1. AP Statistics Chapter 4 Part 1 Displaying and Summarizing Quantitative Data

2. Learning Goals 1.Know how to display the distribution of a quantitative variable with a histogram, a stem-and-leaf display, or a dotplot. 2.Know how to display the relative position of quantitative variable with a Cumulative Frequency Curve and analysis the Cumulative Frequency Curve. 3.Be able to describe the distribution of a quantitative variable in terms of its shape. 4.Be able to describe any anomalies or extraordinary features revealed by the display of a variable.

3. Learning Goals 5.Be able to determine the shape of the distribution of a variable by knowing something about the data. 6.Know the basic properties and how to compute the mean and median of a set of data. 7.Understand the properties of a skewed distribution. 8.Know the basic properties and how to compute the standard deviation and IQR of a set of data.

4. Learning Goals 9.Understand which measures of center and spread are resistant and which are not. 10.Be able to select a suitable measure of center and a suitable measure of spread for a variable based on information about its distribution. 11.Be able to describe the distribution of a quantitative variable in terms of its shape, center, and spread.

5. Learning Goal 1 Know how to display the distribution of a quantitative variable with a histogram, a stem-and-leaf display, or a dotplot

6. Learning Goal 1: Ways to Graph Quantitative Data  Histograms and Stemplots These are summary graphs for a single variable. They are very useful to understand the pattern of variability in the data.  Dotplots Quick and easy graph for small data sets.  Cumulative Frequency Curves (Ogive) Used to compare relative standings of the data.  Line Graphs: Time Plots Use when there is a meaningful sequence, like time. The line connecting the points helps emphasize any change over time.

7. Learning Goal 1: Dealing With a Lot of Numbers…  Summarizing the data will help us when we look at large sets of quantitative data.  Without summaries of the data, it’s hard to grasp what the data tell us.  The best thing to do is to make a picture…  We can’t use bar charts or pie charts for quantitative data, since those displays are for categorical variables.

8. Learning Goal 1: Tabulating Numerical Data What is a Frequency Distribution (table)?  A frequency distribution is a list or a table …  containing class groupings (ranges within which the data fall)...  and the corresponding frequencies with which data fall within each grouping or class.

9. Learning Goal 1: Why Use a Frequency Distribution?  It is a way to summarize numerical data.  It condenses the raw data into a more useful form.  It allows for a quick visual interpretation of the data.

10. HISTOGRAM Quantitative Data

11. Learning Goal 1: Histograms 11 A Histogram is a graph that uses bars to portray the frequencies or the relative frequencies of the possible outcomes for a quantitative variable.

12. Learning Goal 1: Histograms  The most common graph used to display one variable quantitative data.

13. Learning Goal 1: Histograms  To make a histogram we first need to organize the data using a quantitative frequency table.  Two types of quantitative data 1.Discrete – use ungrouped frequency table to organize. 2.Continuous – use grouped frequency table to organize.

14. Learning Goal 1: Quantitative Frequency Tables – Ungrouped What is an ungrouped frequency table? An ungrouped frequency table simply lists the data values with the corresponding frequency counts with which each value occurs. Commonly used with discrete quantitative data.

15. Learning Goal 1: Quantitative Frequency Tables – Ungrouped Example: The at-rest pulse rate for 16 athletes at a meet were 57, 57, 56, 57, 58, 56, 54, 64, 53, 54, 54, 55, 57, 55, 60, and 58. Summarize the information with an ungrouped frequency distribution.

16. Learning Goal 1: Quantitative Frequency Tables – Ungrouped  Example continued: 57, 57, 56, 57, 58, 56, 54, 64, 53, 54, 54, 55, 57, 55, 60, 58. Note: The (ungrouped) classes are the observed values themselves. Class (pulse rate)Frequency, f 531 543 552 562 574 582 590 601 610 620 630 641 TotalN =16

17. Learning Goal 1: Quantitative Relative Freq. Tables - Ungrouped Note: The relative freq. for a class is obtained by computing f/n. Class (pulse rate) Frequency, f Relative Frequency 5310.0625 5430.1875 5520.1250 5620.1250 5740.2500 5820.1250 5900 6010.0625 6100 6200 6300 6410.0625 TotalN =161

18. Learning Goal 1: Relative Freq. Tables – Your Turn TVs per Household Trends in Television, published by the Television Bureau of Advertising, provides information on television ownership. The table gives the number of TV sets per household for 50 randomly selected households. Use classes based on a single value to construct a ungrouped-data relative frequency table for these data.

