Lesson Note on Statistics: Mean, Median and Mode of Ungrouped Data for SSS 2

Class: Senior Secondary School 2 (SS2 / SSS 2)
Term: Third Term
Week: 3
Age: 16 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Statistics
Previous Lesson: Probability: Independent and Complementary Events.
Topic: STATISTICS I
Subject Matter: Meaning and computation of mean, median and mode of ungrouped data; meaning of dispersion, range, variance and standard deviation for ungrouped data; presentation of grouped figures, class interval, determination of class boundaries, grouped data histogram.

Specific Objectives

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

Cognitive Domain:

• Define statistics and explain its importance.
• Calculate the mean, median, and mode for ungrouped data.
• Explain the meaning of dispersion and calculate the range, variance, and standard deviation for ungrouped data.
• Define class interval, class boundaries, and class mark.
• Describe how to construct a grouped frequency table.
• Identify the components of a histogram for grouped data.

Affective Domain:

• Appreciate the need for grouping data when dealing with large datasets.
• Show interest in solving statistical problems accurately.
• Value the use of statistical tools in real-life situations.

Psychomotor Domain:

• Accurately compute measures of central tendency and dispersion for ungrouped data.
• Construct a grouped frequency table from given raw data.
• Draw a simple histogram given a grouped frequency table.

Social Domain:

• Collaborate with peers to solve statistical problems.
• Participate actively in class discussions about data analysis.

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:

• Score charts
• Height samples
• Grouped frequency tables
• Calculators

Rationale for the Lesson

This lesson helps pupils understand how to organize, summarize, and interpret numerical data. This knowledge is important for making sense of information encountered daily in various fields like science, business, and everyday decision-making, enabling them to analyze trends and make informed judgments.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of basic arithmetic operations and simple data collection from their previous classes.

Lesson Content/Board Summary

STATISTICS I

Meaning of Statistics

Statistics is the science of collecting, organizing, presenting, analyzing, and interpreting numerical data to make informed decisions or draw conclusions.

Measures of Central Tendency for Ungrouped Data

Measures of central tendency indicate the central or typical value of a dataset.

• Mean (Arithmetic Mean): The sum of all values divided by the number of values.

  Formula: \(bar{x} = frac{sum x}{n}\)

  Where \(sum x\) is the sum of all values and \(n\) is the number of values.

  Example 1: Find the mean of 5, 8, 12, 15, 20.

  Step 1: Sum the values. \(sum x = 5 + 8 + 12 + 15 + 20 = 60\)

  Step 2: Count the number of values. \(n = 5\)

  Step 3: Calculate the mean. \(bar{x} = frac{60}{5} = 12\)

  Example 2: The scores of 4 students in a test are 7, 9, 10, 14. Calculate the mean score.

  Step 1: Sum the scores. \(sum x = 7 + 9 + 10 + 14 = 40\)

  Step 2: Count the number of scores. \(n = 4\)

  Step 3: Calculate the mean. \(bar{x} = frac{40}{4} = 10\)

• Median: The middle value in a dataset when arranged in ascending or descending order.

  If \(n\) is odd, the median is the \(left(frac{n+1}{2}right)^{th}\) term.

  If \(n\) is even, the median is the average of the \(left(frac{n}{2}right)^{th}\) and \(left(frac{n}{2}+1right)^{th}\) terms.

  Example 1 (Odd \(n\)): Find the median of 10, 5, 12, 8, 15.

  Step 1: Arrange in ascending order: 5, 8, 10, 12, 15

  Step 2: Identify the middle term. \(n=5\), so median is \(left(frac{5+1}{2}right)^{th} = 3^{rd}\) term.

  Result: Median = 10

  Example 2 (Even \(n\)): Find the median of 2, 8, 5, 10.

  Step 1: Arrange in ascending order: 2, 5, 8, 10

  Step 2: Identify the two middle terms. \(n=4\), so median is average of \(left(frac{4}{2}right)^{th} = 2^{nd}\) and \(left(frac{4}{2}+1right)^{th} = 3^{rd}\) terms.

  Result: Median = \(frac{5+8}{2} = frac{13}{2} = 6.5\)

• Mode: The value that appears most frequently in a dataset. A dataset can have one mode (unimodal), more than one mode (multimodal), or no mode.

