Lesson Note on Cumulative Frequency: Formula Method for Median and Percentiles for SSS 2

Class: Senior Secondary School 2 (SS2 / SSS2)
Term: Third Term
Week: 8
Age: 16 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Statistics and Probability
Previous Lesson: Cumulative Frequency: Deciles and Interquartile Range.
Topic: CUMULATIVE FREQUENCY IV
Subject Matter: Meaning of median on cumulative frequency curve percentiles quartiles deciles, determination of median deciles quartiles and percentiles by formula method

Specific Objectives

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

Cognitive Domain:

• Define median, quartiles, deciles, and percentiles based on a cumulative frequency curve (ogive).
• State the formula for calculating median, quartiles, deciles, and percentiles for grouped data.
• Explain the steps involved in determining these measures using both graphical and formula methods.

Affective Domain:

• Appreciate the importance of these statistical measures in analyzing data.
• Demonstrate carefulness when interpreting data from graphs and applying formulas.

Psychomotor Domain:

• Calculate class boundaries for grouped data.
• Draw accurate cumulative frequency curves (ogives).
• Estimate median, quartiles, deciles, and percentiles from a given ogive.
• Determine median, quartiles, deciles, and percentiles using the appropriate formula for grouped data.

Social Domain:

• Participate actively in discussions related to data analysis.
• Collaborate with peers to solve problems involving cumulative frequency.

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:

• Graph board
• Ruler
• Pencil
• Published charts showing cumulative frequency curves

Rationale for the Lesson

This lesson helps pupils understand how to find specific values within a data set, such as the middle value or values that divide the data into equal parts. This knowledge is important for analyzing and interpreting data in various fields like economics, science, and everyday decision-making.

Prerequisite/Previous Knowledge

Pupils are expected to have prior knowledge of frequency distribution, cumulative frequency, class intervals, class boundaries, and how to plot simple graphs.

Lesson Content/Board Summary

CUMULATIVE FREQUENCY IV

Meaning of Median, Quartiles, Deciles, and Percentiles from a Cumulative Frequency Curve (Ogive)

The cumulative frequency curve, also known as an ogive, helps in determining various positional measures of data graphically.

• Median (Q2 or P50): This is the middle value of a data set. From an ogive, it is the value on the horizontal axis corresponding to N/2 on the vertical axis, where N is the total frequency.
• Quartiles: These divide the data into four equal parts.
  • First Quartile (Q1 or P25): The value corresponding to N/4 on the vertical axis.
  • Third Quartile (Q3 or P75): The value corresponding to 3N/4 on the vertical axis.
• Deciles: These divide the data into ten equal parts. The kth decile (Dk) is the value corresponding to kN/10 on the vertical axis. For example, the 6th decile (D6) corresponds to 6N/10.
• Percentiles: These divide the data into one hundred equal parts. The kth percentile (Pk) is the value corresponding to kN/100 on the vertical axis. For example, the 80th percentile (P80) corresponds to 80N/100.

Determination of Median, Quartiles, Deciles, and Percentiles by Formula Method (Grouped Data)

For grouped data, these measures can be calculated using a general formula:

Formula:
Measure = L + [ ( (k/x)N – Cf ) / f ] * C

Where:

• L = Lower class boundary of the class containing the measure.
• N = Total frequency.
• Cf = Cumulative frequency of the class before the class containing the measure.
• f = Frequency of the class containing the measure.
• C = Class size (width) of the class containing the measure.
• k = The specific position (e.g., 1 for Q1, 3 for Q3, 6 for D6, 80 for P80).
• x = The divisor based on the measure (2 for median, 4 for quartiles, 10 for deciles, 100 for percentiles).

Example 1: Finding the Median

Consider the data below:

Find the median.

Solution:

Step 1: Calculate N. N = 50.

Step 2: Determine the position of the median. Median position = N/2 = 50/2 = 25th value.

Step 3: Identify the median class. The 25th value falls in the class 30-39 (since Cf up to 20-29 is 17, and up to 30-39 is 35).

Step 4: Identify the values for the formula from the median class (30-39):

• Lower class boundary (L) = 29.5 (from 30-0.5)
• Cumulative frequency before (Cf) = 17
• Frequency of median class (f) = 18
• Class size (C) = 10 (39.5 – 29.5)

Step 5: Apply the formula:

Median = L + [ ( N/2 – Cf ) / f ] * C

Median = 29.5 + [ ( 25 – 17 ) / 18 ] * 10

Median = 29.5 + [ 8 / 18 ] * 10

Median = 29.5 + 0.4444 * 10

Median = 29.5 + 4.44

Median = 33.94

Example 2: Finding the 75th Percentile (P75)

Using the same data from Example 1, find the 75th percentile (P75).

Solution:

Step 1: Calculate N. N = 50.

Step 2: Determine the position of P75. P75 position = (75/100)N = (75/100) * 50 = 37.5th value.

