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Chapter 2

Chapter 2. Descriptive Statistics. Chapter Outline. 2.1 Frequency Distributions and Their Graphs 2.2 More Graphs and Displays 2.3 Measures of Central Tendency 2.4 Measures of Variation 2.5 Measures of Position. 2. Section 2.1. Frequency Distributions and Their Graphs. 3.

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Chapter 2

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  1. Chapter 2 Descriptive Statistics

  2. Chapter Outline • 2.1 Frequency Distributions and Their Graphs • 2.2 More Graphs and Displays • 2.3 Measures of Central Tendency • 2.4 Measures of Variation • 2.5 Measures of Position 2

  3. Section 2.1 Frequency Distributions and Their Graphs 3

  4. Section 2.1 Objectives • Construct frequency distributions • Construct frequency histograms, frequency polygons, relative frequency histograms, and ogives 4

  5. Summarizing data/Frequency tables • When data is collected from a survey or designed experiment, they must be organized into a manageable form. Data that is not organized is referred to as raw data. • Ways to Organize Data: Tables; Graphs; and Numerical Summaries. • A frequency distribution lists classes (or categories) of values, along with frequencies (or counts) of the number of values that fall into each class. 5

  6. Frequency tables: Definitions • Lower Class Limits: are the smallest numbers that can actually belong to different classes. • Upper Class Limits: are the largest numbers that can actually belong to different classes. • Class Width: is the difference between two consecutive lower class limits or two consecutive upper class limits. 6

  7. Frequency Distribution Frequency Distribution • A table that shows classes or intervals of data with a count of the number of entries in each class. • The frequency, f, of a class is the number of data entries in the class. Class width 6 – 1 = 5 Lower class limits Upper class limits 7

  8. Constructing a Frequency Distribution • Decide on the number of classes. • Usually between 5 and 20; otherwise, it may be difficult to detect any patterns. • Find the class width. • Determine the range of the data. • Divide the range by the number of classes. • Round up to the next convenient number. 8

  9. Constructing a Frequency Distribution • Find the class limits. • You can use the minimum data entry as the lower limit of the first class. • Find the remaining lower limits (add the class width to the lower limit of the preceding class). • Find the upper limit of the first class. Remember that classes cannot overlap. • Find the remaining upper class limits. 9

  10. Constructing a Frequency Distribution • Make a tally mark for each data entry in the row of the appropriate class. • Count the tally marks to find the total frequency f for each class. 10

  11. Guidelines for Frequency tables 1. Be sure that the classes are mutually exclusive. 2. Include all classes, even if the frequency is zero. 3. Use the same width for all classes. 4. Select convenient numbers for class limits. 5. Use between 5 and 20 classes. 6. The sum of the class frequencies must equal the number of original data values. 11

  12. Example: Constructing a Frequency Distribution The following sample data set lists the number of minutes 50 Internet subscribers spent on the Internet during their most recent session. Construct a frequency distribution that has seven classes. 50 40 41 17 11 7 22 44 28 21 19 23 37 51 54 42 86 41 78 56 72 56 17 7 69 30 80 56 29 33 46 31 39 20 18 29 34 59 73 77 36 39 30 62 54 67 39 31 53 44 12

  13. Solution: Constructing a Frequency Distribution • Number of classes = 7 (given) • Find the class width 50 40 41 17 11 7 22 44 28 21 19 23 37 51 54 42 86 41 78 56 72 56 17 7 69 30 80 56 29 33 46 31 39 20 18 29 34 59 73 77 36 39 30 62 54 67 39 31 53 44 Round up to 12 13

  14. Solution: Constructing a Frequency Distribution Use 7 (minimum value) as first lower limit. Add the class width of 12 to get the lower limit of the next class. 7 + 12 = 19 Find the remaining lower limits. Class width = 12 19 31 43 55 67 79 14

  15. Solution: Constructing a Frequency Distribution The upper limit of the first class is 18 (one less than the lower limit of the second class). Add the class width of 12 to get the upper limit of the next class. 18 + 12 = 30 Find the remaining upper limits. Class width = 12 18 30 42 54 66 78 90 15

  16. Solution: Constructing a Frequency Distribution Make a tally mark for each data entry in the row of the appropriate class. Count the tally marks to find the total frequency f for each class. Σf = 50 16

  17. Determining the Midpoint Midpoint of a class Class width = 12 17

  18. Determining the Relative Frequency Relative Frequency of a class • Portion or percentage of the data that falls in a particular class. 18

  19. Determining the Cumulative Frequency Cumulative frequency of a class • The sum of the frequency for that class and all previous classes. 6 + 16 + 29 19

