Mode of grouped data
The modal class has the highest frequency. Then Mode = l + ((f₁ − f₀) ÷ (2f₁ − f₀ − f₂)) × h.
What mode means for grouped data
For ungrouped data, the mode is the value that occurs most often. In grouped data we cannot see single values. So we first find the modal class, the class with the highest frequency. The mode lies inside it. The formula finds where, using the classes just before and just after it.
Reading the symbols (from example 1)
| Symbol | Meaning | Value in example 1 |
|---|---|---|
| l | Lower limit of the modal class | 40 |
| h | Class size | 5 |
| f₁ | Frequency of the modal class | 18 |
| f₀ | Frequency of the class before it | 10 |
| f₂ | Frequency of the class after it | 12 |
Worked example 1: weights of students
The weights (in kg) of 50 students are: 30-35: 4, 35-40: 10, 40-45: 18, 45-50: 12, 50-55: 6. Find the mode.
- The highest frequency is 18, so the modal class is 40-45.
- l = 40, h = 5, f₁ = 18, f₀ = 10, f₂ = 12.
- Formula: Mode = l + ((f₁ − f₀) ÷ (2f₁ − f₀ − f₂)) × h.
- f₁ − f₀ = 18 − 10 = 8; 2f₁ − f₀ − f₂ = 36 − 10 − 12 = 14.
- Mode = 40 + (8 ÷ 14) × 5 = 40 + 40 ÷ 14 = 40 + 2.86 = 42.86.
Mode ≈ 42.86 kg (rounded to 2 decimal places).
Worked example 2: a fun run in Bengaluru
The times (in minutes) taken by 80 runners in a 5 km fun run are: 20-25: 6, 25-30: 10, 30-35: 22, 35-40: 25, 40-45: 17. Find the modal time.
- The highest frequency is 25, so the modal class is 35-40.
- l = 35, h = 5, f₁ = 25, f₀ = 22, f₂ = 17.
- Formula: Mode = l + ((f₁ − f₀) ÷ (2f₁ − f₀ − f₂)) × h.
- f₁ − f₀ = 3; 2f₁ − f₀ − f₂ = 50 − 22 − 17 = 11.
- Mode = 35 + (3 ÷ 11) × 5 = 35 + 15 ÷ 11 = 35 + 1.36 = 36.36.
Modal time ≈ 36.36 minutes.
Mistakes to avoid
- l is the lower limit of the modal class, not of the first class.
- f₀ and f₂ belong to the classes just before and just after the modal class.
- If the modal class is the first class, take f₀ = 0. If it is the last class, take f₂ = 0.
- Classes must be continuous, like 10-20, 20-30. If they are not, make them continuous first.
- The mode must lie inside the modal class. If it does not, recheck your working.
A shoe shop in Chennai stocks more of the modal size, the size most customers buy. A school canteen can plan for the most common lunch-break rush in the same way.
1 marks: “In a frequency table, the class 20-30 has the highest frequency. What is the lower limit of the modal class?” 20.
2 marks: “For a distribution, the modal class is 40-50, with f₁ = 12, f₀ = 8 and f₂ = 10. Find the mode.” Mode = 40 + ((12 − 8) ÷ (24 − 8 − 10)) × 10 = 40 + (4 ÷ 6) × 10 = 46.67.
3 marks: “Find the mode of the data: 0-10: 5, 10-20: 8, 20-30: 15, 30-40: 9, 40-50: 3.” Modal class 20-30; l = 20, f₁ = 15, f₀ = 8, f₂ = 9, h = 10. Mode = 20 + (7 ÷ 13) × 10 = 20 + 5.38 = 25.38.
Top: me minus the one before. Bottom: twice me minus both neighbours. Times h, plus l.