19. Learning Goal 1: Relative Freq. Tables – Solution

20. Learning Goal 1: Quantitative Frequency Tables – Grouped What is a grouped frequency table? A grouped frequency table is obtained by constructing classes (or intervals) for the data, and then listing the corresponding number of values (frequency counts) in each interval. Commonly used with continuous quantitative data.

21.  Class: an interval of values. Example: 61  x  70.  Frequency: the number of data values that fall within a class. “Five data fall within the class 61  x  70”.  Relative Frequency: the proportion of data values that fall within a class. “18% of the data fall within the class 61  x  70”. Learning Goal 1: Quantitative Frequency Tables – Grouped

22. Learning Goal 1: Grouped Frequency Tables – Example  A frequency table  organizes quantitative data.  partitions data into classes (intervals).  shows how many data values are in each class. Test Score Number of Students 61-704 71-808 81-9015 91-1007

23. Learning Goal 1: Grouped Frequency Table Terminology  Class - non-overlapping intervals the data is divided into.  Class Limits –The smallest and largest observed values in a given class.  Class Boundaries – Fall halfway between the upper class limit for the smaller class and the lower class limit for larger class. Used to close the gap between classes.  Class Width – The difference between the class boundaries for a given class.  Class Midpoint or Mark – The midpoint of a class.

24. Learning Goal 1: Grouped Frequency Tables – Classes A grouped frequency table should have a minimum of 5 classes and a maximum of 20 classes. For small data sets, one can use between 5 and 10 classes. For large data sets, one can use up to 20 classes.

25. Learning Goal 1: Number of Classes Too Many Classes - Not summarized enough. Same data set

26. Learning Goal 1: Number of Classes Too Few Classes – summarized too much. Same data set

27. Learning Goal 1: Number of Classes Correct Number of Classes – 5 to 10. Same data set

28. Learning Goal 1: Class Limits Lower Class Limits are the smallest numbers that can actually belong to different classes. Lower Class Limits

29. Learning Goal 1: Class Limits Upper Class Limits are the largest numbers that can actually belong to different classes. Upper Class Limits

30. Learning Goal 1: Class Boundaries  Class Boundaries are the numbers used to separate classes, but without the gaps created by class limits.  Class boundaries split the gap, created by the class limits between two consecutive classes, in half.  Half of the gap is given to the upper class and half given to the lower class. Thus, bringing the bars of the two consecutive classes together, with no gap.

31. A “class” is basically an interval on a number line. It has:  A lower limit a and an upper limit b.  A width.  A lower boundary and an upper boundary (integer data).  A midpoint. Learning Goal 1: Structure of a Data Class (b + 0.5) - (a - 0.5)

32. If a = 60 and b = 69 for integer data, what is the value of the lower boundary? a). 60b). 59.5 c). 9d). 64.5 Learning Goal 1: Structure of a Data Class - Problem (b + 0.5) - (a - 0.5)

33. Learning Goal 1: Class Boundaries Class Boundaries are the number separating classes. Class Boundaries - 0.5 99.5 199.5 299.5 399.5 499.5

34. Learning Goal 1: Class Midpoints or Class Mark  Class Midpoint or Class Mark is the midpoint of each class.  Class midpoints can be found by adding the lower class limit to the upper class limit and dividing the sum by two.

35. Learning Goal 1: Class Midpoints  Class Midpoint is the midpoint of each class. Class Midpoints 49.5 149.5 249.5 349.5 449.5

36. Learning Goal 1: Class Width Class Width is the difference between two consecutive lower class limits or two consecutive lower class boundaries Class Width 100

37. Learning Goal 1: Constructing A Frequency Table 1. Decide on the number of classes (should be between 5 and 20). 2. Calculate (round up). 3. Starting point: Begin by choosing a lower limit of the first class. 4. Using the lower limit of the first class and class width, proceed to list the lower class limits. 5. List the lower class limits in a vertical column and proceed to enter the upper class limits. 6. Go through the data set putting a tally in the appropriate class for each data value. class width  (highest value) – (lowest value) number of classes

38. Learning Goal 1: Constructing A Frequency Table - Example A manufacturer of insulation randomly selects 20 winter days and records the daily high temperature. 24, 35, 17, 21, 24, 37, 26, 46, 58, 30, 32, 13, 12, 38, 41, 43, 44, 27, 53, 27

39. Learning Goal 1: Constructing A Frequency Table - Example  Sort raw data in ascending order: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58  Find range: 58 - 12 = 46  Select number of classes: 5 (usually between 5 and 10)  Compute class interval (width): 10 (46/5 then round up)  Determine lower class (limits): 10, 20, 30, 40, 50. List in a vertical column.  Compute upper class limits 19, 29, 39, 49, 59, and then class midpoints: 14.5, 24.5, 34.5, 44.5, 54.5.  Count observations & assign to classes (continued)

