STATISTICS-CLASS-10-how to teach Mode of Grouped Data-AP/TS/NCERT

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

Hook Activity

To spark interest and connect statistical concepts to real-life situations, let's look at a chart displaying student marks grouped into intervals:

Discussion Questions:

1. Which mark range has the highest frequency?
2. What does this tell us about the performance of most students?
3. If you had to identify a "typical" performance, what would it be?
4. How might we find a more precise value for the most common mark range?

This leads to the conclusion that a new approach is needed to estimate the mode using grouped data.

Introduction to Mode

Mode is defined as the value of a variable which occurs most frequently. It is the value of the variable that corresponds to the maximum frequency of the distribution. In any series, it is the value of the item which is most characteristic or common and is usually repeated the maximum number of times.

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.

Types of Distributions

Unimodal Distribution

If a data set has only one value that occurs most often, the set is called unimodal.

Example: 2, 3, 5, 5, 5, 7, 9

Mode = 5

Bimodal Distribution

A data set that has two values that occur with the greatest frequency is referred to as bimodal.

Example: 2, 3, 5, 5, 5, 7, 8, 8, 8

Modes = 5 and 8

Multimodal Distribution

When a set has more than two values that occur with the same greatest frequency, the set is called multimodal.

Example: 2, 2, 3, 5, 5, 7, 7, 9, 9

Modes = 2, 5, 7, 9

Mode of Grouped Data

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.

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.

Conclusion

Mode helps identify the most frequent value or group in a dataset. For ungrouped data, we can find the mode by simple observation, while for grouped data, we use a specific formula.

Understanding these measures of central tendency allows us to summarize and make sense of large datasets, enabling better decision-making based on data analysis.

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.