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Chapter 2: Frequency Distributions

Chapter 2: Frequency Distributions. Variables and Values. Variable : A characteristic that takes on two or more values among different people or things Other Names for “Values” Scores Data points Categories Other Names for “People” or “Objects” Cases Observations Participants

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Chapter 2: Frequency Distributions

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  1. Chapter 2:Frequency Distributions

  2. Variables and Values • Variable: A characteristic that takes on two or more values among different people or things • Other Names for “Values” • Scores • Data points • Categories • Other Names for “People” or “Objects” • Cases • Observations • Participants • Individuals

  3. Frequencies • Frequency: The number of cases having a given score on a variable • Example: Blood type of 500 randomly selected residents of Finland (“Finns”) • Cases: 500 Finns • Variable: Blood type • Scores/Values: O, A, B, and AB • Frequency Distribution: See next slide

  4. Frequencies Table 1. Frequency Distribution for Blood Type of 500 Finns

  5. Relative Frequencies:Motivating Example • Researchers analyzed 400 personal ads • 165 placed by males, 235 placed by females • In Table 2, they tallied the number of ads where each of 4 characteristics of a potential mate was mentioned • Age • Appearance • Financial status • Emotional attractiveness (e.g., warm, romantic, considerate) • They wanted to know whether there were gender differences in characteristics mentioned in personal ads

  6. Relative Frequencies: Motivating Example Table 2. Frequencies for Personal Ad Characteristics by Gender - How do men and women differ in the characteristics mentioned in personal ads?

  7. Relative Frequencies: Motivating Example • Problem: We cannot directly compare frequencies on each characteristic because the sample sizes differ • 165 males • 235 females • Solution: Compute relative frequencies to make the groups comparable

  8. Two Types of Relative Frequencies • Proportion (P): The frequency for each score on a variable divided by the total number of cases: • f = frequency • N = total number of cases

  9. Two Types of Relative Frequencies • Percentage (%): The proportion multiplied by 100 • It re-expresses frequencies as if there were a total of 100 cases

  10. Relative Frequencies:Application to Motivating Example • What proportion of males mentioned appearance? • What percentage of females mentioned emotional attractiveness?

  11. Relative Frequencies: Application to Motivating Example Table 3. Percentages for Personal Ad Characteristics by Gender • Now what do you conclude about gender differences in characteristics of potential mates mentioned in personal ads?

  12. Finding Frequencies From Relative Frequencies • Motivation: Exposure to Violence • Researchers reported the following statistics about exposure to violence in a sample 178 of Detroit adolescents: • The proportion of adolescents who were shot at was 0.118 • 12.4% of adolescents had witnessed a murder • Sometimes you want to know frequencies instead • How many adolescents were shot at? • How many adolescents witnessed a murder?

  13. Finding Frequencies From Relative Frequencies • Frequency From a Proportion: Multiply the proportion (P) by the total number of cases (N) • P = proportion • N = total number of cases

  14. Finding Frequencies From Relative Frequencies • Frequency From a Percentage: Multiply the percentage (%) by the total number of cases (N), then divide by 100 • % = percentage • N = total number of cases

  15. Finding Frequencies From Relative Frequencies • Application to Motivating Example • Frequency of adolescents who were shot at: • Frequency of adolescents who witnessed a homicide

  16. Motivating Example: Tattoos • We interviewed 40 college students • We asked them to indicate how many tattoos they have • Their responses are provided in Table 4 (next slide) • How would you characterize the distribution of tattoos for these 40 college students?