40. Learning Goal 1: Constructing A Frequency Table - Example Class Frequency 10 - 19 3.15 15% 20 - 29 6.30 30% 30 - 39 5.25 25% 40 - 49 4.20 20% 50 - 59 2.10 10% Total 20 1.00 100% Relative Frequency Percentage Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 (continued)

41.  Use Tally marks to count the data in each class.  Record the frequencies (and relative frequencies if desired) on the table. Learning Goal 1: Tip for Constructing A Frequency Table

42. Learning Goal 1: Histogram Then to make the Histogram, graph the Frequency Table data.

43. Make a frequency table. Choose appropriate scale for vertical axis (freq. or relative freq.) and horizontal axis (based on classes). Label both axis. Place class boundaries on horizontal axis. Place frequencies on vertical axis. For each class, draw a bar with height equal to the class frequency and width equal to the class width. Title the graph. Learning Goal 1: Making a Histogram

44. Temperatures (degrees) Learning Goal 1: Making a Histogram (No gaps between bars) Class 10 - 19 15 3 20 - 29 25 6 30 - 39 35 5 40 - 49 45 4 50 - 59 55 2 Frequency Class Midpoint

45. Learning Goal 1: Frequency Table From a Histogram There are several procedures that one can use to construct a grouped frequency tables. However, because of the many statistical software packages (MINITAB, SPSS etc.) and graphing calculators (TI-84 etc.) available today, it is not necessary to try to construct such distributions using pencil and paper.

46. Learning Goal 1: Frequency Table From a Histogram The weights of 30 female students majoring in Physical Education on a college campus are as follows: 143, 113, 107, 151, 90, 139, 136, 126, 122, 127, 123, 137, 132, 121, 112, 132, 133, 121, 126, 104, 140, 138, 99, 134, 119, 112, 133, 104, 129, and 123. Summarize the data with a frequency distribution using seven classes.

47. Learning Goal 1: Frequency Table From a Histogram The MINITAB statistical software was used to generate the histogram (similar to the histogram on our TI-84) in the next slide. The histogram has seven classes. Classes for the weights are along the x-axis and frequencies are along the y-axis. The number at the top of each rectangular box, represents the frequency for the class.

48. Learning Goal 1: Frequency Table From a Histogram Histogram with 7 classes for the weights.

49. Learning Goal 1: Frequency Table From a Histogram Observations From the histogram, the classes (intervals) are 85 – 95, 95 – 105,105 – 115 etc. with corresponding frequencies of 1, 3, 4, etc. We will use this information to construct the group frequency distribution.

50. Learning Goal 1: Frequency Table From a Histogram Observations (continued) Observe that the upper class limit of 95 for the class 85 – 95 is listed as the lower class limit for the class 95 – 105. Since the value of 95 cannot be included in both classes, we will use the convention that the upper class limit is not included in the class.

51. Learning Goal 1: Frequency Table From a Histogram Observations (continued) That is, the class 85 – 95 should be interpreted as having the values 85 and up to 95 but not including the value of 95. Using these observations, the grouped frequency distribution is constructed from the histogram and is given on the next slide.

52. Learning Goal 1: Frequency Table From a Histogram Class (weight)Frequency 85 – 951 95 – 1053 105 – 1154 115 – 1256 125 – 1359 135 – 1456 145 – 1551 Totaln = 30

53. Learning Goal 1: Using the TI-84 to Make Histograms  Start by entering data into a list (STAT / Edit / L 1 ).  Example: Enter the presidential data on the next slide into list L 1.

54. Learning Goal 1: Using the TI-84 to Make Histograms

55.  Choose 2 nd : Stat Plot to choose a histogram plot.  Caution: Watch out for other plots that might be “turned on” or equations that might be graphed.

56. Learning Goal 1: Using the TI-84 to Make Histograms  Turn the plot “on”,  Choose the histogram plot.  Xlist should point to the location of the data.

57. Learning Goal 1: Using the TI-84 to Make Histograms  Under the “Zoom” menu, choose option 9: ZoomStat

58. Learning Goal 1: Using the TI-84 to Make Histograms  The result is a histogram where the calculator has decided the width and location of the ranges.  You can use the Trace key to get information about the ranges and the frequencies.

59. Learning Goal 1: Using the TI-84 to Make Histograms  You can change the size and location of the ranges by using the Window button.  Use the Xscl to change the class width on the graph.  Press the Graph button to see the results

60. Learning Goal 1: Using the TI-84 to Make Histograms  Voila!  Of course, you can still change the ranges if you don’t like the results.  And you can construct a frequency table from the histogram.

