Correlation and Rank Correlation for SS 3

Class: SS 3
Term: First Term
Week: 8
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Previous Lesson: Equations of Parabolas, Ellipses and Hyperbolas
Topic: CORRELATION
Subject Matter: Concept of correlation as measure of relationships; Rank correlation; Scatter diagrams; Product moment; Tied rank

Specific Objectives

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

Cognitive Domain

• Define correlation as a measure of relationships between variables.
• Describe different types of correlation observed in scatter diagrams.
• State the formula for calculating the product moment correlation coefficient.
• Explain the concept of rank correlation and its formula.
• Describe how to handle tied ranks in rank correlation calculations.

Affective Domain

• Appreciate the importance of correlation in analysing real-life data.
• Develop an interest in statistical analysis and interpretation.

Psychomotor Domain

• Construct scatter diagrams from given data.
• Calculate the product moment correlation coefficient for a set of data.
• Calculate the rank correlation coefficient, including cases with tied ranks.

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:

• Real-life data examples (e.g., students’ scores in two subjects, ages and heights).
• Graph papers or white board for drawing scatter diagrams.
• Calculators.

Rationale for the Lesson

This lesson is essential for students to understand how to quantify and interpret relationships between two variables. It provides fundamental tools for data analysis, which is crucial in various fields like science, economics, and social studies. Mastering correlation helps students make informed predictions and decisions based on data patterns.

Prerequisite/Previous Knowledge

Students should have a basic understanding of data representation, plotting points on a graph, and simple statistical concepts like mean and standard deviation.

Lesson Content/Board Summary

CORRELATION

Concept of Correlation

Correlation is a statistical measure that expresses the extent to which two variables are linearly related (meaning they change together at a constant rate). It indicates both the strength and direction of the linear relationship between two quantitative variables.

The correlation coefficient ranges from -1 to +1:

1. A value of +1 indicates a perfect positive linear relationship.
2. A value of -1 indicates a perfect negative linear relationship.
3. A value of 0 indicates no linear relationship.

Scatter Diagrams

A scatter diagram (or scatter plot) is a graph that displays the relationship between two quantitative variables. Each point on the diagram represents a pair of values for the two variables.

Types of correlation observed from scatter diagrams:

1. Positive Correlation: As one variable increases, the other variable also tends to increase. The points generally rise from left to right.
2. Negative Correlation: As one variable increases, the other variable tends to decrease. The points generally fall from left to right.
3. No Correlation: There is no clear pattern or relationship between the variables. The points are scattered randomly.
4. Strong Correlation: Points cluster closely around a line.
5. Weak Correlation: Points are widely scattered, but a general trend might still be visible.

Product Moment Correlation Coefficient (Pearson’s r)

The product moment correlation coefficient, denoted by (r), measures the strength and direction of the linear relationship between two variables, (X) and (Y). It is calculated using the formula:

Formula

( r = frac{nsum XY – (sum X)(sum Y)}{sqrt{[nsum X^2 – (sum X)^2][nsum Y^2 – (sum Y)^2]}} )

Where:

1. (n) = number of pairs of observations
2. (sum X) = sum of all X values
3. (sum Y) = sum of all Y values
4. (sum XY) = sum of the products of each X and Y pair
5. (sum X^2) = sum of the squares of all X values
6. (sum Y^2) = sum of the squares of all Y values

Example 1

Question: Calculate the product moment correlation coefficient for the following data:

X: 2, 3, 4, 5, 6

Y: 4, 5, 6, 7, 8

Solution:

First, create a table to calculate the necessary sums:

X	Y	XY	X2	Y2
2	4	8	4	16
3	5	15	9	25
4	6	24	16	36
5	7	35	25	49
6	8	48	36	64
∑X=20	∑Y=30	∑XY=130	∑X2=90	∑Y2=190

(n = 5)

Step 1: Write the formula.

( r = frac{nsum XY – (sum X)(sum Y)}{sqrt{[nsum X^2 – (sum X)^2][nsum Y^2 – (sum Y)^2]}} )

Step 2: Substitute the values.

