Lesson Note on Statistics: Calculating Mean of a Given Data from Raw Scores for JSS 3

Class: Junior Secondary School 3 (JSS 3 / JS3)
Term: 3rd Term
Week: 1
Age: 14 years
Duration: 45 minutes
Subject: Mathematics
Curriculum Theme: Mathematics
Previous Lesson: Third Term General Mathematics Examination.
Topic: STATISTICS
Subject Matter: Calculating mean of a given data from raw scores, Obtaining median, Finding mode: Finding the range of any given date. Calculating mean and median from frequency table.

Specific Objectives

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

Cognitive Domain:

• Define mean, median, mode, and range.
• Calculate the mean of raw data.
• Determine the median and mode of raw data.
• Find the range of a given set of raw data.
• Calculate the mean from a frequency table.
• Obtain the median from a frequency table.

Affective Domain:

• Appreciate the importance of statistics in interpreting real-world data.
• Show interest in solving problems related to measures of central tendency.

Psychomotor Domain:

• Accurately organize and present raw data for analysis.
• Solve statistical problems by applying appropriate formulas and steps.

Social Domain:

• Work collaboratively to solve statistical problems.
• Discuss and compare methods for finding mean, median, mode, and range.

Reference Materials

The following resources were used in planning this lesson:

• 9 Years Basic Education Curriculum
• State Unified Scheme of Work
• New General Mathematics for Junior Secondary Schools 3

Instructional Materials

The teacher will teach this lesson with the aid of:

• A chart showing formulas for mean, median, and mode.
• Prepared data sets (raw scores and frequency tables).
• Calculator.

Rationale for the Lesson

This lesson enables pupils to understand how to summarize and interpret data using statistical measures. It helps them to analyze information effectively, which is useful in various aspects of daily life and future studies.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of basic arithmetic operations such as addition, division, and ordering numbers.

Lesson Content/Board Summary

STATISTICS

Introduction to Statistics

Statistics is the branch of mathematics that deals with the collection, organization, analysis, interpretation, and presentation of data. Data refers to raw facts and figures collected for a specific purpose.

Measures of Central Tendency

Measures of central tendency are single values that attempt to describe a set of data by identifying the central position within that set of data. The most common measures are mean, median, and mode.

Mean of Raw Data

The mean (or arithmetic mean) is the sum of all values in a data set divided by the number of values in the set.

Formula: Mean (x̄) = Σx / n

• Σx = Sum of all values
• n = Number of values

Example 1: Find the mean of 5, 8, 10, 12, 15.

• Step 1: Sum the values: Σx = 5 + 8 + 10 + 12 + 15 = 50
• Step 2: Count the number of values: n = 5
• Step 3: Apply the formula: Mean = 50 / 5 = 10

Example 2: Find the mean of 7, 9, 4, 11, 6, 13.

• Step 1: Sum the values: Σx = 7 + 9 + 4 + 11 + 6 + 13 = 50
• Step 2: Count the number of values: n = 6
• Step 3: Apply the formula: Mean = 50 / 6 = 8.33 (to 2 decimal places)

Median of Raw Data

The median is the middle value in a data set when the values are arranged in ascending or descending order.

• If the number of values (n) is odd, the median is the middle value.
• If the number of values (n) is even, the median is the average of the two middle values.

Example 3: Find the median of 5, 8, 10, 12, 15.

• Step 1: Arrange in ascending order (already done): 5, 8, 10, 12, 15
• Step 2: Identify the middle value: The middle value is 10.
• Median = 10

Example 4: Find the median of 7, 9, 4, 11, 6, 13.

• Step 1: Arrange in ascending order: 4, 6, 7, 9, 11, 13
• Step 2: Identify the two middle values: 7 and 9
• Step 3: Calculate their average: Median = (7 + 9) / 2 = 16 / 2 = 8

Mode of Raw Data

The mode is the value that appears most frequently in a data set.

• A data set can have one mode (unimodal), two modes (bimodal), more than two modes (multimodal), or no mode if all values appear with the same frequency.

Example 5: Find the mode of 2, 3, 5, 3, 7, 2, 3.

• Step 1: Count the frequency of each value: 2 appears 2 times, 3 appears 3 times, 5 appears 1 time, 7 appears 1 time.
• Step 2: Identify the value with the highest frequency: 3 appears most often.
• Mode = 3

Example 6: Find the mode of 10, 12, 15, 12, 18, 10.

• Step 1: Count frequencies: 10 appears 2 times, 12 appears 2 times, 15 appears 1 time, 18 appears 1 time.
• Step 2: Identify values with the highest frequency: 10 and 12 both appear twice.
• Mode = 10 and 12 (bimodal)

Range of Raw Data

The range is the difference between the highest value and the lowest value in a data set.

Formula: Range = Highest Value – Lowest Value

Example 7: Find the range of 5, 8, 10, 12, 15.

• Step 1: Identify the highest value = 15
• Step 2: Identify the lowest value = 5
• Step 3: Calculate the range: Range = 15 – 5 = 10

Mean from Frequency Table

For data presented in a frequency table, the mean is calculated by multiplying each value (x) by its frequency (f), summing these products (Σfx), and then dividing by the total number of values (Σf).