  17. Motivating Example: Tattoos Table 4. Number of Tattoos for N = 40 College Students

  18. Motivating Example: Tattoos • Problem: It is difficult to characterize the distribution of tattoos based on Table 4 • What We Need: A way to organize and summarize the data in a systematic and concise way • Solution: We can construct a frequency distribution

  19. Frequency Distribution • Frequency Distribution: A table reporting the number and percentage of cases having each score on a variable • Steps in Constructing a Frequency Distribution • Step 1: Tally the number of cases for each score on a variable (this will give you the frequency) • Step 2: Calculate the percentage of cases for each score on a variable • Step 3: Report the frequencies and percentages in a table

  20. Constructing a Frequency Distribution:Step 1 (Tally to get Frequencies)

  21. Constructing a Frequency Distribution:Step 2 (Calculate Percentages)

  22. Constructing a Frequency Distribution:Step 3 (Construct a Table) Table 5. Frequency Distribution for Number of Tattoos Among 40 College Students

  23. Features of a Frequency Distribution • First Column • Variable name in the first row • Scores/values of the variables below the variable name • Second Column • Number of cases (frequencies) for each score/value • Total number of cases in the last row • Third Column • Percentage of cases for each score/value • Total percentage (100%) in the last row • The total percentage might not be exactly equal 100% due to rounding

  24. Data for Future Examples • Cases: 24 vehicles passing Peach Street on North Locust Street (Hwy. 77) • Variables and Scores • Gender of Driver (1 = Male, 2 = Female) • Size of Vehicle (1 = Small, 2 = Medium, 3 = Large) • Age of Driver (in years) • Speed of Vehicle (mph) • Data: Table 6 (next slide)

  25. Data for Future Examples: Table 6. Characteristics of 24 Vehicles and Drivers

  26. Arrangement of Scores in a Frequency Distribution • Nominal Variables: List the scores in any meaningful order • Ordinal Variables: List the scores from smallest to largest or from largest to smallest • Interval-Ratio Variables • When there are relatively few scores, list them as you would for an ordinal variable • When there are many scores, group the scores into meaningful groups called class intervals

  27. Table 7. Frequency Distribution for Gender of Driver

  28. Table 8. Frequency Distribution for Size of Vehicle

  29. Table 9. Frequency Distribution For Age of Driver (Ungrouped)

  30. Table 10. Frequency Distribution for Age of Driver (Grouped)

  31. Table 11. Frequency Distribution for Vehicle Speed

  32. Cumulative Frequencies and Percentages • Motivation: Sometimes we want to know the number or percentage of cases above or below some score on an ordinal or interval-ratio variable • Example 1: How many people exceed the speed limit (35 mph)? • Example 2: The speed limit will be changed to the value at the 85th percentile. At what speed is the 85th percentile? • Solution: Our work can be made easier by computing cumulative frequencies and percentages

  33. Cumulative Frequency (Cf) • Interpretation: Number of cases at or below a given score on a variable • Types of Variables: Ordinal or interval-ratio • Computation • Begin with the frequency at the lowest score of a variable • Add to it the frequency at the next value of the variable and report it • Continue this until you reach the last score of a variable • The last step will yield the number of cases (N)

  34. Cumulative Percentage (C%) • Interpretation: Percentage of cases at or below a given score on a variable • Types of Variables: Ordinal or interval-ratio • Computation: • Begin with the percentage at the lowest score of a variable • Add to it the percentage at the next value of the variable and report it • Continue this until you reach the last score of a variable • The last step will yield 100% (or close to it, depending on rounding) • Percentile: Another name for cumulative percentage

  35. Cumulative Frequencies and Percentages: Example of Vehicle Speed (Data in Table 6)

  36. Table 12. Frequencies and Percentages for Vehicle Speed

  37. Cumulative Frequencies and Percentages: Example Questions • Example 1: How many people exceed the speed limit (35 mph)? • Method 1: Sum up the number of people above 35: 6 + 3 = 9 • Method 2: Use cumulative frequencies: 24 – 15 = 9 • Example 2: The speed limit will be changed to the value at the 85th percentile. At what speed is the 85th percentile? • Answer: This is the speed where the cumulative percent goes above 85%, which is 40 mph

  38. Rates • Rate: The number of occurrences of an event divided by the number of possible occurrences • Multiplied by Large Number: A rate is often multiplied by a large number (1,000 or 10,000 or 100,000) to make it easier to interpret

  39. Example of Computing a Rate:U.S. Suicide Rate in 2005* *Source: American Association of Suicidology (http://www.suicidology.org)

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