Lecture on Histograms

Introduction to Histograms

• A histogram is a type of graph consisting of bars of equal width, drawn adjacent to each other without gaps.
• Displays frequency distribution of quantitative data.
• Horizontal scale: Represents classes of quantitative data values (labelled with class boundaries or midpoints).
• Vertical scale: Represents frequencies (height of the bar corresponds to frequency).

Example of Creating a Histogram

• Using Class Boundaries:
  • Class boundaries are the centers of gaps between classes.
  • Example: First class ends at 19, second begins at 20; boundary = 19.5 (found by averaging 19 and 20).
  • Continue this method to find all class boundaries.
  • First boundary: 20 less than 19.5 = -0.5
  • Last boundary: 20 more than 119.5 = 139.5
• Constructing the Histogram:
  • Horizontal scale: x-values using class boundaries.
  • Vertical scale: Frequencies, accommodates up to 12 (example scale up to 15).
  • Bar heights correspond to class frequencies (e.g., first class with frequency of 5, bar height = 5).
• Resulting histogram shows approximate normal distribution (bell-shaped).

Relative Frequency Histogram

• Same shape and horizontal scale as a histogram, but vertical scale uses relative frequencies.
• Steps to Create:
  • Calculate class midpoints (e.g., 0 and 9 = 4.5 midpoint).
  • Calculate relative frequency (class frequency divided by total frequency).
  • Example: Total frequencies = 99, Relative frequency for first class = 20/99 ≈ 20%.
  • Construct bars proportional to relative frequencies.
• Relative frequency histogram may not be normal; different distribution shapes possible.

Common Histogram Shapes

• Normal Distribution:
  • Bell-shaped, symmetric.
• Uniform Distribution:
  • Bars of approximately equal height, similar frequencies across classes.
• Skewness:
  • Skewed if not symmetric; extends more to one side.
  • Right Skewed (Positively Skewed): Longer right tail.
  • Left Skewed (Negatively Skewed): Longer left tail.
  • Memory aid: Toes on feet (right foot toes smaller to right for positive skew, left foot toes smaller to left for negative skew).

Conclusion

• Understanding histograms and their properties is crucial for analyzing the distribution of data.
• Different histogram shapes can provide insights into the nature of the data distribution.