Frequency Distributions
Frequency Distributions: Key Concepts
Vocabulary Introduced
Lower Limits (LL): Starting values of the data classes or bins in a frequency distribution.
Upper Limits (UL): Ending values of the data classes or bins in a frequency distribution.
Class Midpoints: The middle value of each class, calculated as the average of the lower and upper limits for that class.
Class Width: The difference between the lower limits of consecutive classes.
Cumulative Frequency: The running total of frequencies, showing the total number of observations that fall within each class up to that point.
Relative Frequency: The frequency of each class expressed as a proportion of the total number of observations, often represented as a fraction, decimal, or percentage.
Example Scenario
A fictional exam has student scores ranging from 41 to 99.
Notable scores:
Lowest score: 41
Highest score: 99
Part A: Constructing a Frequency Distribution
Bins (Classes): Organizing scores by ranges:
40 - 49
50 - 59
60 - 69
70 - 79
80 - 89
90 - 99
Frequency Calculation for each bin:
40 - 49: 2 students (scores: 41, 46)
50 - 59: 2 students
60 - 69: 6 students
70 - 79: 5 students
80 - 89: 3 students
90 - 99: 2 students
Part B: Lower and Upper Limits
Lower Limits: 40, 50, 60, 70, 80, 90
Upper Limits: 49, 59, 69, 79, 89, 99
Part C: Class Width
Calculation of Class Width:
Class Width = Lower limit of the next class - Lower limit of the current class
Example: 50 - 40 = 10
Result: Class Width = 10
Part D: Class Boundaries
Definition of Class Boundaries: Values that represent the range of the classes, helping to address scores that may fall precisely on the boundary of two classes.
Class Boundary Calculation:
For class 40-49, the boundaries are 39.5 to 49.5
Subsequent boundaries: 49.5 + 10 = 59.5, …, up to 99.5.
List of Class Boundaries:
39.5 - 49.5
49.5 - 59.5
59.5 - 69.5
69.5 - 79.5
79.5 - 89.5
89.5 - 99.5
Part E: Class Midpoints
Class Midpoint Calculation:
Midpoint = (Lower Limit + Upper Limit) / 2
Example for 40 - 49: (40 + 49) / 2 = 44.5
Class Midpoints: 44.5, 54.5, 64.5, 74.5, 84.5, 94.5
Part F: Number of Students
To find total students, sum the frequencies:
Total = 2 + 2 + 6 + 5 + 3 + 2 = 20 students
Part G: Cumulative Frequency
Cumulative Frequency Definition and Calculation:
First class cumulative frequency = frequency of that class.
Cumulative frequency for each class:
40 - 49: 2
50 - 59: 2 + 2 = 4
60 - 69: 4 + 6 = 10
70 - 79: 10 + 5 = 15
80 - 89: 15 + 3 = 18
90 - 99: 18 + 2 = 20
Total cumulative frequency matches total surveyed students: 20.
Part H: Relative Frequency Distribution
Relative Frequency Definition: Indicates what fraction or percentage each class contributes to the total:
Relative Frequency Calculation for each class:
40 - 49: 2/20 = 0.10 = 10%
50 - 59: 2/20 = 0.10 = 10%
60 - 69: 6/20 = 0.30 = 30%
70 - 79: 5/20 = 0.25 = 25%
80 - 89: 3/20 = 0.15 = 15%
90 - 99: 2/20 = 0.10 = 10%
Summary of Key Concepts
Cumulative Frequencies: Important for understanding distributions and supplementing data analysis, especially in advanced statistics and probability distributions.
Relative Frequencies: Crucial for determining proportions of data points within specified ranges, helping contextualize performance across categories (e.g., grades).