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Describing Data with Tables and Graphs A frequency distribution is a collection of observations produced by sorting observations into classes and showing their frequency (f) of occurrence in each class. Ungrouped data Grouped data ◦ Group data ◦ Lose detail ◦ Gain simplified picture of the data 1. Each observation should be included in one, and only one, class. 2. List all classes, even those with zero frequencies. 3. All classes should have equal intervals. Optional 4. All classes should have both an upper boundary and a lower boundary 5. Select the class interval from convenient numbers, such as 1, 2, 3, … 10, particularly 5 and 10 or multiples of 5 and 10. 6. The lower boundary of each class interval should be a multiple of the class interval. 7. Aim for a total of approximately 10 classes. 1. 2. 3. 4. 5. 6. 7. 8. 9. Find the range. Find the class interval required to span the range. Round off to the nearest convenient interval. Determine where the lowest class should begin. Determine where the lowest class should end. Working upward, list as many equivalent classes as are required to include the largest observation. Indicate with a tally the class in which each observation falls. Replace the tally count with each class with a frequency (f) and show the total of all frequencies. Supply the headings for both columns and a title for the table. Highest – Lowest data value. Range/desired number of classes Keep the intervals equal The lowest score should be a multiple of the class interval. Add the interval value to the lowest value and subtract 1 Working upward, list as many equivalent classes as are required to include the largest observation Indicate with a tally the class in which each observation falls Replace the tally count with each class with a frequency (f) and show the total of all frequencies. Supply the headings for both columns and a title for the table Progress Check 2.2 (page 32) ◦ Construct a frequency distribution for grouped data. 91 85 84 79 80 87 96 75 86 104 95 71 105 90 77 123 80 100 93 108 98 69 99 95 90 110 109 94 100 103 112 90 90 98 89 Relative frequency distribution (page 35) ◦ Relative frequency distributions show the frequency of each class as a part or fraction of the total frequency for the entire distribution GRE f 725-749 1 700-724 3 675-699 14 650-674 30 625-649 34 600-624 42 575-599 30 550-574 27 525-549 13 500-524 4 475-499 2 Total 200 Cumulative frequency distribution (page 36) ◦ Cumulative frequency distributions show the number of observations in each class and in all lower-ranked classes. ◦ Use the data from progress check 2.5 and create a cumulative frequency distribution. See table 2.6 page 37. Percentile ranks (page 38) ◦ Percentile rank of a score indicates the percentage of scores in the entire distribution with similar or smaller values than that score. Examples of Frequency distributions for qualitative data (p 38) Figure 1. Pie chart of iMac purchases illustrating frequencies of previous computer ownership. Figure 2. Bar chart of iMac purchases as a function of previous computer ownership. Figure 5. A line graph of the number of people playing different card games on Sunday and Wednesday. ◦ A line graph for quantitative data which also emphasizes the continuity of continuous variables. Histograms (page 41) Equal units along the horizontal axis (X) Equal units along the vertical axis (Y) Intersection of axes define the origin (0) Numerical scales increase from left to right, bottom to top. ◦ Body reflects frequency for the classes reflected by height of bars. ◦ ◦ ◦ ◦ Stem and leaf display ◦ A device for sorting quantitative data on the basis of leading and trailing digits. (progress check 2.10 page 45) Construct a stem and leaf display for the following IQ scores obtained from a group of four-year-old children. 120 126 108 102 98 85 141 132 118 88 123 109 117 124 137 106 99 104 78 143 111 113 96 Frequency polygons ◦ Common shapes Normal Bimodal Positively skewed Negatively skewed Decide on the appropriate type of graph. Draw the horizontal axis, then the vertical axis. Identify the string of class intervals for horizontal axis. Superimpose the string of class intervals (with gaps for bar graphs) along the horizontal axis. Superimpose progression of numbers along vertical axis. Construct bars to reflect the frequency. Supply labels for both axes and a title. 2.9% 3.0% 3.0% 3.0% Q1 Q2 Q3 2.6% 0.1% Q2 Q3 2007 Q4 2008 This link to a histogram applet shows the duration, in minutes, for geyser eruptions of Old Faithful in Yellowstone National Park. To see how varying bin (bar) widths affect the shape of the data, change the width by using your mouse to drag the arrow underneath the bin width scale at the bottom of the histogram. Progress Check 2.13 (p49)