Class: Senior Secondary School 1 (SS1, SS 1, SSS1, SSS 1)
Term: Second Term
Week: 11
Age: 15 years
Duration: 45 minutes
Subject: Further Maths
Curriculum Theme: Further Mathematics
Previous Lesson: .
Topic: MEASURE OF DISPERSION
Subject Matter: Mean deviation, standard deviation, coefficient of variation

Specific Objectives

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

Cognitive Domain:

Define measures of dispersion.
State the formulas for mean deviation, standard deviation, and coefficient of variation.
Calculate the mean deviation for ungrouped and grouped data.
Calculate the standard deviation for ungrouped and grouped data.
Determine the coefficient of variation for a given set of data.

Affective Domain:

Appreciate the importance of measures of dispersion in data analysis.
Show interest in solving problems related to dispersion.
Demonstrate accuracy in calculations involving measures of dispersion.

Psychomotor Domain:

Apply the correct formulas to solve problems involving mean deviation, standard deviation, and coefficient of variation.
Use calculators effectively to compute statistical measures.
Present solutions clearly and systematically.

Collaborate with peers to solve problems during group activities.
Share ideas and strategies for calculating measures of dispersion.

Reference Materials

The following resources were used in planning this lesson:

9 Years Basic Education Curriculum for Senior Secondary School.
State Unified Scheme of Work for Further Mathematics SSS 1.
New Further Mathematics for Senior Secondary Schools Book 1.
https://www.education.gov.ng
https://www.nerdc.gov.ng

Instructional Materials

The teacher will teach this lesson with the aid of:

Charts displaying dispersion formulas and examples.
Whiteboard or chalkboard.
Markers or chalk.
Scientific calculators.
Textbooks.

Rationale for the Lesson

This lesson helps pupils understand how data points spread out from their average value. Knowing about measures of dispersion is important because it enables pupils to analyze data more completely, moving beyond just averages to understand the variability within a dataset, which is useful in many real-world applications.

Prerequisite/Previous Knowledge

Pupils are expected to have prior knowledge of measures of central tendency (mean, median, mode), summation notation, and basic algebraic operations.

Lesson Content/Board Summary

MEASURE OF DISPERSION

Introduction to Measures of Dispersion

Measures of dispersion indicate how spread out a set of data is. They describe the variability or scatter of the data points around a central value.

The purpose of measures of dispersion is to:

Provide information about the spread of data.
Help in comparing the variability of different datasets.
Complement measures of central tendency for a complete data description.

Mean Deviation

The mean deviation is the average of the absolute differences between each data point and the mean of the dataset.

To calculate mean deviation:

For Ungrouped Data: Find the mean (x̄). Subtract the mean from each data point, take the absolute value of the differences (|x – x̄|), sum these absolute differences, and divide by the number of data points (n).
Formula: MD = (Σ|x – x̄|) / n
For Grouped Data: Find the mean (x̄). Find the midpoint (x) for each class. Subtract the mean from each midpoint, take the absolute value (|x – x̄|), multiply by the frequency (f|x – x̄|), sum these products, and divide by the total frequency (Σf).
Formula: MD = (Σf|x – x̄|) / Σf

Standard Deviation

The standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the data points tend to be close to the mean of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.

The variance is the square of the standard deviation.

To calculate standard deviation:

For Ungrouped Data: Find the mean (x̄). Subtract the mean from each data point (x – x̄). Square each difference ((x – x̄)²). Sum the squared differences (Σ(x – x̄)²). Divide by the number of data points (n) to get the variance. Take the square root of the variance to get the standard deviation.
Formula: SD = √[Σ(x – x̄)² / n] (for population) or √[Σ(x – x̄)² / (n-1)] (for sample)
For Grouped Data: Find the mean (x̄). Find the midpoint (x) for each class. Subtract the mean from each midpoint (x – x̄). Square each difference ((x – x̄)²). Multiply by the frequency (f(x – x̄)²). Sum these products (Σf(x – x̄)²). Divide by the total frequency (Σf) to get the variance. Take the square root of the variance.
Formula: SD = √[Σf(x – x̄)² / Σf] (for population) or √[Σf(x – x̄)² / (Σf-1)] (for sample)

Coefficient of Variation (CV)

The coefficient of variation is a standardized measure of dispersion of a probability distribution or frequency distribution. It is often expressed as a percentage and is useful for comparing the extent of variation between datasets with different units or widely different means.

