Lecture on Graphs of Frequency Distributions

Overview

In this lecture, we discussed various types of graphs used to represent frequency distributions, including:

• Frequency Histograms
• Frequency Polygons
• Relative Frequency Histograms
• Ogives (Cumulative Frequency Graphs)

Frequency Histogram

• Definition: A bar graph representing frequency distribution.
• Horizontal Scale: Quantitative, measures data values.
• Vertical Scale: Measures frequencies of classes.
• Bars: Must touch, indicating class boundaries (no gaps).
• Class Boundaries: Numbers separating classes without gaps.
  • Example:
    • Upper limit of first class = 190
    • Lower limit of second class = 191
    • Difference = 1, half = 0.5
    • First class boundary: 154.5 to 190.5
• Construction:
  • Calculate class boundaries.
  • Draw bars from boundaries (e.g., 154.5 to 190.5 with frequency of 3).
  • Consider midpoints for alternative construction (e.g., midpoint of first class is 172.5).

Frequency Polygon

• Definition: A line graph emphasizing continuous frequency change.
• Construction:
  • Use same scales as histogram, label with class midpoints.
  • Start and end on x-axis.
  • Example midpoint: 172.5
  • Class width: 36
  • Plot and connect midpoints, ensuring start and end touch x-axis.

Relative Frequency Histogram

• Definition: Similar to frequency histogram, but vertical scale shows relative frequencies.
• Construction:
  • Use relative frequencies instead of absolute frequencies.
  • Maintain same class boundaries.
  • Example: Relative frequency of first class is 0.1.

Ogive (Cumulative Frequency Graph)

• Definition: Line graph displaying cumulative frequencies at each class's upper boundary.
• Construction:
  • Upper boundaries on horizontal axis, cumulative frequencies on vertical.
  • Connect points from left to right, starting at zero.
  • Start graph at lower boundary of first class.
  • Example:
    • First class lower boundary: 154.5, start at 0.
    • Plot cumulative points (e.g., at 190.5 cumulative frequency is 3).
    • Final point (406.5) should equal sample size (e.g., 30).
• Insights:
  • Shows cumulative totals (e.g., 10 adults spent $262.50 or less).
  • Indicates greatest increases in frequency.

Conclusion

• Understanding these graphs provides insight into data distribution and cumulative trends in datasets.
• Each type of graph serves different purposes, useful in different analytical contexts.