Teach Mode of Grouped Data

Mode of Grouped Data

Grade 10 Mathematics | AP/TS/NCERT Curriculum

Period 13.5: (LO-5) Student will be able to find the mode of a grouped data

Introduction

This section builds upon previous learning of finding the Mean, by introducing how to calculate the Mode and Median for grouped data.

Recall from Class IX, a mode is that value among the observations which occurs most often, that is, the value of the observation having the maximum frequency.

In a grouped frequency distribution, it is not possible to determine the mode by looking at the frequencies. Here, we can only locate a class with the maximum frequency, called the modal class.

Formula for Mode of Grouped Data

The mode for grouped data is calculated using the formula:

Mode = l +

f₁ - f₀2f₁ - f₀ - f₂

× h

Where:

• l = lower boundary of the modal class
• h = size of the class interval (assuming all class sizes to be equal)
• f₁ = frequency of the modal class
• f₀ = frequency of the class preceding the modal class
• f₂ = frequency of the class succeeding the modal class

Teacher Modeling (I Do)

Let's work through an example step by step:

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.

Solution:

1. Identify the modal class (class with highest frequency): 3 - 5 (frequency = 8)
2. Find the values:
   • l = lower boundary of modal class = 3
   • h = class size = 2
   • f₁ = frequency of modal class = 8
   • f₀ = frequency of class before modal class = 7
   • f₂ = frequency of class after modal class = 2
3. Apply the formula:
   Mode = 3 +

   8 - 72×8 - 7 - 2

   × 2
4. Simplify:
   Mode = 3 +

   116 - 9

   × 2 = 3 +

   17

   × 2
5. Calculate:
   Mode = 3 +

   27

   = 3 + 0.286 = 3.286

Therefore, the mode of the data is 3.286.

Group Work (We Do)

Let's practice with another example together. Try to solve this problem in groups.

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.

Solution:

1. Modal class: 35 - 45 (highest frequency = 23)
2. Values:
   • l = 35
   • h = 10
   • f₁ = 23
   • f₀ = 21
   • f₂ = 14
3. Apply formula:
   Mode = 35 +

   23 - 212×23 - 21 - 14

   × 10
4. Simplify:
   Mode = 35 +

   246 - 35

   × 10 = 35 +

   211

   × 10
5. Calculate:
   Mode = 35 +

   2011

   = 35 + 1.818 = 36.818

Therefore, the mode of the data is 36.818 years.

Independent Work (You Do)

Now try to solve these problems on your own:

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.

Solution:

1. Modal class: 60 - 80 (highest frequency = 61)
2. Values:
   • l = 60
   • h = 20
   • f₁ = 61
   • f₀ = 52
   • f₂ = 38
3. Apply formula:
   Mode = 60 +

   61 - 522×61 - 52 - 38

   × 20
4. Simplify:
   Mode = 60 +

   9122 - 90

   × 20 = 60 +

   932

   × 20
5. Calculate:
   Mode = 60 +

   18032

   = 60 + 5.625 = 65.625

Therefore, the modal lifetime of the components is 65.625 hours.

Exercise 6

A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarized it in the table given below:

No. of cars	0 - 10	10 - 20	20 - 30	30 - 40	40 - 50	50 - 60	60 - 70	70 - 80
Frequency	7	14	13	12	20	11	15	8

Find the mode of the data.

Solution:

1. Modal class: 40 - 50 (highest frequency = 20)
2. Values:
   • l = 40
   • h = 10
   • f₁ = 20
   • f₀ = 12
   • f₂ = 11
3. Apply formula:
   Mode = 40 +

   20 - 122×20 - 12 - 11

   × 10
4. Simplify:
   Mode = 40 +

   840 - 23

   × 10 = 40 +

   817

   × 10
5. Calculate:
   Mode = 40 +

   8017

   = 40 + 4.705 = 44.705

Therefore, the mode of the data is 44.705 cars.

Homework

Complete all problems in Exercise 13.2 from your textbook. Make sure to:

• Show all steps clearly
• Write the formula first
• Substitute values correctly
• Simplify fractions properly
• Box your final answer

Think deeply about how the mode represents the most frequent value in the data and what it tells us about the distribution.