Lesson Note on Cumulative Frequency: Ogive, Median and Quartiles for SSS 2

Class: Senior Secondary School 2 (SS2 / SSS2)
Term: 3rd Term
Week: 6
Age: 16 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Statistics and Probability
Previous Lesson: Cumulative Frequency: Tables, Histogram and Frequency Polygon.
Topic: CUMULATIVE FREQUENCY II
Subject Matter: Cumulative frequency curve (ogive), plotting of cumulative frequency graph, definition of median, quartiles, percentiles, and using the curve to find median, quartiles, interquartile range, and semi-interquartile range.

Specific Objectives

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

Cognitive Domain:

• Define a cumulative frequency curve (ogive), median, quartiles, and percentiles.
• Explain the steps involved in plotting an ogive.
• State how to determine the median, quartiles, and percentiles from an ogive.

Affective Domain:

• Appreciate the importance of cumulative frequency curves in data analysis.
• Develop interest in interpreting statistical graphs.

Psychomotor Domain:

• Plot a cumulative frequency curve (ogive) accurately from a given table.
• Use an ogive to determine the median, lower quartile (Q1), upper quartile (Q3), and any given percentile.
• Calculate the interquartile range and semi-interquartile range from values obtained from an ogive.

Social Domain:

• Work collaboratively to plot graphs and interpret data.

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
• No specific links provided in the input.

Instructional Materials

The teacher will teach this lesson with the aid of:

• Graph board
• Graph book
• Pencil
• Ruler

Rationale for the Lesson

This lesson helps pupils to understand how to visually represent cumulative data and extract important statistical measures such as the median, quartiles, and percentiles. This knowledge is important for interpreting large datasets and making informed decisions in various fields.

Prerequisite/Previous Knowledge

Pupils should have prior knowledge of frequency distribution, class intervals, class boundaries, cumulative frequency, and basic graph plotting.

Lesson Content/Board Summary

Cumulative Frequency II

1. Cumulative Frequency Curve (Ogive)

An ogive is a graph that shows the cumulative frequency plotted against the upper class boundaries of a frequency distribution. It is used to estimate statistical measures like median, quartiles, and percentiles.

2. Steps for Plotting an Ogive

The following are the steps to plot an ogive:

• Step 1: Prepare a cumulative frequency table that includes the upper class boundaries.
• Step 2: Choose appropriate scales for the axes. Plot upper class boundaries on the horizontal (x) axis and cumulative frequency on the vertical (y) axis.
• Step 3: Plot the points corresponding to the upper class boundaries and their respective cumulative frequencies.
• Step 4: Join the plotted points with a smooth curve. The curve should start from the lower class boundary of the first class interval with a cumulative frequency of 0.

3. Definitions of Statistical Measures from an Ogive

• Median (Q2): The median is the middle value of a dataset when arranged in order. From an ogive, it is the value on the x-axis corresponding to N/2 on the y-axis (where N is the total frequency).
• Quartiles: Quartiles divide a dataset into four equal parts.
  • Lower Quartile (Q1): The value on the x-axis corresponding to N/4 on the y-axis.
  • Upper Quartile (Q3): The value on the x-axis corresponding to 3N/4 on the y-axis.
• Percentiles (Pk): Percentiles divide a dataset into 100 equal parts. The Pk percentile is the value on the x-axis corresponding to (kN/100) on the y-axis. For example, the 70th percentile (P70) corresponds to (70N/100).
• Interquartile Range (IQR): The difference between the upper quartile and the lower quartile (IQR = Q3 – Q1).
• Semi-Interquartile Range (SIQR): Half the interquartile range (SIQR = (Q3 – Q1) / 2).

4. Worked Examples

Example 1: Plotting an Ogive and Finding the Median

The table below shows the marks scored by 50 students in a Mathematics test.

(a) Plot an ogive for the data.

(b) Use the ogive to estimate the median mark.

Solution:

(a) Plotting the Ogive:

• Step 1: Draw the horizontal axis (x-axis) for “Marks (Upper Class Boundary)” and the vertical axis (y-axis) for “Cumulative Frequency”.
• Step 2: Choose suitable scales. For example, 2cm to represent 10 marks on the x-axis and 2cm to represent 10 students on the y-axis.
• Step 3: Plot the points (0.5, 0), (10.5, 2), (20.5, 7), (30.5, 15), (40.5, 27), (50.5, 37), (60.5, 44), (70.5, 48), (80.5, 50). Note: Start from (0.5, 0) for the first class boundary.
• Step 4: Join the points with a smooth curve.

(b) Finding the Median:

• Step 1: Total frequency (N) = 50.
• Step 2: Median position = N/2 = 50/2 = 25th value.
• Step 3: Locate 25 on the cumulative frequency (y) axis.
• Step 4: Draw a horizontal line from 25 to meet the ogive.
• Step 5: From the point on the ogive, draw a vertical line down to the x-axis (Marks).
• Step 6: Read the value on the x-axis. (From a typical plot, this would be approximately 38.5 marks).

Example 2: Finding Quartiles, Interquartile Range, and Semi-Interquartile Range

Using the ogive plotted in Example 1, find:

(a) The lower quartile (Q1) and upper quartile (Q3).

(b) The interquartile range (IQR).

(c) The semi-interquartile range (SIQR).

