9 14.1 25.3 35.7 45.8 n Example n =8 (ordered) 2.8 3.2 13.9 14.1 17.5 25.3 35.7 45.8 m = 14.1 m = (14.1+17.5)/2 = 15.8

45. Below are the annual tuition charges at 7 public universities. What is the median tuition? 4429 4960 4971 5245 5546 7586 1. 5245 2. 4965.5 3. 4960 4. 4971 Countdown 10

46. Below are the annual tuition charges at 7 public universities. What is the median tuition? 4429 4960 5245 5546 4971 5587 7586 1. 5245 2. 4965.5 3. 5546 4. 4971 Countdown 10

47. Measures of Spread n The range and interquartile range

48. Ways to measure variability range=largest-smallest  OK sometimes; in general, too crude; sensitive to one large or small data value  The range measures spread by examining the ends of the data  A better way to measure spread is to examine the middle portion of the data

49. m = median = 3.4 Q 1 = first quartile = 2.3 Q 3 = third quartile = 4.2 Quartiles: Measuring spread by examining the middle The first quartile, Q 1, is the value in the sample that has 25% of the data at or below it (Q 1 is the median of the lower half of the sorted data). The third quartile, Q 3, is the value in the sample that has 75% of the data at or below it (Q 3 is the median of the upper half of the sorted data).

50. Quartiles and median divide data into 4 pieces Q1 M Q3 Q1 M Q3 1/4 1/41/4 1/4

51. Quartiles are common measures of spread n http://www2.acs.ncsu.edu/UPA/admissi ons/fresprof.htm http://www2.acs.ncsu.edu/UPA/admissi ons/fresprof.htm n http://www2.acs.ncsu.edu/UPA/peers/cu rrent/ncsu_peers/sat.htm http://www2.acs.ncsu.edu/UPA/peers/cu rrent/ncsu_peers/sat.htm n University of Southern California University of Southern California

52. Rules for Calculating Quartiles Step 1: find the median of all the data (the median divides the data in half) Step 2a: find the median of the lower half; this median is Q 1 ; Step 2b: find the median of the upper half; this median is Q 3. Important: when n is odd include the overall median in both halves; when n is even do not include the overall median in either half.

53. Example n 2 4 6 8 10 12 14 16 18 20 n = 10 n Median n m = (10+12)/2 = 22/2 = 11 n Q 1 : median of lower half 2 4 6 8 10 Q 1 = 6 n Q 3 : median of upper half 12 14 16 18 20 Q 3 = 16 11

54. Quartile example: odd no. of data values n HR’s hit by Babe Ruth in each season as a Yankee  54 59 35 41 46 25 47 60 54 46 49 46 41 34 22  Ordered values:  22 25 34 35 41 41 46 46 46 47 49 54 54 59 60  Median: value in ordered position 8. median = 46  Lower half (including overall median):  22 25 34 35 41 41 46 46  Upper half (including overall median):  46 46 47 49 54 54 59 60

55. Pulse Rates n = 138 Median: mean of pulses in locations 69 & 70: median= (70+70)/2=70 Q 1 : median of lower half (lower half = 69 smallest pulses); Q 1 = pulse in ordered position 35; Q 1 = 63 Q 3 median of upper half (upper half = 69 largest pulses); Q 3 = pulse in position 35 from the high end; Q 3 =78

56. Below are the weights of 31 linemen on the NCSU football team. What is the value of the first quartile Q 1 ? #stemleaf 22255 42357 62426 7257 1026257 122759 (4)281567 152935599 1030333 73145 532155 2336 1340 1. 287 2. 257.5 3. 263.5 4. 262.5 Countdown 10

57. Interquartile range n lower quartile Q 1 n middle quartile: median n upper quartile Q 3 n interquartile range (IQR) IQR = Q 3 – Q 1 measures spread of middle 50% of the data

58. Example: beginning pulse rates n Q 3 = 78; Q 1 = 63 n IQR = 78 – 63 = 15

59. Below are the weights of 31 linemen on the NCSU football team. The first quartile Q 1 is 263.5. What is the value of the IQR? #stemleaf 22255 42357 62426 7257 1026257 122759 (4)281567 152935599 1030333 73145 532155 2336 1340 1. 23.5 2. 39.5 3. 46 4. 69.5 Countdown 10

60. 5-number summary of data n Minimum Q 1 median Q 3 maximum n Pulse data 45 63 70 78 111

