Lesson Note on Cumulative Frequency: Deciles and Interquartile Range for SSS 2

Class: Senior Secondary School 2 (SS2 / SSS2)
Term: 3rd Term
Week: 7
Age: 16 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Statistics and Probability
Previous Lesson: Cumulative Frequency: Ogive, Median and Quartiles.
Topic: CUMULATIVE FREQUENCY III
Subject Matter: Meaning of deciles, examples showing median and quartiles from graph, interquartile range quarter deviation by formula.

Specific Objectives

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

Cognitive Domain:

• Define deciles, interquartile range, and quartile deviation.
• State the formulas for calculating deciles, quartiles, interquartile range, and quartile deviation for grouped data.
• Calculate deciles and quartiles using the appropriate formulas.
• Determine the interquartile range and quartile deviation for a given data set.

Affective Domain:

• Appreciate the importance of deciles and quartiles in analyzing data distribution.
• Develop a systematic approach to solving problems involving cumulative frequency.

Psychomotor Domain:

• Read and interpret median, quartiles, and deciles from a given cumulative frequency curve.
• Accurately apply formulas to calculate measures of position for grouped data.

Social Domain:

• Collaborate with peers to solve problems related to cumulative frequency.
• Participate actively in classroom discussions and problem-solving sessions.

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
• https://www.studywithjosh.com/2019/07/cumulative-frequency-curve-median.html
• https://www.nagwa.com/en/explainers/824151740925/

Instructional Materials

The teacher will teach this lesson with the aid of:

• Graph board
• Graph book
• Markers/Chalk
• Ruler
• Calculator
• Cumulative frequency curves (pre-drawn or projected)

Rationale for the Lesson

This lesson enables pupils to understand how to further analyze data using deciles, quartiles, interquartile range, and quartile deviation. These measures help pupils interpret the spread and position of values within a data set, which is useful in various real-world applications.

Prerequisite/Previous Knowledge

Pupils are expected to have prior knowledge of constructing cumulative frequency tables, drawing cumulative frequency curves, and determining the median and quartiles from a graph.

Lesson Content/Board Summary

CUMULATIVE FREQUENCY III

Deciles

Deciles are values that divide a data set into ten equal parts. There are nine deciles, denoted D1, D2, …, D9. D1 separates the lowest 10% of the data, D2 separates the lowest 20%, and so on.

For grouped data, the k-th decile (Dk) is calculated using the formula:

Dk = L + [ (kN/10 – cf_b) / f_k ] * c

• L = Lower class boundary of the decile class
• N = Total frequency
• cf_b = Cumulative frequency before the decile class
• f_k = Frequency of the decile class
• c = Class width

Quartiles by Formula

Quartiles divide a data set into four equal parts. There are three quartiles: Q1 (Lower Quartile), Q2 (Median), and Q3 (Upper Quartile).

For grouped data, the k-th quartile (Qk) is calculated using the formula:

Qk = L + [ (kN/4 – cf_b) / f_k ] * c

• L = Lower class boundary of the quartile class
• N = Total frequency
• cf_b = Cumulative frequency before the quartile class
• f_k = Frequency of the quartile class
• c = Class width

Note: Q2 is the median.

Interquartile Range (IQR)

The Interquartile Range (IQR) is a measure of statistical dispersion, representing the range of the middle 50% of the data. It is the difference between the upper quartile (Q3) and the lower quartile (Q1).

IQR = Q3 – Q1

Quartile Deviation (QD)

The Quartile Deviation (QD), also known as the semi-interquartile range, is half of the interquartile range. It measures the average deviation of the quartile values from the median.

QD = (Q3 – Q1) / 2

Examples Showing Median and Quartiles from Graph

To find the median, quartiles, or deciles from a cumulative frequency curve:

1. Calculate the position for the required value on the cumulative frequency axis (e.g., N/2 for median, N/4 for Q1, 3N/4 for Q3, kN/10 for Dk).
2. Locate this position on the vertical (cumulative frequency) axis.
3. Draw a horizontal line from this point to intersect the cumulative frequency curve.
4. From the intersection point on the curve, draw a vertical line down to the horizontal (class boundary/score) axis.
5. Read the value on the horizontal axis; this is the median, quartile, or decile.

Worked Example: Calculation by Formula

The table below shows the distribution of scores of 50 students in a mathematics test:

Calculate Q1, Q3, IQR, and QD.

Solution:

First, adjust class intervals to obtain class boundaries:

Total frequency N = 50. Class width (c) = 10.5 – 0.5 = 10.

