Welcome back, class! In Chapter 3: Visualizing Data, we move away from raw lists of numbers and start creating pictures that tell a story. Whether you are looking at political polls, sales figures, or biological data, the ability to organize and graph data is fundamental to Statistics. Here is a breakdown of the key concepts from our recent lecture notes.

Part 1: Qualitative Data (Categorical)

When dealing with categories (like political affiliation or customer ratings), we focus on grouping and counting.

  • Frequency Distribution: This is a summary technique that organizes data into classes and provides a count of observations in each class.
  • Relative Frequency: This represents the percentage of the total data that falls into a specific class. It is calculated as: $$ \text{Relative Frequency} = \frac{\text{Frequency}}{\text{Total Sample Size}} $$
  • Bar Charts: These can be vertical or horizontal. A specific type of bar chart, the Pareto Chart, arranges the bars in decreasing order of frequency to highlight the most significant categories.
  • Pie Charts: These are useful for showing parts of a whole, where the slices represent relative frequencies (percentages).

Part 2: Quantitative Data (Numerical)

When dealing with numbers (like heart rates or test scores), constructing a frequency distribution requires a bit more math. We have to create our own "bins" or classes.

The Recipe for a Quantitative Frequency Distribution:

  1. Determine the Number of Classes: Usually between 5 and 20. Too few classes compress the data too much; too many provide too little summary.
  2. Calculate Class Width: This is a crucial step. Use the following formula: $$ \text{Class Width} \approx \frac{\text{Maximum Value} - \text{Minimum Value}}{\text{Number of Classes}} $$ Note: Always round this result up to the next convenient number (usually an integer) to ensure all data is covered.
  3. Set Class Limits: establish your Lower Class Limits and Upper Class Limits so that they do not overlap.

Advanced Calculations:

  • Cumulative Frequency: The sum of the frequency for a particular class and all preceding classes. By the final class, this should equal your total sample size ($n$).
  • Cumulative Relative Frequency: The proportion of observations that fall into a specific class and all preceding classes. This sums up to 1.0 (or 100%).

Part 3: Graphing Quantitative Data

Once we have our tables, we visualize them using specific tools:

  • Histograms: These look like bar charts, but for continuous quantitative data, the bars must touch. The horizontal axis represents the classes, and the vertical axis represents frequency.
  • Distribution Shapes: When looking at a histogram, identifying the shape is key:
    • Symmetric (Bell-Shaped): The left and right sides are mirror images.
    • Skewed Left (Negatively Skewed): The "tail" of the graph extends to the left.
    • Skewed Right (Positively Skewed): The "tail" extends to the right.
    • Uniform: All bars are roughly the same height.
  • Stem-and-Leaf Plots: These are excellent because they preserve the actual data values while showing the shape of the distribution. Remember to always include a Key (e.g., $9|0 = 90$) so the reader knows the place value of the stems.

Mastering these visualizations is the first step toward analyzing trends and making predictions. Be sure to review the attached PDF notes for the detailed examples on heart rates and Olympic medal counts!