61. End of General Numerical Summaries Next: Numerical Summaries of Symmetric Data

62. Numerical Summaries of Symmetric Data. Measure of Center: Mean Measure of Variability: Standard Deviation

63. Symmetric Data Body temp. of 93 adults

64. Recall: 2 characteristics of a data set to measure n center measures where the “middle” of the data is located n variability measures how “spread out” the data is

65. Measure of Center When Data Approx. Symmetric n mean (arithmetic mean) n notation

67. Connection Between Mean and Histogram n A histogram balances when supported at the mean. Mean x = 140.6

68. Mean: balance point Median: 50% area each half right histo: mean 55.26 yrs, median 57.7yrs

69. Properties of Mean, Median 1.The mean and median are unique; that is, a data set has only 1 mean and 1 median (the mean and median are not necessarily equal). 2.The mean uses the value of every number in the data set; the median does not.

70. Example: class pulse rates n 53 64 67 67 70 76 77 77 78 83 84 85 85 89 90 90 90 90 91 96 98 103 140

71. 2010, 2014 baseball salaries n 2010 n = 845  = $3,297,828 median = $1,330,000 max = $33,000,000 n 2014 n = 848  = $3,932,912 median = $1,456,250 max = $28,000,000

72. Disadvantage of the mean n Can be greatly influenced by just a few observations that are much greater or much smaller than the rest of the data

73. Mean, Median, Maximum Baseball Salaries 1985 - 2014

74. Skewness: comparing the mean, and median n Skewed to the right (positively skewed) n mean>median

75. Skewed to the left; negatively skewed n Mean < median n mean=78; median=87;

76. Symmetric data n mean, median approx. equal

77. DESCRIBING VARIABILITY OF SYMMETRIC DATA

78. Describing Symmetric Data (cont.) n Measure of center for symmetric data: n Measure of variability for symmetric data?

79. Example n 2 data sets: x 1 =49, x 2 =51 x=50 y 1 =0, y 2 =100 y=50

80. On average, they’re both comfortable 0 100 49 51

81. Ways to measure variability 1. range=largest-smallest ok sometimes; in general, too crude; sensitive to one large or small obs.

82. Previous Example

83. The Sample Standard Deviation, a measure of spread around the mean n Square the deviation of each observation from the mean; find the square root of the “average” of these squared deviations

84. Calculations … Mean = 63.4 Sum of squared deviations from mean = 85.2 (n − 1) = 13; (n − 1) is called degrees freedom (df) s 2 = variance = 85.2/13 = 6.55 inches squared s = standard deviation = √6.55 = 2.56 inches Women height (inches)

85. 1. First calculate the variance s 2. 2. Then take the square root to get the standard deviation s. Mean ± 1 s.d. We’ll never calculate these by hand, so make sure to know how to get the standard deviation using your calculator, Excel, or other software.

86. Population Standard Deviation

87. Remarks 1. The standard deviation of a set of measurements is an estimate of the likely size of the chance error in a single measurement

88. Remarks (cont.) 2. Note that s and  are always greater than or equal to zero. 3. The larger the value of s (or  ), the greater the spread of the data. When does s=0? When does  =0? When all data values are the same.

89. Remarks (cont.) 4. The standard deviation is the most commonly used measure of risk in finance and business –Stocks, Mutual Funds, etc. 5. Variance  s 2 sample variance   2 population variance  Units are squared units of the original data  square $, square gallons ??

90. Remarks 6):Why divide by n-1 instead of n? n degrees of freedom n each observation has 1 degree of freedom however, when estimate unknown population parameter like , you lose 1 degree of freedom

91. Remarks 6) (cont.):Why divide by n-1 instead of n? Example n Suppose we have 3 numbers whose average is 9 nx1=x2=nx1=x2= n then x 3 must be n once we selected x 1 and x 2, x 3 was determined since the average was 9 n 3 numbers but only 2 “degrees of freedom” Since the average (mean) is 9, x 1 + x 2 + x 3 must equal 9*3 = 27, so x 3 = 27 – (x 1 + x 2 ) Choose ANY values for x 1 and x 2

92. Computational Example

93. class pulse rates

94. Review: Properties of s and  s and  are always greater than or equal to 0 when does s = 0?  = 0? The larger the value of s (or  ), the greater the spread of the data n the standard deviation of a set of measurements is an estimate of the likely size of the chance error in a single measurement

95. Summary of Notation

96. End of Chapter 4