Step 1: Calculate Q1 (Lower Quartile)

• Position of Q1 = N/4 = 50/4 = 12.5th position.
• The 12.5th position falls in the class with cf = 13 (class 10.5 – 20.5). This is the Q1 class.
• L = 10.5
• cf_b = 5 (cf before the Q1 class)
• f_k = 8 (frequency of the Q1 class)
• Q1 = 10.5 + [ (12.5 – 5) / 8 ] * 10
• Q1 = 10.5 + [ 7.5 / 8 ] * 10
• Q1 = 10.5 + 0.9375 * 10
• Q1 = 10.5 + 9.375 = 19.875

Step 2: Calculate Q3 (Upper Quartile)

• Position of Q3 = 3N/4 = 3 * 50 / 4 = 150 / 4 = 37.5th position.
• The 37.5th position falls in the class with cf = 40 (class 30.5 – 40.5). This is the Q3 class.
• L = 30.5
• cf_b = 25 (cf before the Q3 class)
• f_k = 15 (frequency of the Q3 class)
• Q3 = 30.5 + [ (37.5 – 25) / 15 ] * 10
• Q3 = 30.5 + [ 12.5 / 15 ] * 10
• Q3 = 30.5 + 0.8333 * 10
• Q3 = 30.5 + 8.333 = 38.833

Step 3: Calculate IQR (Interquartile Range)

• IQR = Q3 – Q1
• IQR = 38.833 – 19.875 = 18.958

Step 4: Calculate QD (Quartile Deviation)

• QD = (Q3 – Q1) / 2
• QD = 18.958 / 2 = 9.479

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 greets the pupils and reviews the previous lesson on constructing cumulative frequency curves and finding median and quartiles from graphs.
Pupils’ Activity: Pupils respond to greetings and briefly recall how to find median and quartiles from a graph.
Learning Point: Pupils connect previous knowledge to the new topic.

Step 2: Meaning of Deciles

Time: 5 minutes
Teaching Skill: Explanation
Teacher’s Activity: The teacher introduces the concept of deciles, explaining that they divide data into ten equal parts, and presents the formula for calculating deciles for grouped data.
Pupils’ Activity: Pupils listen attentively, ask questions for clarification, and copy the definition and formula into their notes.
Learning Point: Pupils understand the definition and formula for deciles.

Step 3: Calculating Quartiles by Formula

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher revisits the concept of quartiles and explains how to calculate them using the formula for grouped data, emphasizing the meaning of each variable in the formula.
Pupils’ Activity: Pupils observe the explanation, ask questions, and take down the quartile formula and its components.
Learning Point: Pupils learn the formula for calculating quartiles for grouped data.

Step 4: Interquartile Range and Quartile Deviation

Time: 5 minutes
Teaching Skill: Explanation
Teacher’s Activity: The teacher defines interquartile range (IQR) and quartile deviation (QD), explaining their significance as measures of spread and presenting their respective formulas.
Pupils’ Activity: Pupils listen and copy the definitions and formulas for IQR and QD.
Learning Point: Pupils understand how to calculate and interpret IQR and QD.

Step 5: Worked Example (Calculation by Formula)

Time: 10 minutes
Teaching Skill: Problem Solving/Demonstration
Teacher’s Activity: The teacher works through a comprehensive example on the board, demonstrating step-by-step how to calculate Q1, Q3, IQR, and QD from a given grouped frequency distribution table using the formulas.
Pupils’ Activity: Pupils actively follow the steps, participate in calculations, and copy the example into their notebooks.
Learning Point: Pupils gain practical experience in applying the formulas to solve problems.

Step 6: Reading from Graph

Time: 5 minutes
Teaching Skill: Illustration
Teacher’s Activity: The teacher briefly demonstrates or reminds pupils how to read median, quartiles, and deciles from a pre-drawn cumulative frequency curve, emphasizing the steps involved.
Pupils’ Activity: Pupils observe and recall the procedure for extracting values from a cumulative frequency graph.
Learning Point: Pupils reinforce their understanding of reading values from a cumulative frequency curve.

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 deciles.
2. State the formula for calculating the k-th quartile for grouped data.
3. If Q1 = 25 and Q3 = 75, calculate the Interquartile Range and Quartile Deviation.
4. List two uses of cumulative frequency curves.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 5 minutes
Teaching Skill: Summarization/Assignment
Teacher’s Activity: The teacher summarizes the key points of the lesson, reiterating the importance of deciles, quartiles, IQR, and QD in data analysis. The teacher then assigns homework.
Pupils’ Activity: Pupils listen to the summary and copy down the homework assignment.
Learning Point: Pupils consolidate their understanding and are given tasks to practice.

Lesson Keywords

• Deciles – Values that divide a data set into ten equal parts.
• Quartiles – Values that divide a data set into four equal parts (Q1, Q2, Q3).
• Interquartile Range (IQR) – The difference between the third and first quartiles (Q3 – Q1).
• Quartile Deviation (QD) – Half of the interquartile range ((Q3 – Q1) / 2).
• Cumulative Frequency – The running total of frequencies.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide additional complex problems involving deciles or percentiles. For struggling pupils, the teacher can offer more guided practice with simpler data sets and provide individual attention or peer tutoring opportunities.

Note for teachers using this lesson plan

Ensure that pupils are comfortable with basic cumulative frequency curve construction before delving into deciles and detailed formula calculations. Emphasize the practical application of these measures to real-world data. Encourage pupils to use calculators effectively during calculations. Provide clear, step-by-step solutions for all examples.