Step 3: Identify the P75 class. The 37.5th value falls in the class 40-49 (since Cf up to 30-39 is 35, and up to 40-49 is 45).

Step 4: Identify the values for the formula from the P75 class (40-49):

• Lower class boundary (L) = 39.5
• Cumulative frequency before (Cf) = 35
• Frequency of P75 class (f) = 10
• Class size (C) = 10

Step 5: Apply the formula:

P75 = L + [ ( (75/100)N – Cf ) / f ] * C

P75 = 39.5 + [ ( 37.5 – 35 ) / 10 ] * 10

P75 = 39.5 + [ 2.5 / 10 ] * 10

P75 = 39.5 + 0.25 * 10

P75 = 39.5 + 2.5

P75 = 42.0

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 cumulative frequency curves and asks pupils to recall how to plot an ogive.
Pupils’ Activity: Pupils respond to questions and briefly recap plotting ogives.
Learning Point: Pupils recall prior knowledge of cumulative frequency curves.

Step 2: Definition of Median, Quartiles, Deciles, Percentiles from Ogive

Time: 7 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher defines median, quartiles, deciles, and percentiles, explaining how each can be located on a cumulative frequency curve by finding the corresponding cumulative frequency value.
Pupils’ Activity: Pupils listen attentively, take notes, and ask questions for clarification.
Learning Point: Pupils understand the meaning of these measures in relation to an ogive.

Step 3: Graphical Determination

Time: 8 minutes
Teaching Skill: Demonstration
Teacher’s Activity: The teacher demonstrates on the graph board how to draw an ogive from a given frequency table and then how to estimate the median, quartiles, and possibly a decile or percentile from the drawn ogive.
Pupils’ Activity: Pupils observe the demonstration, follow the steps, and practice drawing on their own graph sheets if available.
Learning Point: Pupils learn to estimate positional measures from an ogive.

Step 4: Introduction to Formula Method

Time: 5 minutes
Teaching Skill: Explanation
Teacher’s Activity: The teacher introduces the general formula for calculating median, quartiles, deciles, and percentiles for grouped data, explaining each variable in the formula.
Pupils’ Activity: Pupils write down the formula and the meaning of each variable.
Learning Point: Pupils are introduced to the formulaic approach for these measures.

Step 5: Worked Example 1 (Median by Formula)

Time: 8 minutes
Teaching Skill: Problem Solving/Guidance
Teacher’s Activity: The teacher presents a grouped data set and guides pupils step-by-step through the calculation of the median using the formula, ensuring they identify the correct class and values for L, Cf, f, and C.
Pupils’ Activity: Pupils copy the example, follow the teacher’s steps, and perform calculations.
Learning Point: Pupils learn to apply the formula to find the median of grouped data.

Step 6: Worked Example 2 (Quartile/Decile by Formula)

Time: 7 minutes
Teaching Skill: Problem Solving/Application
Teacher’s Activity: Using the same or a similar data set, the teacher guides pupils through finding a quartile (e.g., Q1 or Q3) or a decile (e.g., D6) or a percentile (e.g., P75) using the formula, reinforcing the steps.
Pupils’ Activity: Pupils work through the second example, applying the formula with teacher guidance.
Learning Point: Pupils strengthen their ability to apply the formula for various positional measures.

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 the median on a cumulative frequency curve?
2. How do you locate the third quartile (Q3) on an ogive?
3. State the formula for calculating the 8th decile (D8) for grouped data.
4. Explain the meaning of ‘L’ and ‘Cf’ in the formula for calculating percentiles.

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 points of the lesson, emphasizing the importance of both graphical and formula methods for determining median, quartiles, deciles, and percentiles. The teacher assigns homework involving practice problems.
Pupils’ Activity: Pupils listen to the summary and copy down the assigned homework.
Learning Point: Pupils consolidate their learning and identify areas for further practice.

Lesson Keywords

• Median – The middle value of a data set.
• Quartiles – Values that divide a data set into four equal parts.
• Deciles – Values that divide a data set into ten equal parts.
• Percentiles – Values that divide a data set into one hundred equal parts.
• Cumulative Frequency – The running total of frequencies.
• Ogive – A graph of a cumulative frequency distribution.
• Grouped Data – Data organized into classes or intervals.
• Formula – A mathematical rule expressed in symbols.

Differentiation

For pupils who grasp the concepts quickly, the teacher can provide more complex data sets or challenge them to explain the differences between the graphical and formula methods. For pupils needing more support, the teacher will provide additional guided practice, focus on one measure at a time, and use simplified examples with smaller data sets.

Note for teachers using this lesson plan

Ensure that pupils have access to graph paper for plotting ogives. Emphasize the importance of accurate calculations of class boundaries and careful reading of values from both tables and graphs. Encourage pupils to understand the ‘why’ behind each step in the formula rather than just memorizing it. Provide varied practice questions to cover all aspects of median, quartiles, deciles, and percentiles.