Frequency Distribution Notes

Frequency Distribution

Construction of a Frequency Distribution

• Frequency distribution is a method for organizing and summarizing data.
• For qualitative data, selecting categories is relatively straightforward.
• For quantitative data, selecting categories is more arbitrary. Careful consideration is crucial.

Steps for Construction:

1. Choose the Number of Classes:

   • The number of classes impacts the granularity of the distribution.
   • Use more classes when dealing with larger datasets.
   • Generally, aim for somewhere between 4 and 20 classes.

2. Determine the Class Width:

   • Each class should ideally have the same width to maintain consistency.
   • Exceptions can be made for the beginning and ending intervals, if necessary.
   • Formula to determine class width:
     $Width = \frac{Maximum\ Value - Minimum\ Value}{Number\ of\ Classes}$
   • Round the calculated width to a convenient integer value to simplify the class intervals.

3. Find the Limits of Each Class:

   • Start with choosing the lower limit of the first class. Selection can be arbitrary but should be logical.
   • Repeatedly add the chosen width to the lower limit to determine the starting point of the subsequent classes.

4. Determine the Frequency:

   • Tally the number of data values that fall into each class interval.
   • This process results in the frequency for each class.

Example: Heart Rates of 50 Students

• Dataset: Heart rates per minute of 50 students.
• Values range from 62 bpm to 98 bpm.
• Goal: Construct a frequency distribution for this data.

Step 1: Choose the number of classes

• Arbitrarily choose 5 classes.

Step 2: Choose the class width

• Using the formula:
  $Width = \frac{Maximum\ Value - Minimum\ Value}{Number\ of\ Classes}$
  $Width = \frac{98 - 62}{5} = \frac{36}{5} = 7.2$
• Round 7.2 up to 10 (arbitrarily chosen).

Step 3: Find the class limits

• Arbitrarily starting point of 57 bpm.
• Repeatedly add 10 to get the next class starting point.

Step 4: Determine Frequencies by Tallying Data

• Final frequency distribution table:
  • Interval 57 to 66: Frequency = 2
  • Interval 67 to 76: Frequency = 10
  • Interval 77 to 86: Frequency = 32
  • Interval 87 to 96: Frequency = 5
  • Interval 97 to 106: Frequency = 1

Other Distributions

• From the frequency distribution, we can derive other distributions:

Relative Frequency

• Calculation: Number of observations in a class divided by the total number of observations.

Cumulative Frequency

• Calculation: The sum of the frequency of the class and all preceding classes.

Cumulative Relative Frequency

• Calculation: Cumulative frequency divided by the total number of observations.

Example Table