Variance of the Different Probability Distributions for SS 3

Class: SS 3
Term: First Term
Week: 10
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Probability and Statistics
Previous Lesson: Binomial and Poisson Probability Distributions
Topic: VARIANCE OF THE DIFFERENT PROBABIILTY DISTRIBUTIONS
Subject Matter: Mean; Variance; Coefficient of variance

Specific Objectives

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

Cognitive Domain

• Define the mean of a probability distribution.
• State the formula for calculating the mean of a discrete probability distribution.
• Calculate the mean of a given discrete probability distribution.
• Define the variance of a probability distribution.
• State the formula for calculating the variance of a discrete probability distribution.
• Calculate the variance of a given discrete probability distribution.
• Define the coefficient of variance.
• State the formula for calculating the coefficient of variance.
• Calculate the coefficient of variance for a given probability distribution.

Affective Domain

• Appreciate the importance of mean, variance, and coefficient of variance in describing probability distributions.
• Show interest in solving problems involving these statistical measures.

Psychomotor Domain

• Accurately apply the formulas to solve problems related to mean, variance, and coefficient of variance.
• Construct tables to aid in the calculation of these measures.

Reference Materials

The following resources were used in planning this lesson:

• 2025 Revised 9 Years Basic Education Curriculum
• Relevant State Unified Scheme of Work
• Further Mathematics for Senior Secondary Schools Textbook
• The HeadTeacher Scheme of work

Instructional Materials

The teacher will teach this lesson with the aid of:

• A chart showing the formulas for mean, variance, and coefficient of variance of probability distributions.
• Whiteboard and markers.
• Calculators.
• Textbooks.

Rationale for the Lesson

This lesson is fundamental to understanding the characteristics and spread of random variables in probability and statistics. Knowledge of mean, variance, and coefficient of variance allows students to quantify the central tendency, dispersion, and relative variability of data, which are essential skills for advanced statistical analysis and real-world problem-solving.

Prerequisite/Previous Knowledge

Students should have prior knowledge of basic probability concepts, discrete random variables, probability distributions, summation notation, and measures of central tendency.

Lesson Content/Board Summary

VARIANCE OF THE DIFFERENT PROBABIILTY DISTRIBUTIONS

Mean of a Probability Distribution

The mean of a probability distribution, also known as the expected value (E(X)), represents the average value of a random variable. It indicates the central tendency of the distribution.

Formula for Mean (Expected Value)

For a discrete random variable (X) with probability mass function (P(X=x)), the mean is given by:

( mu = E(X) = sum x P(X=x) )

Where:

1. (x) = a possible value of the random variable.
2. (P(X=x)) = the probability of (X) taking the value (x).
3. (sum) = summation over all possible values of (x).

Example 1 (Mean)

Question: Find the mean of the following probability distribution:

(x)	0	1	2	3
(P(X=x))	0.1	0.3	0.4	0.2

Solution:

Step 1: Write the formula for the mean.

( E(X) = sum x P(X=x) )

Step 2: Substitute the values and calculate.

( E(X) = (0 times 0.1) + (1 times 0.3) + (2 times 0.4) + (3 times 0.2) )

( E(X) = 0 + 0.3 + 0.8 + 0.6 )

Step 3: Simplify and write the answer.

( E(X) = 1.7 )

Answer: ( E(X) = 1.7 )

Variance of a Probability Distribution

The variance of a probability distribution measures the spread or dispersion of the random variable’s values around its mean. A larger variance indicates greater spread, while a smaller variance indicates that the values are clustered closer to the mean.

Formula for Variance

For a discrete random variable (X) with mean (mu), the variance is given by:

( Var(X) = E(X^2) – [E(X)]^2 )

Where:

1. (E(X^2) = sum x^2 P(X=x))
2. (E(X)) = the mean of the distribution.

Alternatively, the variance can be calculated as:

( Var(X) = sum (x – mu)^2 P(X=x) )

The standard deviation (sigma) is the square root of the variance: ( sigma = sqrt{Var(X)} ).