  Example 1: Find the mode of 7, 8, 7, 9, 10, 7, 8.

  Step 1: Count the frequency of each value.

  7 appears 3 times.

  8 appears 2 times.

  9 appears 1 time.

  10 appears 1 time.

  Result: The mode is 7 (it appears most frequently).

  Example 2: Find the mode of 3, 5, 5, 6, 7, 7, 8.

  Step 1: Count the frequency of each value.

  3 appears 1 time.

  5 appears 2 times.

  6 appears 1 time.

  7 appears 2 times.

  8 appears 1 time.

  Result: The modes are 5 and 7 (both appear twice, which is the highest frequency).

Measures of Dispersion for Ungrouped Data

Measures of dispersion describe how spread out the data points are from the central value.

• Meaning of Dispersion: Dispersion refers to the degree to which data values are spread or scattered around the average value.
• Range: The difference between the highest and lowest values in a dataset.

  Formula: Range = Highest value – Lowest value

  Example 1: Find the range of 15, 10, 20, 5, 12.

  Step 1: Identify the highest value = 20

  Step 2: Identify the lowest value = 5

  Step 3: Calculate the range. Range = 20 – 5 = 15

• Variance (\(s^2\)): The average of the squared differences from the mean.

  Formula: \(s^2 = frac{sum (x - bar{x})^2}{n}\) or \(s^2 = frac{sum x^2}{n} - (bar{x})^2\)

  Example: Find the variance of 2, 4, 6, 8.

  Step 1: Calculate the mean (\(bar{x}\)). \(bar{x} = frac{2+4+6+8}{4} = frac{20}{4} = 5\)

  Step 2: Calculate \((x - bar{x})\) for each value:

  (2-5) = -3

  (4-5) = -1

  (6-5) = 1

  (8-5) = 3

  Step 3: Calculate \((x - bar{x})^2\) for each value:

  \((-3)^2 = 9\)

  \((-1)^2 = 1\)

  \((1)^2 = 1\)

  \((3)^2 = 9\)

  Step 4: Sum the squared differences. \(sum (x - bar{x})^2 = 9 + 1 + 1 + 9 = 20\)

  Step 5: Calculate variance. \(s^2 = frac{20}{4} = 5\)

• Standard Deviation (\(s\)): The square root of the variance. It measures the typical distance of data points from the mean.

  Formula: \(s = sqrt{frac{sum (x - bar{x})^2}{n}}\)

  Example: Using the data from the variance example (2, 4, 6, 8), where variance (\(s^2\)) = 5.

  Step 1: Take the square root of the variance.

  Result: Standard Deviation \(s = sqrt{5} approx 2.24\) (to 2 decimal places)

Grouped Data

When dealing with a large number of observations, it is often more convenient to group the data into classes or intervals.

• Need for Grouping: Grouping data simplifies presentation and analysis, especially for large datasets, by organizing raw data into a more manageable form.
• Class Interval: A range of values within which observations fall. For example, 10-19, 20-29.
• Class Boundaries: The true limits of a class interval, obtained by adjusting the stated class limits. They bridge the gap between consecutive classes.

  Lower Class Boundary = Lower Class Limit – 0.5 (for whole numbers)

  Upper Class Boundary = Upper Class Limit + 0.5 (for whole numbers)

  Example: For a class interval of 10-19:

  Lower Class Boundary = 10 – 0.5 = 9.5

  Upper Class Boundary = 19 + 0.5 = 19.5

  Thus, the class boundary is 9.5 – 19.5

• Class Mark (Midpoint): The midpoint of a class interval.

  Formula: Class Mark = \(frac{text{Lower Class Limit} + text{Upper Class Limit}}{2}\)

  Example: For a class interval of 10-19:

  Class Mark = \(frac{10+19}{2} = frac{29}{2} = 14.5\)

• Grouped Frequency Table: A table that shows the frequency of data points falling within specific class intervals. It typically includes class intervals, tally marks, frequencies, and sometimes class marks and cumulative frequencies.
• Histogram: A graphical representation of a grouped frequency distribution. It consists of adjacent vertical bars, where the width of each bar represents the class interval (using class boundaries) and the height represents the frequency of that class. There are no gaps between the bars in a histogram.

