Understanding Cumulative Frequency Analysis

Apr 23, 2025

Cumulative Frequency and Cumulative Frequency Graphs

Definition

  • Cumulative Frequency: The running sum of frequencies.
    • First cumulative frequency = First frequency.
    • Subsequent cumulative frequencies are obtained by adding the current frequency to the previous cumulative frequency.
    • Example Calculation:
      • Frequencies: 1, 2, 4, 3
      • Cumulative Frequencies: 1, 3, 7, 10
  • Characteristics:
    • Always increases.
    • If it decreases, a mistake has been made.

Interpretation

  • Cumulative frequency indicates how many values are less than a certain point.
    • E.g., If the cumulative frequency is 7 at a value of 16, then 7 values are less than 16.

Drawing a Cumulative Frequency Graph

  1. Plot Points:
    • Use the lowest boundary for the first point (x = boundary, y = 0).
    • For each class:
      • x = upper boundary of the class.
      • y = cumulative frequency of the class.
  2. Example:
    • Boundaries: 1, 3, 8, 16, 18
    • Cumulative Frequencies: 0, 1, 3, 7, 10
    • Points: (1, 0), (3, 1), (8, 3), (16, 7), (18, 10)
  3. Connect Points:
    • Use a smooth curve, not straight lines.
    • Y-axis: Cumulative Frequency
    • X-axis: Represents data (e.g., Length)

Insights from the Graph

  • Reflects the information from the cumulative frequency table.
  • Always rises (never falls) due to the nature of cumulative frequency.

Finding Statistical Measures

  • Median: Locate y = n/2 on the graph to find corresponding x.
    • Example: n = 10, median is x at y = 5 (approximately 13.1).
  • Lower Quartile (LQ): Locate y = n/4 on the graph.
    • Example: LQ is approx. 6.6.
  • Upper Quartile (UQ): Locate y = 3n/4 on the graph.
    • Example: UQ is approx. 16.4.
  • Percentiles: E.g., 90th percentile where y = 0.9n.
    • Example: 90th percentile is approx. 17.3.

Greater Than/Less Than Questions

  • Find y for a given x on the graph to determine how many values are greater/less.
  • Example: For x = 10, approximately 7 values are greater than 10.

Box and Whisker Plot

  • Components:
    • Median, LQ, UQ, minimum, maximum.
    • Whiskers extend from minimum to LQ and maximum to UQ.
  • Link to Cumulative Frequency Graph:
    • Convert between the two if n is known.

Summary

  • Cumulative frequency graph provides a visual representation of group data distribution.
  • Offers an easy view of large data sets but is an approximation, not an exact measure.
  • In class, we explored drawing the graph, interpreting, and converting it to/from a box and whisker plot.

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