Example 2 (Variance)

Question: Using the probability distribution from Example 1, calculate the variance.

(x)	0	1	2	3
(P(X=x))	0.1	0.3	0.4	0.2

Solution:

Step 1: Calculate (E(X)) (mean). From Example 1, (E(X) = 1.7).

Step 2: Calculate (E(X^2)).

( E(X^2) = sum x^2 P(X=x) )

( E(X^2) = (0^2 times 0.1) + (1^2 times 0.3) + (2^2 times 0.4) + (3^2 times 0.2) )

( E(X^2) = (0 times 0.1) + (1 times 0.3) + (4 times 0.4) + (9 times 0.2) )

( E(X^2) = 0 + 0.3 + 1.6 + 1.8 )

( E(X^2) = 3.7 )

Step 3: Apply the variance formula.

( Var(X) = E(X^2) – [E(X)]^2 )

( Var(X) = 3.7 – (1.7)^2 )

( Var(X) = 3.7 – 2.89 )

Step 4: Simplify and write the answer.

( Var(X) = 0.81 )

Answer: ( Var(X) = 0.81 )

Coefficient of Variance (CV)

The coefficient of variance (CV) is a measure of relative variability. It expresses the standard deviation as a percentage of the mean, allowing for the comparison of variability between different data sets, even if they have different units or vastly different means.

Formula for Coefficient of Variance

The coefficient of variance is given by:

( CV = frac{sigma}{mu} times 100% )

Where:

1. (sigma) = standard deviation ((sqrt{Var(X)})).
2. (mu) = mean ((E(X))).

Example 3 (Coefficient of Variance)

Question: Using the probability distribution from Example 1, calculate the coefficient of variance.

Solution:

Step 1: Identify the mean ((mu)) and variance from previous calculations.

( mu = E(X) = 1.7 )

( Var(X) = 0.81 )

Step 2: Calculate the standard deviation ((sigma)).

( sigma = sqrt{Var(X)} = sqrt{0.81} )

( sigma = 0.9 )

Step 3: Apply the coefficient of variance formula.

( CV = frac{sigma}{mu} times 100% )

( CV = frac{0.9}{1.7} times 100% )

( CV approx 0.5294 times 100% )

Step 4: Simplify and write the answer.

( CV approx 52.94% )

Answer: ( CV approx 52.94% )

Teaching Methods/Instructional Techniques

Explanation, Discussion, Demonstration, Guided Practice, Problem Solving, Question and Answer, Individual Practice.

Instructional Procedures

Step 1: Introduction

Time: 5 minutes

Teaching Skill: Recalling/Engaging

Teacher’s Activity: The teacher greets the students and reviews previous knowledge on discrete random variables and probability distributions. The teacher asks students to recall what a probability distribution is and how to identify one.

Pupils’ Activity: Students respond to questions about probability distributions and discrete random variables.

Learning Point: Probability distribution recall

Step 2: Introduction to Mean of Probability Distribution

Time: 7 minutes

Teaching Skill: Explaining/Demonstrating

Teacher’s Activity: The teacher defines the mean of a probability distribution (expected value) and explains its significance. The teacher then writes the formula for the mean of a discrete probability distribution on the board and demonstrates Example 1.

Pupils’ Activity: Students listen, ask questions, and observe the calculation of the mean.

Learning Point: Mean definition and calculation

Step 3: Introduction to Variance of Probability Distribution

Time: 7 minutes

Teaching Skill: Defining/Explaining

Teacher’s Activity: The teacher defines variance as a measure of spread and explains why it is important to understand the dispersion of data. The teacher then writes the formula for variance on the board.

Pupils’ Activity: Students listen attentively and write down the definition and formula for variance.

Learning Point: Variance definition and formula

Step 4: Calculating Variance

Time: 8 minutes

Teaching Skill: Demonstrating/Guiding

Teacher’s Activity: The teacher demonstrates how to calculate variance step-by-step using Example 2, emphasizing the calculation of (E(X^2)) and the application of the formula (Var(X) = E(X^2) – [E(X)]^2). The teacher guides students through the calculation.