( r = frac{5(130) – (20)(30)}{sqrt{[5(90) – (20)^2][5(190) – (30)^2]}} )

( r = frac{650 – 600}{sqrt{[450 – 400][950 – 900]}} )

( r = frac{50}{sqrt{[50][50]}} )

( r = frac{50}{sqrt{2500}} )

Step 3: Simplify and write the answer.

( r = frac{50}{50} )

Answer: ( r = 1 )

Rank Correlation Coefficient (Spearman’s Rho)

Spearman’s rank correlation coefficient, denoted by (rho) (rho), measures the strength and direction of the monotonic relationship between two ranked variables. It is used when data are in ranks or can be converted to ranks.

Formula

( rho = 1 – frac{6sum d^2}{n(n^2 – 1)} )

Where:

1. (n) = number of pairs of ranks
2. (d) = difference between the ranks of corresponding pairs
3. (sum d^2) = sum of the squares of the differences in ranks

Tied Ranks

When two or more observations have the same value, they are said to be tied. To handle tied ranks:

1. Assign the average of the ranks they would have occupied if they were not tied.
2. For example, if two values are tied for the 3rd and 4th ranks, both are assigned a rank of ((3+4)/2 = 3.5).
3. If three values are tied for the 5th, 6th, and 7th ranks, all three are assigned a rank of ((5+6+7)/3 = 6).

The formula for Spearman’s rank correlation coefficient remains the same even with tied ranks.

Example 2

Question: Two judges, A and B, ranked 5 contestants in a beauty pageant as follows:

Contestant	Judge A (Rank X)	Judge B (Rank Y)
1	1	2
2	2	1
3	3	3
4	4	5
5	5	4

Calculate Spearman’s rank correlation coefficient.

Solution:

First, create a table to calculate (d) and (d^2):

Rank X	Rank Y	d = (X – Y)	d2
1	2	-1	1
2	1	1	1
3	3	0	0
4	5	-1	1
5	4	1	1
		∑d=0	∑d2=4

(n = 5)

Step 1: Write the formula.

( rho = 1 – frac{6sum d^2}{n(n^2 – 1)} )

Step 2: Substitute the values.

( rho = 1 – frac{6(4)}{5(5^2 – 1)} )

( rho = 1 – frac{24}{5(25 – 1)} )

( rho = 1 – frac{24}{5(24)} )

( rho = 1 – frac{24}{120} )

Step 3: Simplify and write the answer.

( rho = 1 – 0.2 )

Answer: ( rho = 0.8 )

This indicates a strong positive correlation between the judges’ rankings.

Teaching Methods/Instructional Techniques

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

Instructional Procedures

Step 1: Introduction

Time: 5 minutes

Teaching Skill: Activating Prior Knowledge

Teacher’s Activity: The teacher asks students about situations where two things seem to change together (e.g., study time and exam scores, rainfall and crop yield). The teacher then introduces the idea of measuring such relationships.

Pupils’ Activity: Students share examples and discuss possible relationships between variables.

Learning Point: Introduction to correlation

Step 2: Concept of Correlation and Scatter Diagrams

Time: 8 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher explains the concept of correlation as a measure of the relationship between variables. Using a whiteboard or graph paper, the teacher demonstrates how to draw scatter diagrams and interpret different types of correlation (positive, negative, no correlation, strong, weak) from their patterns.

Pupils’ Activity: Students observe the demonstration, ask questions, and identify types of correlation from given diagrams.

Learning Point: Understanding correlation types

Step 3: Product Moment Correlation Formula

Time: 7 minutes

Teaching Skill: Explanation

Teacher’s Activity: The teacher introduces the formula for the product moment correlation coefficient (Pearson’s r), explaining each variable and its significance. The teacher emphasizes that it measures linear relationships.

Pupils’ Activity: Students copy the formula and ask for clarification on terms.

Learning Point: Pearson’s r formula

Step 4: Worked Example on Product Moment Correlation

Time: 7 minutes

Teaching Skill: Guided Practice

Teacher’s Activity: The teacher guides students through a step-by-step calculation of the product moment correlation coefficient using a simple real-life data set (e.g., given scores). The teacher ensures students understand how to construct the table for calculations.

