Descriptive Statistics 2 Descriptive Statistics Elementary Statistics

Frequency Distributions and Their Graphs Section 2.1 Frequency Distributions and Their Graphs

Frequency Distributions Minutes Spent on the Phone 102 124 108 86 103 82 71 104 112 118 87 95 103 116 85 122 87 100 105 97 107 67 78 125 109 99 105 99 101 92 This data will be used for several examples. You might want to duplicate it for use with slides that follow. Make a frequency distribution table with five classes. Minimum value = Maximum value = 67 Key values: 125

Steps to Construct a Frequency Distribution 1. Choose the number of classes Should be between 5 and 15. (For this problem use 5) 2. Calculate the Class Width Find the range = maximum value – minimum. Then divide this by the number of classes. Finally, round up to a convenient number. (125 - 67) / 5 = 11.6 Round up to 12. 3. Determine Class Limits The lower class limit is the lowest data value that belongs in a class and the upper class limit is the highest. Use the minimum value as the lower class limit in the first class. (67) 4. Mark a tally | in appropriate class for each data value. After all data values are tallied, count the tallies in each class for the class frequencies.

Construct a Frequency Distribution Minimum = 67, Maximum = 125 Number of classes = 5 Class width = 12 3 5 8 9 Class Limits Tally 78 90 102 114 126 67 79 91 103 115 Do all lower class limits first.

Frequency Histogram Boundaries Class 3 66.5 - 78.5 67 - 78 78.5 - 90.5 90.5 - 102.5 102.5 -114.5 114.5 -126.5 Class 67 - 78 79 - 90 91 - 102 103 -114 115 -126 3 5 8 9 Time on Phone 9 9 8 8 7 6 5 5 5 A histogram is a bar graph for which the bars touch. To form boundaries, find the distance between consecutive classes. Add half that distance to the lower limits and half to the upper limits. In this case the distance is 1 unit so add .5 to all upper limits and subtract .5 from all lower ones. The data must be quantitative. This histogram is labeled at the class boundaries. Explain that midpoints could have been labeled instead. 4 3 3 2 1 6 6 . 5 7 8 . 5 9 . 5 1 2 . 5 1 1 4 . 5 1 2 6 . 5 minutes

Other Information Midpoint: (lower limit + upper limit) / 2 Relative frequency: class frequency/total frequency Cumulative frequency: number of values in that class or in lower Class Midpoint Relative Frequency Cumulative Frequency 67 - 78 79 - 90 91 - 102 103 - 114 115 - 126 3 5 8 9 72.5 84.5 96.5 108.5 120.5 0.10 0.17 0.27 0.30 3 8 16 25 30 The first two columns reflect the work done in previous slides. Once the first midpoint is calculated, the others can be found by adding the class width to the previous midpoint. Notice the last entry in the cumulative frequency column is equal to the total frequency.

Relative Frequency Histogram Time on Phone Relative frequency A relative frequency histogram will have the same shape as a frequency histogram. minutes Relative frequency on vertical scale

More Graphs and Displays Section 2.2 More Graphs and Displays

Stem-and-Leaf Plot Lowest value is 67 and highest value is 125, so list stems from 6 to 12. 102 124 108 86 103 82 Stem Leaf 6 | 7 | 8 | 9 | 10 | 11 | 12 | 6 2 Divide each data value into a stem and a leaf. The leaf is the rightmost significant digit. The stem consists of the digits to the left. The data shown represent the first line of the ‘minutes on phone’ data used earlier. The complete stem and leaf will be shown on the next slide. 2 8 3 To see complete display, go to next slide. 4

Stem-and-Leaf Plot Key: 6 | 7 means 67 6 | 7 7 | 1 8 8 | 2 5 6 7 7 6 | 7 7 | 1 8 8 | 2 5 6 7 7 9 | 2 5 7 9 9 10 | 0 1 2 3 3 4 5 5 7 8 9 11 | 2 6 8 12 | 2 4 5 Key: 6 | 7 means 67 Stress the importance of using a key to explain the plot. 6|7 could mean 6700 or .067 for a different problem. A stem and leaf should not be used with data when values are very different such as 3, 34,900, 24 etc. The stem-and leaf has the advantage over a histogram of retaining the original values.

NASA budget (billions of $) divided among 3 categories. Pie Chart Used to describe parts of a whole Central Angle for each segment NASA budget (billions of $) divided among 3 categories. Pie charts help visualize the relative proportion of each category. Find the relative frequency for each category and multiply it by 360 degrees to find the central angle. Billions of $ Human Space Flight 5.7 Technology 5.9 Mission Support 2.7 Construct a pie chart for the data.

Pie Chart Human Space Flight 5.7 143 Technology 5.9 149 Billions of $ Degrees Human Space Flight 5.7 143 Technology 5.9 149 Mission Support 2.7 68 14.3 360 Total Mission Support 19% Human Space Flight 40% NASA Budget (Billions of $) Technology 41%