  20. Expanded Frequency Distribution Σf = 50 20

  21. Graphs of Frequency Distributions Frequency Histogram • A bar graph that represents the frequency distribution. • The horizontal scale is quantitative and measures the data values. • The vertical scale measures the frequencies of the classes. • Consecutive bars must touch. frequency data values 21

  22. Class Boundaries Class boundaries • The numbers that separate classes without forming gaps between them. • The distance from the upper limit of the first class to the lower limit of the second class is 19 – 18 = 1. • Half this distance is 0.5. 6.5 – 18.5 • First class lower boundary = 7 – 0.5 = 6.5 • First class upper boundary = 18 + 0.5 = 18.5 22

  23. Class Boundaries 23

  24. Example: Frequency Histogram Construct a frequency histogram for the Internet usage frequency distribution. 24

  25. Solution: Frequency Histogram (using Midpoints) 25

  26. Solution: Frequency Histogram (using class boundaries) 6.5 18.5 30.5 42.5 54.5 66.5 78.5 90.5 You can see that more than half of the subscribers spent between 19 and 54 minutes on the Internet during their most recent session. 26

  27. Graphs of Frequency Distributions Frequency Polygon • A line graph that emphasizes the continuous change in frequencies. • The line segments are extended to the right and left so that the graph begins and ends on the x-axis. frequency data values 27

  28. Example: Frequency Polygon Construct a frequency polygon for the Internet usage frequency distribution. 28

  29. Solution: Frequency Polygon The graph should begin and end on the horizontal axis, so extend the left side to one class width before the first class midpoint and extend the right side to one class width after the last class midpoint. You can see that the frequency of subscribers increases up to 36.5 minutes and then decreases. 29

  30. Graphs of Frequency Distributions Relative Frequency Histogram • Has the same shape and the same horizontal scale as the corresponding frequency histogram. • The vertical scale measures the relative frequencies, not frequencies. relative frequency data values 30

  31. Example: Relative Frequency Histogram Construct a relative frequency histogram for the Internet usage frequency distribution. 31

  32. Solution: Relative Frequency Histogram 6.5 18.5 30.5 42.5 54.5 66.5 78.5 90.5 From this graph you can see that 20% of Internet subscribers spent between 18.5 minutes and 30.5 minutes online. 32

  33. Graphs of Frequency Distributions Cumulative Frequency Graph or Ogive • A line graph that displays the cumulative frequency of each class at its upper class boundary. • The upper boundaries are marked on the horizontal axis. • The cumulative frequencies are marked on the vertical axis. • Ogives are useful for determining the number of values less than some particular class boundary. cumulative frequency data values 33

  34. Constructing an Ogive • Construct a frequency distribution that includes cumulative frequencies as one of the columns. • Specify the horizontal and vertical scales. • The horizontal scale consists of the upper class boundaries. • The vertical scale measures cumulative frequencies. • Plot points that represent the upper class boundaries and their corresponding cumulative frequencies. 34

  35. Constructing an Ogive • Connect the points in order from left to right. • The graph should start at the lower boundary of the first class (cumulative frequency is zero) and should end at the upper boundary of the last class (cumulative frequency is equal to the sample size). 35

  36. Example: Ogive Construct an ogive for the Internet usage frequency distribution. 36

  37. Solution: Ogive 6.5 18.5 30.5 42.5 54.5 66.5 78.5 90.5 From the ogive, you can see that about 40 subscribers spent 60 minutes or less online during their last session. The greatest increase in usage occurs between 30.5 minutes and 42.5 minutes. 37

  38. Practice Questions Q. (2.1) The average quantitative GRE scores for the top 30 graduate schools of engineering are listed below. Construct a frequency distribution with six classes. 38

  39. Practice Questions Q. (2.2) The ages of the signers of the Declaration of Independence are shown below (age is approximate since only the birth year appeared in the source and one has been omitted since his birth year is unknown). Construct a frequency distribution for the data using seven classes. 39

  40. Class limits Frequency 90 – 98 6 99 – 107 22 108 – 116 43 117 – 125 28 126 – 134 9 Practice Questions Q. (2.3) For 108 randomly selected college applicants, the following frequency distribution for entrance exam scores was obtained. Construct a histogram, frequency polygon, and ogive for the data. 40

  41. Practice Questions Q. (2.4) The number of calories per serving for selected ready – to - eat cereals is listed here. Construct a frequency distribution using seven classes. Draw a histogram, frequency polygon, and ogive for the data, using relative frequencies. Describe the shape of the histogram. 41

  42. Section 2.1 Summary • Constructed frequency distributions • Constructed frequency histograms, frequency polygons, relative frequency histograms and ogives 42

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