61. Learning Goal 1: Using the TI-84 to Make Histograms – Your Turn  Using the data given, on sodium in cereals, construct a histogram on your TI – 84 and then using your histogram construct a frequency/relative frequency table.  Use 8 classes, with a lower class limit of 0. Sodium Data: 0 210 260 125 220 290 210 140 220 200 125 170 250 150 170 70 230 200 290 180

62. Learning Goal 1: TI-84 to Make Histogram Using Freq. Table Data 350 to < 450 450 to < 550 550 to < 650 650 to < 750 750 to < 850 850 to < 950 11 10 2 2 2 1  Class Limits  Frequency Same as raw data, using the class midpoint to represent the class.

63. Learning Goal 1: TI-84 to Make Histogram Using Freq. Table Data Enter the data into 2 lists. L 1 is the classes (class midpoint) and L 2 is the frequency.

64. Learning Goal 1: TI-84 to Make Histogram Using Freq. Table Data Turn on Stats Plot1 and select the histogram. Xlist is L 1 the classes and Freq is L 2 the frequencies.

65. Learning Goal 1: TI-84 to Make Histogram Using Freq. Table Data Select ZoomStat to graph the histogram.

66. Learning Goal 1: TI-84 to Make Histogram Using Freq. Table Data Adjust the WINDOW to improve the picture and/or make the values better.

67. Learning Goal 1: TI-84 to Make Histogram Using Freq. Table Data Use the Trace Key to determine values on the graph.

68. Learning Goal 1: Freq. Histogram vs Relative Freq. Histogram Frequency Histogram - a bar graph in which the horizontal scale represents the classes of data values and the vertical scale represents the frequencies.

69. Learning Goal 1: Freq. Histogram vs Relative Freq. Histogram Relative Frequency Histogram - has the same shape and horizontal scale as a histogram, but the vertical scale is marked with relative frequencies.

70. Learning Goal 1: Freq. Histogram vs Relative Freq. Histogram  They look the same with the exception of the vertical axis scale.

71. Learning Goal 1: Freq. Histogram vs Relative Freq. Histogram - Example

72. Learning Goal 1: Histograms - Facts Histograms are useful when the data values are quantitative. A histogram gives an estimate of the shape of the distribution of the population from which the sample was taken. If the relative frequencies were plotted along the vertical axis to produce the histogram, the shape will be the same as when the frequencies are used.

73. Title Number of occurrences (frequencies) are shown on the vertical axis. Label both horizontal and vertical axes. Height of each bar represents the frequency in each class. Note that there are no spaces between bars. (continuous data) Each bar represents a class. The number of classes is usually between 5 and 20. Here, there are 17 classes. The width of each class is determined by dividing the range of the data set by the number of classes, and rounding up. In this data set, the range is 82. 82/17 = 4.8, rounded up to 5. This class goes from 5 to 10. The numbers shown on the horizontal axis are the boundaries of each class. Number of observations. Empty Class: No data were recorded between 75 and 80. NOTE: Sometimes the numbers shown on the horizontal axis are the midpoints of each class. (A class midpoint is also referred to as the mark of the class.) Learning Goal 1: Anatomy of a Histogram

74. STEM AND LEAF PLOT Quantitative Data

75. Learning Goal 1: Stem-and-Leaf Plots What is a stem-and-leaf plot? A stem-and- leaf plot is a data plot that uses part of a data value as the stem to form groups or classes and part of the data value as the leaf. Most often used for small or medium sized data sets. For larger data sets, histograms do a better job. Note: A stem-and-leaf plot has an advantage over a grouped frequency table or histogram, since a stem-and-leaf plot retains the actual data by showing them in graphic form.

76. Learning Goal 1: Stem-and-Leaf Plots  Stem-and-leaf plots are used for summarizing quantitative variables.  Separate each observation into a stem (first part of the number) and a leaf (typically the last digit of the number).  Write the stems in a vertical column ordered from smallest to largest, including empty stems; draw a vertical line to the right of the stems.  Write each leaf in the row to the right of its stem in order. 76

77. Learning Goal 1: Stem and Leaf Plot Construction

78. How to make a stemplot: 1)Separate each observation into a stem, consisting of all but the final (rightmost) digit, and a leaf, which is that remaining final digit. Stems may have as many digits as needed. Use only one digit for each leaf—either round or truncate the data values to one decimal place after the stem. 2)Write the stems in a vertical column with the smallest value at the top, and draw a vertical line at the right of this column. 3)Write each leaf in the row to the right of its stem, in increasing order out from the stem. Title and include key. Original data: 9, 9, 22, 32, 33, 39, 39, 42, 49, 52, 58, 70. Learning Goal 1: Stem-and-Leaf Plots STEM LEAVES Include key – how to read the stemplot. 0|9 = 9

