Mode of Grouped Data
Grade 10 Mathematics | AP/TS/NCERT - Period 13.5
Formula for Mode of Grouped Data
Where: l = lower boundary of modal class, h = class size, f₁ = frequency of modal class,
f₀ = frequency of class preceding modal class, f₂ = frequency of class succeeding modal class
Worked Examples
Example 5
A survey conducted on 20 households in a locality by a group of students resulted in the following frequency table:
| Family size | 1-3 | 3-5 | 5-7 | 7-9 | 9-11 |
|---|---|---|---|---|---|
| Number of families | 7 | 8 | 2 | 2 | 1 |
Find the mode of this data.
Step 1: Identify the modal class (class with highest frequency)
The maximum class frequency is 8, and the class corresponding to this frequency is 3-5.
So, the modal class is 3-5.
Step 2: Identify the values needed for the formula:
l = lower boundary of modal class = 3
h = class size = 2
f₁ = frequency of modal class = 8
f₀ = frequency of class preceding modal class = 7
f₂ = frequency of class succeeding modal class = 2
Step 3: Apply the formula:
Step 4: Interpretation
Therefore, the mode of the data is 3.286.
Practice Exercises
Exercise 1
The following table shows the ages of the patients admitted in a hospital during a year:
| Age (in years) | 5-15 | 15-25 | 25-35 | 35-45 | 45-55 | 55-65 |
|---|---|---|---|---|---|---|
| No. of patients | 6 | 11 | 21 | 23 | 14 | 5 |
Find the mode of the data.
Step 1: Identify the modal class
The highest frequency is 23, which corresponds to the class 35-45.
So, the modal class is 35-45.
Step 2: Identify the values:
l = lower boundary of modal class = 35
h = class size = 10
f₁ = frequency of modal class = 23
f₀ = frequency of class preceding modal class = 21
f₂ = frequency of class succeeding modal class = 14
Step 3: Apply the formula:
Step 4: Interpretation
Therefore, the mode of the data is 36.818.
Exercise 2
The following data gives the information on the observed lifetimes (in hours) of 225 electrical components:
| Lifetimes (in hours) | 0-20 | 20-40 | 40-60 | 60-80 | 80-100 | 100-120 |
|---|---|---|---|---|---|---|
| Frequency | 10 | 35 | 52 | 61 | 38 | 29 |
Determine the modal lifetimes of the components.
Step 1: Identify the modal class
The highest frequency is 61, which corresponds to the class 60-80.
So, the modal class is 60-80.
Step 2: Identify the values:
l = lower boundary of modal class = 60
h = class size = 20
f₁ = frequency of modal class = 61
f₀ = frequency of class preceding modal class = 52
f₂ = frequency of class succeeding modal class = 38
Step 3: Apply the formula:
Step 4: Interpretation
Therefore, the modal lifetime of the components is 65.625 hours.