Exploring Data with Histograms and Stem-and-Leaf Plots

1. Chapter 3Displaying and Summarizing Quantitative Data Display: Histograms, Stem and Leaf Plots Numerical Summaries: Median, Mean, Quartiles, Standard Deviation

2. Relative Frequency Histogram of Exam Grades .30 .25 .20 Relative frequency .15 .10 .05 0 40 50 60 70 80 90 100 Grade

3. Frequency Histograms

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

5. All 200 m Races 20.2 secs or less

6. Histograms Showing Different Centers

7. Histograms - Same Center, Different Spread

8. Frequency and Relative Frequency Histograms • identify smallest and largest values in data set • divide interval between largest and smallest values into between 5 and 20 subintervals called classes * each data value in one and only one class * no data value is on a boundary

9. How Many Classes?

10. Histogram Construction (cont.) * compute frequency or relative frequency of observations in each class * x-axis: class boundaries; y-axis: frequency or relative frequency scale * over each class draw a rectangle with height corresponding to the frequency or relative frequency in that class

11. Example. Number of daily employee absences from work • 106 obs; approx. no of classes= {2(106)}1/3 = {212}1/3 = 5.69 1+ log(106)/log(2) = 1 + 6.73 = 7.73 • There is no single “correct” answer for the number of classes • For example, you can choose 6, 7, 8, or 9 classes; don’t choose 15 classes

12. EXCEL Histogram

13. Absences from Work (cont.) • 6 classes • class width: (158-121)/6=37/6=6.17 7 • 6 classes, each of width 7; classes span 6(7)=42 units • data spans 158-121=37 units • classes overlap the span of the actual data values by 42-37=5 • lower boundary of 1st class: (1/2)(5) units below 121 = 121-2.5 = 118.5

14. EXCEL histogram

15. 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

16. 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

17. 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

18. Relative Frequency Histogram of Grades .30 .25 .20 Relative frequency .15 .10 .05 0 40 50 60 70 80 90 100 Grade

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

20. Stem and leaf displays • Have the following general appearance stem leaf 1 8 9 2 1 2 8 9 9 3 2 3 8 9 4 0 1 5 6 7 6 4

21. Stem and Leaf Displays • Partition each no. in data into a “stem” and “leaf” • 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

22. 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 • 18: stem=1; leaf=8; 18 = 1 | 8 stem leaf 1 8 9 2 1 2 8 9 9 3 2 3 8 9 4 0 1 5 6 7 6 4

23. Suppose a 95 yr. old is hired stem leaf 1 8 9 2 1 2 8 9 9 3 2 3 8 9 4 0 1 5 6 7 6 4 7 8 9 5

24. Number of TD passes by NFL teams: 2010 season(stems are 10’s digit)

25. Pulse Rates n = 138

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

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

28. Back-to-back stem-and-leaf displays. TD passes by NFL teams: 1999, 2009multiply stems by 10

29. 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 • 4 • 6 • 8 • 10 • 12 10 Countdown

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

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

32. Shape (cont.): Outliers An important kind of deviation is an outlier. Outliersare 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. Alaska Florida

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

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

35. Other Graphical Methods for Economic Data • 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.)

36. Unemployment Rate, by Educational Attainment

37. Water Use During Super Bowl

38. Winning Times 100 M Dash

39. Annual Mean Temperature

40. End of Histograms, Stem and Leaf plots

41. Describing Distributions Numerically:Medians and Quartiles

42. 2 characteristics of a data set to measure • center measures where the “middle” of the data is located • variability measures how “spread out” the data is

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

44. 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

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

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

47. Examples • Example: n = 7 17.5 2.8 3.2 13.9 14.1 25.3 45.8 • Example n = 7 (ordered): • 2.8 3.2 13.9 14.1 17.5 25.3 45.8 • Example: n = 8 17.5 2.8 3.2 13.9 14.1 25.3 35.7 45.8 • 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

48. Below are the annual tuition charges at 7 public universities. What is the median tuition? 4429 4960 4960 4971 5245 5546 7586 • 5245 • 4965.5 • 4960 • 4971 5

49. Below are the annual tuition charges at 7 public universities. What is the median tuition? 4429 4960 5245 5546 4971 5587 7586 • 5245 • 4965.5 • 5546 • 4971 6

50. Measures of Spread The range and interquartile range