Pupils’ Activity: Students follow the steps, perform calculations with the teacher, and verify the result.

Learning Point: Calculating Pearson’s r

Step 5: Rank Correlation Concept and Formula

Time: 6 minutes

Teaching Skill: Explanation

Teacher’s Activity: The teacher explains the concept of rank correlation (Spearman’s rho) and when it is appropriate to use it (e.g., when data is ordinal or not normally distributed). The teacher then presents the formula for rank correlation.

Pupils’ Activity: Students listen, understand the application of rank correlation, and copy the formula.

Learning Point: Spearman’s rho formula

Step 6: Handling Tied Ranks and Worked Example

Time: 5 minutes

Teaching Skill: Demonstration/Problem Solving

Teacher’s Activity: The teacher explains and demonstrates how to handle tied ranks by assigning average ranks. The teacher then works through an example of calculating Spearman’s rho, including a scenario with tied ranks.

Pupils’ Activity: Students observe the ranking process and calculation, asking questions about tied ranks.

Learning Point: Calculating Spearman’s rho

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. What is correlation, and what does its value tell us?
2. Draw a scatter diagram showing a strong negative correlation.
3. State the formula for product moment correlation coefficient.
4. How do you handle tied ranks when calculating Spearman’s rank correlation coefficient?

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Correlation concepts assessed

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 on correlation, scatter diagrams, and the formulas for product moment and rank correlation into their notebooks.

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

Learning Point: Recording lesson notes

Step 9: Conclusion

Time: 3 minutes

Teaching Skill: Consolidation

Teacher’s Activity: The teacher summarises the key takeaways of the lesson, reiterating the importance of correlation in understanding relationships between variables and its application in real-world scenarios.

Pupils’ Activity: Students listen and ask any final questions.

Learning Point: Correlation understanding reinforced

Continuous Assessment/Further Study

Type: Homework

Instruction: Solve the following problems in your notebook.

1. The table below shows the scores of 6 students in Further Mathematics (X) and Physics (Y). Calculate the product moment correlation coefficient between their scores.

Student	X	Y
A	70	65
B	85	80
C	60	55
D	75	70
E	90	88
F	50	45

2. A panel of judges ranked 7 contestants in a talent show. Calculate the Spearman’s rank correlation coefficient for the following ranks given by Judge 1 and Judge 2.

Contestant	Judge 1 Rank	Judge 2 Rank
1	1	2
2	2	1
3	3	4
4	4	3
5	5	6
6	6	5
7	7	7

Lesson Keywords

• Correlation – A statistical measure indicating the extent to which two variables are linearly related.
• Scatter Diagram – A graph showing the relationship between two variables, where each point represents a pair of values.
• Product Moment Correlation Coefficient (Pearson’s r) – A measure of the linear correlation between two sets of data.
• Rank Correlation (Spearman’s rho) – A measure of the monotonic relationship between two ranked variables.
• Tied Ranks – When two or more observations have the same value and are assigned the average of the ranks they would have occupied.

Differentiation

For weaker learners, provide pre-calculated tables for the product moment and rank correlation examples, focusing only on the formula substitution and final calculation. For faster learners, provide more complex data sets or ask them to interpret the meaning of the calculated correlation coefficients in real-world contexts, or research other types of correlation.

Suggested Lesson Videos

Search YouTube for “Product Moment Correlation SS3” or “Spearman’s Rank Correlation SS3”.

Teacher Guide for Using This Lesson Plan

Before the lesson, ensure you have clear examples of real-life data for both product moment and rank correlation. Prepare blank tables for calculations to guide students efficiently. Begin by engaging students with real-world scenarios to introduce the concept of correlation. When teaching scatter diagrams, draw clear examples of different types of correlation on the board. For product moment and rank correlation, walk through the formulas step-by-step, explaining each component. Guide students through one worked example for each method, allowing them to participate in the calculations. Pay special attention to explaining how to handle tied ranks, as this can be a common point of confusion. Students should copy the Board Summary notes after the main teaching points have been covered and evaluated. Encourage students to use calculators for precision. Provide individual support to students struggling with the arithmetic or formula application, and challenge advanced learners with additional interpretation tasks.