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 asks them what they understand by data and how they might organize large amounts of information. The teacher then introduces the topic “STATISTICS I” as a way to manage and understand data.
Pupils’ Activity: Pupils respond to the questions and listen attentively to the introduction.
Learning Point: Pupils recall basic concepts of data and are introduced to statistics.

Step 2: Ungrouped Data – Measures of Central Tendency

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher revises the meaning and computation of mean, median, and mode for ungrouped data, providing clear examples on the board and guiding pupils through calculations both manually and with a calculator.
Pupils’ Activity: Pupils actively participate in the revision, calculate alongside the teacher, and ask questions for clarification.
Learning Point: Pupils understand and can compute mean, median, and mode for ungrouped data.

Step 3: Ungrouped Data – Measures of Dispersion

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher explains the meaning of dispersion and demonstrates how to calculate the range, variance, and standard deviation for ungrouped data using suitable examples.
Pupils’ Activity: Pupils listen to the explanations, copy notes, and practice calculating the measures of dispersion.
Learning Point: Pupils understand dispersion and can calculate range, variance, and standard deviation for ungrouped data.

Step 4: Need for Grouping and Grouped Frequency Table

Time: 5 minutes
Teaching Skill: Discussion/Questioning
Teacher’s Activity: The teacher asks pupils to suggest a large number of scores and leads a discussion on the difficulty of finding mean, median, and mode from such raw data, thus showing the need for grouping data. The teacher then explains how to construct a simple grouped frequency table.
Pupils’ Activity: Pupils suggest scores, appreciate the need for grouping, and observe how to construct a grouped frequency table.
Learning Point: Pupils understand why data grouping is necessary and how to start constructing a grouped frequency table.

Step 5: Class Interval, Boundaries, and Mark

Time: 5 minutes
Teaching Skill: Explanation/Illustration
Teacher’s Activity: The teacher teaches the concepts of class interval, determination of class boundaries, and calculation of class marks (midpoints) with examples, using the previously suggested scores or a new dataset.
Pupils’ Activity: Pupils calculate class boundaries and class marks from given class intervals.
Learning Point: Pupils learn to identify and calculate class interval components.

Step 6: Presentation of Grouped Data – Histogram

Time: 5 minutes
Teaching Skill: Explanation/Visual Aid
Teacher’s Activity: The teacher explains what a histogram is and demonstrates how it is drawn using class boundaries from a grouped frequency table, possibly using a chart or drawing a simple sketch on the board.
Pupils’ Activity: Pupils observe and understand the characteristics of a histogram.
Learning Point: Pupils understand the visual representation of grouped data through a histogram.

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 statistics.
2. Given the scores: 3, 5, 8, 5, 9. Calculate the mean, median, and mode.
3. What is dispersion in statistics?
4. For the class interval 20-29, state the lower class boundary, upper class boundary, and class mark.
5. Briefly explain what a histogram represents.

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 statistics and the different ways to analyze ungrouped and grouped data. The teacher then assigns homework from the textbook.
Pupils’ Activity: Pupils listen to the summary and copy down the assigned homework.
Learning Point: Pupils consolidate their learning and are given tasks for further practice.

Lesson Keywords

• Statistics – The science of collecting, organizing, presenting, analyzing, and interpreting numerical data.
• Mean – The average of a set of numbers.
• Median – The middle value in an ordered dataset.
• Mode – The most frequently occurring value in a dataset.
• Dispersion – The spread or variability of data points.
• Range – The difference between the highest and lowest values.
• Variance – The average of the squared differences from the mean.
• Standard Deviation – The square root of the variance, indicating data spread.
• Class Interval – A range of values in grouped data.
• Class Boundaries – The true limits of a class interval.
• Class Mark – The midpoint of a class interval.
• Histogram – A bar graph representing a frequency distribution.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide more complex datasets for calculation or challenge them to find real-world examples of statistical applications. For pupils who struggle, the teacher can provide simpler datasets, offer one-on-one guidance, and use more visual aids or step-by-step guides for calculations, focusing on basic definitions and computations before moving to more complex aspects.

Note for teachers using this lesson plan

Ensure pupils have access to calculators for efficient computation of variance and standard deviation. Emphasize the importance of arranging data for median calculation. When explaining grouped data, clearly distinguish between class limits and class boundaries, as this is a common point of confusion. Encourage pupils to practice drawing histograms neatly and accurately.