Solution:

(a) Finding Q1 and Q3:

• Step 1: For Q1, position = N/4 = 50/4 = 12.5th value.
  • Locate 12.5 on the CF axis. Draw a horizontal line to the ogive, then a vertical line to the x-axis. Read the value. (Approximately 27.0 marks).
• Step 2: For Q3, position = 3N/4 = 3 * (50/4) = 3 * 12.5 = 37.5th value.
  • Locate 37.5 on the CF axis. Draw a horizontal line to the ogive, then a vertical line to the x-axis. Read the value. (Approximately 50.8 marks).

(b) Calculating Interquartile Range (IQR):

• Step 1: IQR = Q3 – Q1
• Step 2: IQR = 50.8 – 27.0 = 23.8 marks.

(c) Calculating Semi-Interquartile Range (SIQR):

• Step 1: SIQR = (Q3 – Q1) / 2
• Step 2: SIQR = 23.8 / 2 = 11.9 marks.

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 tables and asks pupils to recall what cumulative frequency is. The teacher then introduces the topic “Cumulative Frequency II,” explaining that they will learn to plot and use the cumulative frequency curve.
Pupils’ Activity: Pupils respond to questions about cumulative frequency and listen attentively to the introduction of the new topic.
Learning Point: Pupils recall prior knowledge and are prepared for the new lesson.

Step 2: Plotting the Cumulative Frequency Curve (Ogive)

Time: 10 minutes
Teaching Skill: Demonstration/Explanation
Teacher’s Activity: The teacher guides pupils on how to plot a cumulative frequency curve using a prepared cumulative frequency table (like the one in the board summary). The teacher demonstrates how to choose scales, plot points (upper class boundary vs. cumulative frequency), and join them with a smooth curve, starting from (lower class boundary of first interval, 0).
Pupils’ Activity: Pupils observe carefully, ask questions for clarification, and begin to plot their own ogives in their graph books using the provided data.
Learning Point: Pupils learn the practical steps of constructing an ogive.

Step 3: Finding the Median from the Ogive

Time: 8 minutes
Teaching Skill: Guided Practice
Teacher’s Activity: The teacher explains how to determine the median from the ogive. The teacher guides pupils to locate the median position (N/2) on the cumulative frequency axis, draw a horizontal line to the curve, and then a vertical line to the x-axis to read the median value.
Pupils’ Activity: Pupils follow the teacher’s instructions to find the median from the ogive they have plotted.
Learning Point: Pupils understand how to graphically determine the median.

Step 4: Finding Quartiles from the Ogive

Time: 7 minutes
Teaching Skill: Explanation/Application
Teacher’s Activity: The teacher explains what quartiles are (Q1 and Q3) and guides pupils to find their positions (N/4 and 3N/4) on the cumulative frequency axis. The teacher demonstrates how to draw lines from these positions to the curve and then to the x-axis to read Q1 and Q3 values.
Pupils’ Activity: Pupils locate Q1 and Q3 on their ogives and read the corresponding values.
Learning Point: Pupils learn to graphically determine the lower and upper quartiles.

Step 5: Finding Percentiles from the Ogive

Time: 7 minutes
Teaching Skill: Explanation/Application
Teacher’s Activity: The teacher explains what percentiles are and how to find any percentile (Pk) from the ogive by locating its position (kN/100) on the cumulative frequency axis. The teacher gives an example, such as finding the 70th percentile (P70).
Pupils’ Activity: Pupils practice finding a given percentile from their ogives following the teacher’s guidance.
Learning Point: Pupils understand how to graphically determine percentiles.

Step 6: Calculating Interquartile and Semi-Interquartile Range

Time: 3 minutes
Teaching Skill: Calculation/Formula Application
Teacher’s Activity: The teacher defines the interquartile range (IQR = Q3 – Q1) and the semi-interquartile range (SIQR = (Q3 – Q1) / 2). The teacher guides pupils to calculate these values using the Q1 and Q3 values they obtained from the ogive.
Pupils’ Activity: Pupils calculate the IQR and SIQR based on their graphical readings.
Learning Point: Pupils learn to calculate measures of spread from graphically determined quartiles.

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 an ogive?
2. State two steps involved in plotting a cumulative frequency curve.
3. How do you find the median from an ogive?
4. If the total frequency (N) is 80, what positions on the cumulative frequency axis correspond to Q1 and Q3?
5. Define the interquartile range and semi-interquartile range.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 1 minute
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key points of the lesson, reinforcing how to plot an ogive and use it to find the median, quartiles, percentiles, and calculate the interquartile and semi-interquartile ranges.
Pupils’ Activity: Pupils listen and make notes.
Learning Point: Pupils consolidate their understanding of the lesson.

Lesson Keywords

• Ogive – A cumulative frequency curve.
• Median – The middle value in a dataset, corresponding to N/2 on the cumulative frequency axis.
• Quartiles – Values that divide a dataset into four equal parts (Q1, Q2, Q3).
• Percentiles – Values that divide a dataset into 100 equal parts.
• Interquartile Range – The difference between the upper and lower quartiles (Q3 – Q1).
• Semi-Interquartile Range – Half of the interquartile range.

Differentiation

For pupils who grasp the concepts quickly, the teacher can provide more complex data sets or ask them to find specific percentiles (e.g., P10, P90). For pupils needing more support, the teacher can provide pre-drawn axes or guide them step-by-step through the plotting and reading processes, offering individual assistance.

Note for teachers using this lesson plan

Ensure pupils have graph papers and drawing instruments. Emphasize the importance of choosing appropriate scales for accurate plotting. Encourage neatness in graph drawing and precise reading of values from the ogive. Practical plotting and reading are essential for this lesson.