Lecture Notes on Frequency Distributions

Key Concepts

• Frequency Distribution: A table showing classes or intervals of data with a count of entries in each class.
  • Frequency (f): Number of data entries in a class.
  • Lower Class Limit: The smallest number that can belong to a class.
  • Upper Class Limit: The largest number that can belong to a class.
  • Class Width: The difference between lower or upper limits of consecutive classes.
  • Range: Difference between the maximum and minimum data entries.

Constructing a Frequency Distribution

1. Decide on the Number of Classes: Typically between 5 and 20.
2. Find Class Width:
   • Calculate range: Max - Min.
   • Divide by number of classes, round up if necessary.
3. Determine Class Limits:
   • Use the smallest data value as the lower limit of the first class.
   • Add class width to find subsequent lower limits.
   • Upper limit of first class is one less than the lower limit of the next class.
   • Ensure classes do not overlap.
4. Create Tally Marks for Data Entries: Count and record frequencies.

Example Problem

• Given data set represents prescription expenses of 30 U.S. adults.
• Task: Construct a frequency distribution with 7 classes.
  • Class Width Calculation:
    • Max: 405, Min: 155
    • Range: 405 - 155 = 250
    • Width: 250 / 7 = 35.71, round up to 36.
  • Lower Limits: Start at 155, add 36 successively.
  • Upper Limits: One less than the next lower limit or add class width.

Calculation of Midpoint, Relative Frequency, and Cumulative Frequency

• Midpoint:
  • Calculated as (Lower limit + Upper limit) / 2.
  • Alternatively, add class width to previous midpoint.
• Relative Frequency:
  • Percentage of data in each class: Class frequency / Total entries.
• Cumulative Frequency:
  • Sum of frequencies for that class and all previous classes.

Expanded Frequency Distribution

• Patterns Observed:
  • Most common expense range: $299 to $334.
  • About half of expenses are less than $299.
• Verification:
  • Frequency column sums to total data entries (30).
  • Relative frequencies sum to 1.
  • Last cumulative frequency equals total data entries.

Importance of Frequency Distributions

• Helps in identifying patterns in data.
• More effective in tables, charts, and graphs than raw data.

These notes cover the essentials of understanding and constructing frequency distributions, including calculations for class limits, midpoint, relative frequency, and cumulative frequency. The practical example provided aids in comprehending how these concepts are applied in real-world scenarios.