Lesson Note on Statistics: Mean, Median and Mode of Grouped Data for SSS 2

Class: Senior Secondary School 2 (SS2 / SSS2)
Term: 3rd Term
Week: 4
Age: 16 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Data Handling and Statistics
Previous Lesson: Statistics: Mean, Median and Mode of Ungrouped Data.
Topic: STATISTICS (II)
Subject Matter: Mean of grouped data, median of grouped data, mode of grouped data

Specific Objectives

By the end of the lesson, pupils should be able to:

Cognitive Domain:

• Define mean, median, and mode for grouped data.
• State the formulas for calculating mean, median, and mode of grouped data.
• Explain the steps involved in calculating each measure of central tendency.

Affective Domain:

• Appreciate the importance of statistics in real-world data analysis.
• Show interest in solving problems involving grouped data.

Psychomotor Domain:

• Construct grouped frequency tables.
• Calculate the mean of grouped data using the appropriate formula.
• Calculate the median of grouped data using the appropriate formula.
• Calculate the mode of grouped data using the appropriate formula.

Social Domain:

• Collaborate effectively with peers when solving statistical problems.

Reference Materials

The following resources were used in planning this lesson:

• Senior Secondary Schools Education Curriculum
• State Unified Scheme of Work
• New General Mathematics for Senior Secondary Schools 2

Instructional Materials

The teacher will teach this lesson with the aid of:

• Score charts
• Computer software (for demonstration, if available)
• Whiteboard/chalkboard
• Markers/chalk
• Calculators

Rationale for the Lesson

This lesson helps pupils understand how to summarize and interpret large datasets by calculating measures of central tendency for grouped data. This skill is important for making informed decisions and understanding statistical information encountered in various fields and daily life.

Prerequisite/Previous Knowledge

Pupils are expected to have prior knowledge of basic statistics, including calculating mean, median, and mode for ungrouped data, constructing frequency tables, and understanding class intervals.

Lesson Content/Board Summary

STATISTICS (II)

1. Mean of Grouped Data

The mean of grouped data is an estimate of the average value when individual data points are not known, but their frequencies within class intervals are. It is calculated by finding the midpoint of each class interval, multiplying it by its frequency, summing these products, and then dividing by the total frequency.

Formula:

Mean (x̄) = Σfx / Σf

Where:

• Σfx = sum of the products of class mark (x) and frequency (f)
• Σf = sum of frequencies (total number of data points)
• x = class mark (midpoint of a class interval) = (Lower Class Limit + Upper Class Limit) / 2

Example 1: Calculate the mean for the following distribution.

Class Interval	Frequency (f)
10-19	5
20-29	10
30-39	15
40-49	12
50-59	8

Solution:

Class Interval	Frequency (f)	Class Mark (x)	fx
10-19	5	(10+19)/2 = 14.5	5 * 14.5 = 72.5
20-29	10	(20+29)/2 = 24.5	10 * 24.5 = 245.0
30-39	15	(30+39)/2 = 34.5	15 * 34.5 = 517.5
40-49	12	(40+49)/2 = 44.5	12 * 44.5 = 534.0
50-59	8	(50+59)/2 = 54.5	8 * 54.5 = 436.0
Total	Σf = 50		Σfx = 1805.0

Mean (x̄) = Σfx / Σf = 1805.0 / 50 = 36.1

2. Median of Grouped Data

The median of grouped data is the middle value in a dataset when arranged in order. For grouped data, it is estimated using interpolation within the median class.

Formula:

Median = L + [(N/2 – cf_b) / f_m] * c

Where:

• L = Lower class boundary of the median class
• N = Total frequency (Σf)
• cf_b = Cumulative frequency of the class immediately before the median class
• f_m = Frequency of the median class
• c = Class size (width) of the median class

Example 2: Calculate the median for the data from Example 1.

Class Interval	Frequency (f)	Cumulative Frequency (cf)
10-19	5	5
20-29	10	15
30-39	15	30
40-49	12	42
50-59	8	50

Solution:

Step 1: Find N/2. N = 50, so N/2 = 25.

Step 2: Identify the median class. This is the first class where the cumulative frequency is greater than or equal to N/2 (25). The median class is 30-39.

Step 3: Identify the values for the formula:

• L = Lower class boundary of median class (30-39). The lower boundary is 30 – 0.5 = 29.5.
• N/2 = 25
• cf_b = Cumulative frequency before the median class (cf of 20-29) = 15
• f_m = Frequency of the median class = 15
• c = Class size = 39.5 – 29.5 = 10 (or 19-10+1 = 10)

Step 4: Substitute values into the formula:

Median = 29.5 + [(25 – 15) / 15] * 10

Median = 29.5 + [10 / 15] * 10

Median = 29.5 + (0.6667 * 10)

Median = 29.5 + 6.67

Median = 36.17

3. Mode of Grouped Data

The mode of grouped data is the class interval with the highest frequency (modal class). The exact mode is then estimated using a formula that considers the frequencies of the modal class and its adjacent classes.