Pupils’ Activity: Students follow along with the calculation, ask questions for clarification, and attempt to solve it in their notebooks.

Learning Point: Step-by-step variance calculation

Step 5: Introduction to Coefficient of Variance

Time: 6 minutes

Teaching Skill: Explaining/Formula presentation

Teacher’s Activity: The teacher defines the coefficient of variance and explains its use in comparing variability between different datasets. The teacher writes the formula for the coefficient of variance on the board.

Pupils’ Activity: Students listen, understand the concept of relative variability, and write down the formula.

Learning Point: Coefficient of variance concept

Step 6: Calculating Coefficient of Variance

Time: 5 minutes

Teaching Skill: Demonstrating/Applying

Teacher’s Activity: The teacher demonstrates how to calculate the coefficient of variance using Example 3, showing the steps to find the standard deviation first and then applying the CV formula. The teacher ensures students understand the percentage representation.

Pupils’ Activity: Students follow the calculation, ensuring they understand each step and the final interpretation.

Learning Point: CV calculation and interpretation

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 mean of a probability distribution.
2. State the formula for the variance of a discrete probability distribution.
3. What is the significance of the coefficient of variance?
4. Given a mean of 10 and a standard deviation of 2, calculate the coefficient of variance.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Mean, variance, CV understanding

Step 8: Note-Taking

Time: 4 minutes

Teaching Skill: Guided Writing

Teacher’s Activity: The teacher guides pupils/students to copy the essential Board Summary notes, including definitions and formulas for mean, variance, and coefficient of variance, into their notebooks.

Pupils’ Activity: Pupils/students copy the notes carefully into their notebooks.

Learning Point: Key statistical measures recorded

Step 9: Conclusion

Time: 1 minute

Teaching Skill: Summarising

Teacher’s Activity: The teacher briefly recaps the importance of mean, variance, and coefficient of variance in describing probability distributions and encourages students to practice more problems.

Pupils’ Activity: Students listen and prepare for the next lesson.

Learning Point: Lesson concepts reinforced

Continuous Assessment/Further Study

Type: Homework

Instruction: Solve the following problems in your notebook:

1. A discrete random variable (Y) has the following probability distribution:
   

   (y)	1	2	3	4
   (P(Y=y))	0.2	0.3	0.4	0.1

   Calculate:

   1. The mean (E(Y)).
   2. The variance (Var(Y)).
   3. The standard deviation (sigma).
   4. The coefficient of variance (CV).
2. Explain in your own words why variance is a better measure of spread than simply looking at the range of values.

Lesson Keywords

• Mean – The average value or expected value of a random variable.
• Variance – A measure of the spread or dispersion of a random variable’s values around its mean.
• Standard Deviation – The square root of the variance, indicating the typical deviation from the mean.
• Coefficient of Variance – A measure of relative variability, expressing standard deviation as a percentage of the mean.
• Probability Distribution – A function that describes the probabilities of all possible outcomes of a random variable.

Differentiation

For weaker learners, provide additional guided practice with simpler probability distributions and step-by-step calculation sheets. For faster learners, challenge them to research the mean and variance formulas for specific probability distributions like the Binomial or Poisson distribution and compare their properties.

Suggested Lesson Videos

For further understanding, students can search YouTube for: “Mean and Variance of Discrete Probability Distribution SS3” or “Expected Value and Variance of Random Variable”.

Teacher Guide for Using This Lesson Plan

Before the lesson, ensure you have a clear chart displaying the formulas for mean, variance, and coefficient of variance. Prepare additional simple probability distribution examples for extra practice if needed. Start by reviewing foundational probability concepts to ensure all students are on the same page. When teaching the formulas, explain each component clearly. Guide students through the worked examples, encouraging them to participate in each step. Pay close attention to common errors in calculation, especially with squaring terms for variance. Students should copy the Board Summary notes during Step 8. Check for understanding frequently through questions and short exercises. For students struggling, provide one-on-one support during practice. Challenge advanced learners with more complex problems or by asking them to interpret the results in a real-world context.