Formula: Mean (x̄) = Σfx / Σf

• x = Value or score
• f = Frequency of the value
• Σfx = Sum of (value × frequency)
• Σf = Sum of all frequencies (total number of values)

Example 8: Calculate the mean from the frequency table below:

• Step 1: Create an ‘fx’ column by multiplying each score (x) by its frequency (f).
• Step 2: Sum the ‘f’ column to get Σf = 11.
• Step 3: Sum the ‘fx’ column to get Σfx = 34.
• Step 4: Apply the formula: Mean = Σfx / Σf = 34 / 11 ≈ 3.09 (to 2 decimal places)

Median from Frequency Table (for discrete data)

To find the median from a frequency table, first calculate the cumulative frequency, then determine the position of the median.

• Step 1: Calculate the cumulative frequency (cf) for each class/score.
• Step 2: Find the total number of observations, N = Σf.
• Step 3: The median position is at (N + 1) / 2.
• Step 4: Locate the score (x) corresponding to the cumulative frequency that is equal to or just greater than the median position.

Example 9: Find the median from the frequency table in Example 8.

• Step 1: Calculate N = Σf = 11.
• Step 2: Calculate the median position = (N + 1) / 2 = (11 + 1) / 2 = 12 / 2 = 6th position.
• Step 3: Look at the cumulative frequency column. The 6th position falls within the score of 3 (since cf=3 for score 2, and cf=8 for score 3, meaning the 4th, 5th, 6th, 7th, 8th values are 3).
• Median = 3

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 asks pupils what they understand by ‘data’ and how they think we can make sense of a large amount of information.
Pupils’ Activity: Pupils share their ideas about data and ways to organize information.
Learning Point: Pupils recall prior knowledge related to data and are introduced to the concept of statistics.

Step 2: Mean of Raw Data

Time: 8 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher defines the mean and demonstrates how to calculate it using the formula Σx / n with examples from the board summary.
Pupils’ Activity: Pupils listen attentively, copy the definition and examples, and solve a similar problem given by the teacher.
Learning Point: Pupils learn to define and calculate the mean of a given set of raw data.

Step 3: Median and Mode of Raw Data

Time: 8 minutes
Teaching Skill: Explanation/Guided Practice
Teacher’s Activity: The teacher explains how to find the median by arranging data and identifying the middle value(s), and defines the mode with examples.
Pupils’ Activity: Pupils practice arranging data and identifying the median and mode for various raw data sets.
Learning Point: Pupils understand how to obtain the median and find the mode of raw scores.

Step 4: Range of Raw Data

Time: 5 minutes
Teaching Skill: Explanation
Teacher’s Activity: The teacher defines the range and shows how to calculate it by subtracting the lowest value from the highest value.
Pupils’ Activity: Pupils define the range and practice calculating it for simple data sets.
Learning Point: Pupils learn to find the range of any given data.

Step 5: Mean from Frequency Table

Time: 8 minutes
Teaching Skill: Demonstration/Problem Solving
Teacher’s Activity: The teacher explains how to calculate the mean from a frequency table using the formula Σfx / Σf, working through an example on the board.
Pupils’ Activity: Pupils observe the steps, ask questions, and attempt to solve a similar problem in their notebooks.
Learning Point: Pupils learn to calculate the mean from a frequency table.

Step 6: Median from Frequency Table

Time: 6 minutes
Teaching Skill: Explanation/Guided Practice
Teacher’s Activity: The teacher demonstrates the steps to obtain the median from a frequency table, including calculating cumulative frequency and locating the median position.
Pupils’ Activity: Pupils follow the steps shown by the teacher and solve a provided example.
Learning Point: Pupils learn to obtain the median from a frequency table.

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 mean, median, mode, and range.
2. Calculate the mean of the following scores: 6, 10, 4, 12, 8.
3. Find the median and mode of the scores: 3, 5, 2, 5, 4, 3, 5.
4. Determine the range of the scores: 15, 22, 10, 30, 18.

Pupils’ Activity: Pupils answer orally and in writing.
Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: Remaining minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key concepts covered: mean, median, mode, and range for raw data and from frequency tables. The teacher assigns homework for practice.
Pupils’ Activity: Pupils listen to the summary and copy the assigned homework.
Learning Point: Pupils consolidate their understanding of statistics and its measures.

Lesson Keywords

• Mean – The average of a set of numbers.
• Median – The middle value in an ordered data set.
• Mode – The most frequently occurring value in a data set.
• Range – The difference between the highest and lowest values in a data set.
• Data – Raw facts and figures.
• Frequency – The number of times a particular value occurs.
• Statistics – The study of collecting, analyzing, interpreting, and presenting data.

Differentiation

For pupils who are struggling, the teacher will provide simpler data sets and more direct guidance, focusing on one measure at a time. For advanced pupils, the teacher will provide more complex data sets, including those with decimals or larger numbers, and encourage them to explore grouped frequency tables.

Note for teachers using this lesson plan

Ensure pupils have a solid grasp of ordering numbers before introducing the median. Emphasize the importance of clear, step-by-step working for all calculations to avoid errors. Encourage pupils to use calculators for larger numbers but also to show their steps clearly.