Formula:

Mode = L + [(f_m – f_b) / (2f_m – f_b – f_a)] * c

Where:

• L = Lower class boundary of the modal class
• f_m = Frequency of the modal class
• f_b = Frequency of the class before the modal class
• f_a = Frequency of the class after the modal class
• c = Class size (width) of the modal class

Example 3: Calculate the mode for the data from Example 1.

Class Interval	Frequency (f)
10-19	5
20-29	10
30-39	15
40-49	12
50-59	8

Solution:

Step 1: Identify the modal class. This is the class with the highest frequency. The modal class is 30-39 (frequency = 15).

Step 2: Identify the values for the formula:

• L = Lower class boundary of modal class (30-39). The lower boundary is 30 – 0.5 = 29.5.
• f_m = Frequency of the modal class = 15
• f_b = Frequency of the class before the modal class (20-29) = 10
• f_a = Frequency of the class after the modal class (40-49) = 12
• c = Class size = 10

Step 3: Substitute values into the formula:

Mode = 29.5 + [(15 – 10) / (2*15 – 10 – 12)] * 10

Mode = 29.5 + [5 / (30 – 10 – 12)] * 10

Mode = 29.5 + [5 / 8] * 10

Mode = 29.5 + (0.625 * 10)

Mode = 29.5 + 6.25

Mode = 35.75

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher reviews the previous lesson on ungrouped data and frequency tables. The teacher then shows score charts and asks pupils to identify the highest and lowest marks, leading to the need for grouping data.
Pupils’ Activity: Pupils recall previous knowledge, observe the score charts, and identify extreme values.
Learning Point: Pupils connect previous knowledge to the new topic and understand the need for grouped data analysis.

Step 2: Presentation of Mean of Grouped Data

Time: 8 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher introduces the concept of the mean for grouped data, presents the formula, and demonstrates how to calculate it using a worked example on the board, guiding pupils through constructing the table and calculations.
Pupils’ Activity: Pupils listen attentively, copy the formula, and follow the teacher’s steps in calculating the mean.
Learning Point: Pupils understand the formula and procedure for calculating the mean of grouped data.

Step 3: Further Practice on Mean

Time: 5 minutes
Teaching Skill: Guided Practice
Teacher’s Activity: The teacher provides another simple grouped data set and guides pupils to calculate its mean, providing assistance where needed.
Pupils’ Activity: Pupils attempt to calculate the mean of the new data set, asking questions for clarification.
Learning Point: Pupils reinforce their understanding and application of the mean formula for grouped data.

Step 4: Presentation of Median of Grouped Data

Time: 8 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher introduces the concept of the median for grouped data, presents its formula, and demonstrates step-by-step calculation using a new example, emphasizing how to identify the median class and other parameters.
Pupils’ Activity: Pupils pay attention, copy the formula, and follow the calculation steps, including setting up the cumulative frequency table.
Learning Point: Pupils learn the formula and method for calculating the median of grouped data.

Step 5: Presentation of Mode of Grouped Data

Time: 8 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher introduces the concept of the mode for grouped data, presents its formula, and demonstrates step-by-step calculation using an example, focusing on identifying the modal class and its adjacent frequencies.
Pupils’ Activity: Pupils listen, copy the formula, and follow the calculation steps, understanding how to identify the modal class.
Learning Point: Pupils learn the formula and method for calculating the mode of grouped data.

Step 6: Class Activity/Practice

Time: 3 minutes
Teaching Skill: Problem Solving
Teacher’s Activity: The teacher assigns a short practice problem to the class, asking them to calculate all three measures of central tendency for a given grouped data set.
Pupils’ Activity: Pupils work individually or in small groups to solve the problem.
Learning Point: Pupils practice and consolidate their understanding of calculating mean, median, and mode for grouped data.

Step 7: Evaluation/Review

Time: 5 minutes
Teaching Skill: Questioning/Assessment
Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. Define the term ‘mean’ in grouped data.
2. State the formula for calculating the median of grouped data.
3. Given a frequency distribution, how would you identify the modal class?
4. Calculate the mean for the following data: Class Interval (0-9, 10-19), Frequency (3, 7).

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 3 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key learning points of the lesson, emphasizing the importance of correctly identifying the class intervals, class marks, and frequencies for accurate calculations. The teacher then assigns homework.
Pupils’ Activity: Pupils listen to the summary and copy down the homework assignment.
Learning Point: Pupils recap the lesson and consolidate their understanding.

Lesson Keywords

• Mean – The average value of a set of numbers.
• Median – The middle value in a sorted dataset.
• Mode – The value that appears most often in a dataset.
• Grouped Data – Data organized into class intervals.
• Frequency Table – A table showing the frequency of data points in each class interval.
• Class Interval – A range of values within which data points fall.
• Class Mark – The midpoint of a class interval.
• Cumulative Frequency – The running total of frequencies.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide more complex grouped data problems or introduce the concept of assumed mean. For struggling pupils, the teacher can offer more guided practice with simpler datasets and provide step-by-step templates for calculations.

Note for teachers using this lesson plan

Encourage pupils to use calculators efficiently. Emphasize the practical applications of these statistical measures in various fields. Ensure pupils understand the difference between class limits and class boundaries, as this is crucial for accurate calculations of median and mode.