Histograms: A Method for Displaying Continuous Data

Introduction

• Why histograms?
  • Means, median, standard deviations don't tell the whole story.
  • Distribution shapes are important and not captured by single summaries.

What is a Histogram?

• Definition: Displays distribution of data by charting the number or percentage of observations within predefined numerical ranges.
• Similarity to Bar Charts: Histograms are similar to bar charts but focus on data distribution.

Example: Age Data from 1995 Statistical Abstract (US)

• Dataset: Proportions of individuals over 65 years for the 50 states.
• Interesting Findings:
  • Smallest percentage: Alaska (4.6%)
  • Largest percentage: Florida (18.4%)

Steps to Create a Histogram

• Step 1: Break data into mutually exclusive, equally sized bins.
• Step 2: Count observations in each bin.
• Notation: Use brackets and parentheses to define precise ranges.

Example Breakdown

• Bin Setup: Bins from 4% to 19% with 1% width.
• Observation Counts:
  • 4-5%: 1 observation
  • 5-6%: 0 observations
  • 6-7%: 0 observations
  • 8-9%: 1 observation
  • More action within 10-16% range.
• Graphical Summary: Histogram visualizes where values are centered and spread out.

Blood Pressure Data Example

• Dataset: Blood pressure data from 113 men.
• Statistics: Mean = 123.6 mmHg, Standard deviation = 12.9 mmHg.
• Histogram Properties:
  • Bin Width: 5 mmHg, height represents number of men.
  • Shape: Symmetric, bell-shaped around mean and median.
  • Arbitrary Bin Width:
    • 20 mmHg width: Too crude.
    • 1 mmHg width: Too fine.
  • Percentage Representation: Vertical axis can represent percentages instead of counts.

Choosing Bin Width and Number

• General Guidance:
  • Dependent on sample size and data spread.
  • Rough rule: Number of intervals ≈ √sample size.
  • Example: 10 observations ≈ 3 bins, 50 observations ≈ 7 bins, 100 observations ≈ 10 bins.
• Computer Selection: Computers typically choose the optimal number of bins for you.

Conclusion

• Histograms are useful for summarizing continuous data and understanding distribution shapes.
• Other visualization options will be explored in the next section.