79. Learning Goal 1: Stem-and-Leaf Plots – Picking Stems  Here, use the 10’s digit for the stem unit: Data in ordered array: 21, 24, 24, 26, 27, 27, 30, 32, 38, 41 21 is shown as 38 is shown as 41 is shown as Stem Leaf 2 1 3 8 4 1

80. Learning Goal 1: Stem-and-Leaf Plots – Picking Stems  Completed stem-and-leaf diagram: StemLeaves 21 4 4 6 7 7 30 2 8 41 (continued) Data in ordered array: 21, 24, 24, 26, 27, 27, 30, 32, 38, 41 Key 3 ⃓ 0 = 30

81. Stem Leaf Learning Goal 1: Stem-and-Leaf Plots - Using Other Stem Units  Using the 100’s digit as the stem:  Round off the 10’s digit to form the leaves  613 would become (610) 6 1  776 would become (780) 7 8 ...  1224 becomes (1220) 12 2

82. Learning Goal 1: Stem-and-Leaf Plots - Using Other Stem Units  Using the 100’s digit as the stem:  The completed stem-and-leaf display: (continued) Stem Leaves 6 1 3 6 7 2 2 5 8 8 3 4 6 6 9 9 9 1 3 3 6 8 10 3 5 6 11 4 7 12 2 Data: 613, 632, 658, 717, 722, 750, 776, 827, 841, 859, 863, 891, 894, 906, 928, 933, 955, 982, 1034, 1047,1056, 1140, 1169, 1224 Key 6 ⃓ 3 = 630

83. Learning Goal 1: Stem-and-Leaf Plots - Example Construct a stem-and-leaf diagram, which simultaneously groups the data and provides a graphical display similar to a histogram.

84. Learning Goal 1: Stem-and-Leaf Plots - Example  Put the data in a List in the TI – 84 (STAT/EDIT/L 1 ).  Order the data using sort ascending function (STAT/EDIT/2:SortA(… ) and List 1.

85. Learning Goal 1: Stem-and-Leaf Plots - Example Return to the list (STAT/EDIT) to view ordered data.

86. Learning Goal 1: Stem-and-Leaf Plots - Example  First, list the leading digits of the numbers in the table (3, 4,..., 9) in a column, as shown to the left of the vertical rule.  Next, write the final digit of each number from the table to the right of the vertical rule in the row containing the appropriate leading digit. Do not forget the title and key.

87. Learning Goal 1: Stem-and-Leaf Plots - Variation  Splitting Stems – (too few stems or classes) Split stems to double the number of stems when all the leaves would otherwise fall on just a few stems.  Each stem appears twice.  Leaves 0-4 go on the 1 st stem.  Leaves 5-9 go on the 2 nd stem.

88. Learning Goal 1: Stem-and-Leaf Plots – Split Stems Example  A pediatrician tested the cholesterol levels of several young patients and was alarmed to find that many had levels higher than 200 mg per 100 mL. The table below presents the readings of 20 patients with high levels. Construct a stem-and-leaf diagram for these data by using a. one line per stem. b. Split Stems - two lines per stem.

89. Learning Goal 1: Stem-and-Leaf Plots – Split Stems Example The stem-and-leaf diagram in (a) is only moderately helpful because there are so few stems. (b) is a better stem-and-leaf diagram for these data. It uses Split Stems - two lines for each stem, with the first line for the leaf digits 0-4 and the second line for the leaf digits 5-9. Cholesterol Levels Key 19 ⃓ 9 = 199 Key 19 ⃓ 9 = 199

90. Learning Goal 1: Stem-and-Leaf Plots - Your Turn A sample of the number of admissions to a psychiatric ward at a local hospital during the full phases of the moon is as follows: 22, 30, 21, 27, 31, 36, 20, 28, 25, 33, 21, 38, 32, 35, 26, 19, 43, 30, 30, 34, 27, and 41. Display the data in an appropriate stem-and-leaf plot.

91. Learning Goal 1: Stemplots versus Histograms Stemplots are quick and dirty histograms that can easily be done by hand, therefore, very convenient for back of the envelope calculations. However, they are rarely found in scientific or laymen publications.

92. Learning Goal 1: Stemplots versus Histograms  Stem-and-leaf displays show the distribution of a quantitative variable, like histograms do, while preserving the individual values.  Stem-and-leaf displays contain all the information found in a histogram and, when carefully drawn, satisfy the area principle and show the distribution.