Frequency Distribution Tables Explained

Frequency Distribution Table Construction

• Frequency distribution tables organize observations by sorting them into classes, displaying their occurrence frequency in each class.

Types of Frequency Distribution

• Categorical frequency distributions are for data placed in distinct categories, such as nominal or ordinal level data.
• Grouped frequency distributions are used for grouped data scores.

Categorical Frequency Distribution Example

• In a sample survey, letters A, B, and C represent three categories.
• Category A appears 6 times.
• Category B appears 9 times.
• Category C appears 15 times.
• The total number of observations in this example is 30 (6 + 9 + 15).

Terminology for Grouped Data

• The lower class limit represents the smallest data value within a class.
• The upper class limit represents the largest data value within a class.
• Class boundaries separate classes without any gaps.
• The class mark is the midpoint of the classes, determined by (lower limit + upper limit) / 2.
• Class width measures the difference between two consecutive lower class limits.
• Cumulative frequency represents the sum of frequencies for the current class and all preceding classes, in increasing order.

Steps to Construct a Frequency Distribution Table

• Begin by arranging the scores from the lowest to the highest.
• Compute the range by subtracting the lowest score from the highest score.
• Determine the number of classes (k) using the formula: 1 + 3.322 * log(n), with n being the total number of observations; always round up to the nearest whole number.
• Calculate the class width by dividing the range by k and round up to the nearest whole number.
• Select a starting point, either the lowest score or a lower class limit, and add the class width to determine subsequent lower class limits.
• Determine upper class limits for each class.
• Calculate class boundaries by subtracting 0.5 from each lower class limit and adding 0.5 to each upper class limit.
• Tally the frequency, or count, for each class.
• Determine the class mark for each class interval using the formula: (lower limit + upper limit) / 2.
• Compute the cumulative frequency by summing the frequencies for each class and all preceding classes.

Example: Math Quiz Scores of 40 Students

• Arrange scores from lowest to highest.
• The highest score is 98, and the lowest score is 40.
• The range is calculated as 98 - 40 = 58.
• Determine the number of classes (k): 1 + 3.322 * log(40) ≈ 6.3229, rounded up to 7.
• The class width is calculated as 58 / 7 ≈ 8.286, rounded up to 9.
• Assign Classes and Determine Class Limits:
  • Start with the lowest score of 40 and add the class width (9) to find the subsequent lower class limits: 40, 49, 58, 67, 76, 85, 94.
• Determine Upper Class Limits and Class Intervals:
  • The resulting class intervals are: 40-48, 49-57, 58-66, 67-75, 76-84, 85-93, 94-102.
• Find Class Boundaries:
  • Subtract 0.5 from lower class limits and add 0.5 to upper class limits: 39.5-48.5, 48.5-57.5, 57.5-66.5, 66.5-75.5, 75.5-84.5, 84.5-93.5, 93.5-102.5
• Tally the frequency of each class:
  • 40-48: 9
  • 49-57: 9
  • 58-66: 13
  • 67-75: 3
  • 76-84: 4
  • 85-93: 1
  • 94-102: 1
• Class Mark (Midpoint) Calculation:
  • Use (lower limit + upper limit)/2 to find the midpoint of each class.
  • With a class width of 9, successively add 9 to the previous midpoint: 44, 53, 62, 71, 80, 89, 98.
• Find the cumulative frequency for each class:
  • 40-48: 9
  • 49-57: 9 + 9 = 18
  • 58-66: 18 + 13 = 31
  • 67-75: 31 + 3 = 34
  • 76-84: 34 + 4 = 38
  • 85-93: 38 + 1 = 39
  • 94-102: 39 + 1 = 40
• The last cumulative frequency (40) matches the total number of observations, confirming the calculations.