Calculating the Mean from a Group Frequency Table

Introduction

• Discussing how to calculate the mean of a group frequency table using student grades.
• Grade intervals:
  • 40-49, 50-59, 60-69, 70-79, 80-89, 90-100.
• Frequency column indicates the number of students in each grade interval.

Understanding the Calculation

• Exact mean cannot be computed due to lack of individual data values.
• Mean is estimated by using the formula for group frequency tables:
  • Mean = (Sum of (Frequency × Midpoint)) / (Sum of Frequency)
• Midpoint is the sum of the lower and upper boundaries of the interval divided by 2.

Calculating Midpoints

• Examples of calculating midpoints for each grade interval:
  • 40-49: (40+49)/2 = 44.5
  • 50-59: (50+59)/2 = 54.5
  • Pattern observed in midpoints: 64.5, 74.5, 84.5, 95.

Sum of Frequency

• Calculating total number of students = 3 + 5 + 6 + 9 + 8 + 7 = 38.

Frequency Times Midpoint

• Create an extra column for Frequency × Midpoint (F × M):
  • Examples:
    • 3 × 44.5 = 133.5
    • 5 × 54.5 = 272.5
• Sum of this column = 2804.5.

Calculating the Mean

• Mean = 2804.5 / 38 = 73.8.
• Mean likely falls in the 70-79 interval where most students scored.

Identifying Median and Mode

• Mode: Occurs in the interval with the highest frequency (70-79).
• Median:
  • Use Cumulative Frequency to determine: 3, 8, 14, 23, 31, 38.
  • Median is between 14 and 23, so it lies in the 70-79 interval.

Additional Example: Student Weights

Cumulative Frequency Calculation

• Example intervals: 120-139, 140-159, etc.
• Cumulative Frequency: 6, 14, 26, 33, 36.

Midpoint Calculations

• Examples:
  • First interval (120-139): (120+139)/2 = 129.5
  • Second interval (140-159): (140+159)/2 = 149.5

Frequency Times Midpoint

• Multiply frequency by midpoint:
  • 6 × 129.5 = 777
  • 8 × 149.5 = 1196
• Sum = 5963.5.

Calculating the Mean

• Mean = 5963.5 / 36 = 165.7, falls in the 160-179 interval.

Identifying Median and Mode

• Mode: Highest frequency interval (160-179).
• Median:
  • 18th student is in the interval 160-179.

Conclusion

• Mean, median, and mode help identify central tendencies in grouped data.
• Practical application in analyzing student grades and weights.