Formula: CV = (Standard Deviation / Mean) × 100%

Teaching Methods/Instructional Techniques

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

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher greets the pupils and reminds them of the previous lesson on measures of central tendency. The teacher then asks pupils to recall what the mean, median, and mode tell us about a dataset. The teacher introduces the topic “Measures of Dispersion” by explaining that while central tendency gives us an average, we also need to know how spread out the data is.
Pupils’ Activity: Pupils respond to questions about previous lessons and listen attentively to the introduction of the new topic.
Learning Point: Pupils recall prior knowledge and are introduced to the concept of data spread.

Step 2: Explanation of Measures of Dispersion

Time: 7 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher defines measures of dispersion and explains their importance using simple examples, such as comparing the consistency of scores from two different students. The teacher uses the charts of dispersion formulas to highlight what will be covered.
Pupils’ Activity: Pupils listen, take notes, and ask questions for clarification.
Learning Point: Pupils understand the definition and purpose of measures of dispersion.

Step 3: Mean Deviation

Time: 8 minutes
Teaching Skill: Demonstration/Explanation
Teacher’s Activity: The teacher explains mean deviation, states its formula for both ungrouped and grouped data, and works through a simple example on the board to demonstrate its calculation. The teacher emphasizes the use of absolute values.
Pupils’ Activity: Pupils observe the demonstration, copy notes and examples, and ask questions.
Learning Point: Pupils learn how to calculate mean deviation.

Step 4: Standard Deviation

Time: 10 minutes
Teaching Skill: Demonstration/Explanation
Teacher’s Activity: The teacher explains standard deviation and variance, states their formulas for both ungrouped and grouped data, and demonstrates the calculation with another example on the board. The teacher highlights the steps involved in squaring differences and taking the square root.
Pupils’ Activity: Pupils pay close attention to the calculations, take notes, and ensure they understand each step.
Learning Point: Pupils understand standard deviation and variance and can calculate them.

Step 5: Coefficient of Variation

Time: 5 minutes
Teaching Skill: Explanation
Teacher’s Activity: The teacher explains the concept of coefficient of variation, states its formula, and discusses why it is useful for comparing variability between different datasets.
Pupils’ Activity: Pupils listen, take notes, and understand the application of CV.
Learning Point: Pupils understand the concept and application of coefficient of variation.

Step 6: Class Activity/Problem Solving

Time: 5 minutes
Teaching Skill: Problem Solving/Group Work
Teacher’s Activity: The teacher provides a short problem (e.g., a small set of ungrouped data) and asks pupils to calculate the mean deviation and standard deviation, either individually or in small groups.
Pupils’ Activity: Pupils attempt to solve the given problem, applying the learned formulas and techniques.
Learning Point: Pupils practice applying the formulas for measures of dispersion.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

Define measures of dispersion.
State the formula for mean deviation of ungrouped data.
Mention two steps involved in calculating the standard deviation.
Explain why the coefficient of variation is useful.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 2 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key learning points of the lesson, reiterating the importance of understanding mean deviation, standard deviation, and coefficient of variation in analyzing data. The teacher then assigns relevant homework from the textbook.
Pupils’ Activity: Pupils listen to the summary and copy down the homework.
Learning Point: Pupils consolidate their understanding and are given tasks for further practice.

Lesson Keywords

Dispersion – The spread or variability of data points.
Mean Deviation – The average of the absolute differences between each data point and the mean.
Standard Deviation – A measure of the average distance between each data point and the mean.
Variance – The square of the standard deviation.
Coefficient of Variation – A relative measure of dispersion, useful for comparing variability across different datasets.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide more complex problems involving grouped data or ask them to compare the dispersion of two different datasets. For pupils who need more support, the teacher will provide simplified examples, offer one-on-one guidance, and encourage peer tutoring. Visual aids and step-by-step demonstrations will benefit all learning styles.

Note for teachers using this lesson plan

Ensure that pupils have a strong foundation in calculating the mean before introducing measures of dispersion. Emphasize the practical applications of these measures to make the lesson more relatable. Encourage pupils to use scientific calculators accurately, as calculations can be extensive. Provide ample practice problems for both ungrouped and grouped data.