Lesson Notes By Weeks and Term v3 - Junior Secondary 3
Measure of central tendency
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Measure of central tendency
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Subject: General Mathematics
Class: Junior Secondary 3
Term: 2nd Term
Week: 5
Theme: Everyday Statistics
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Review the ir previous work on mean, mode and median Calculate the median of a given data Find the mode of given data Calculate the mean of any given data Find the range of any given data Apply measures of central tendency to analyze any given in for mation.
middle values. First middle position = $\frac{n}{2} = \frac{8}{2} = 4^{th}$ position. Second middle position = $(\frac{n}{2} + 1) = (4 + 1) = 5^{th}$ position.
Step 4: Identify the two middle values and calculate their average. The 4th value is
6
8. The 5th value is
7
0. Median = $\frac{(68 + 70)}{2} = \frac{138}{2} = 69$ Interpretation: The median score of the students is
6
9. C. The Mode The mode is the value that appears most frequently in a dataset.
Definition: The value that occurs with the highest frequency in a dataset.
Steps to find the Mode:
1. Observe the data and count the frequency of each distinct value.
2. The value (or values) with the highest frequency is the mode.
Types of Mode: Unimodal: A dataset with only one mode.
Bimodal: A dataset with two modes (two values appear with the same highest frequency).
Multimodal: A dataset with more than two modes.
No Mode: A dataset where all values appear with the same frequency (i.e., no value occurs more frequently than others).
When to use: Best for categorical data (non-numerical) or when needing to know the most common item or value.
Worked Example 4 (Unimodal): The number of children in 10 families in a village are: 3, 4, 2, 3, 5, 3, 2, 6, 3,
4. Find the mode.
Step 1: List the distinct values and their frequencies. 2 appears 2 times 3 appears 4 times 4 appears 2 times 5 appears 1 time 6 appears 1 time Step 2: Identify the value with the highest frequency. The value '3' appears 4 times, which is the highest frequency.
Interpretation: The mode is 3 children.
Worked Example 5 (Bimodal): The sizes of shoes sold by a retailer in a day are: 38, 39, 40, 38, 41, 40, 37, 38, 40,
4
2. Find the mode.
Step 1: List the distinct values and their frequencies. 37 appears 1 time 38 appears 3 times 39 appears 1 time 40 appears 3 times 41 appears 1 time 42 appears 1 time Step 2: Identify the value(s) with the highest frequency. Values '38' and '40' both appear 3 times, which is the highest frequency.
Interpretation: The modes are 38 and
4
0. D. The Range The range is a simple measure of variability or dispersion. It indicates the spread of the data.
Definition: The difference between the highest (maximum) value and the lowest (minimum) value in a dataset.
Formula: Range = Maximum Value - Minimum Value When to use: Gives a quick, albeit rough, idea of the spread of the data. Highly affected by outliers.
Worked Example 6: Using the pure water sales data from Worked Example 1: N250, N300, N270, N320, N
2
6
0. Find the range.
Step 1: Identify the maximum value. Maximum value = N320 Step 2: Identify the minimum value. Minimum value = N250 Step 3: Apply the formula. Range = $320 - 250 = N70$ * Interpretation:** The daily sales vary by N70, indicating the difference between the highest and lowest sales amount.
Introduction to Data and Statistics: Statistics is the branch of mathematics that deals with the collection, organization, analysis, interpretation, and presentation of data. Data refers to raw facts, figures, or observations collected for a specific purpose. In JSS3, the focus is primarily on ungrouped data, which is a collection of individual observations.
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. The three primary measures are Mean, Median, and Mode. The Range, while not a measure of central tendency, is often taught alongside them as a simple measure of data spread. A. The Mean (Arithmetic Mean) The mean is the most common measure of central tendency. It is simply the average of all the numbers in a dataset.
Definition: The sum of all the values in a dataset divided by the number of values in the dataset.
Formula: Mean ($\bar{x}$) = $\frac{\sum x}{n}$ Where $\sum x$ (read as "sigma x") represents the sum of all values (x) in the dataset. And $n$ represents the total number of values in the dataset.
When to use: Best for data that is not skewed (symmetrical distribution) and does not have extreme outliers.
Worked Example 1: The daily sales (in Naira) of pure water by a vendor for five days are: N250, N300, N270, N320, N
2
6
0. Calculate the mean daily sales.
Step 1: Sum the values. $\sum x = 250 + 300 + 270 + 320 + 260 = N1400$ Step 2: Count the number of values. $n = 5$ (since there are 5 days of sales)
Step 3: Apply the formula. Mean ($\bar{x}$) = $\frac{1400}{5} = N280$ Interpretation: The average daily sales of pure water by the vendor is N
2
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0. B. The Median The median is the middle value in a dataset when the data is arranged in ascending or descending order.
Definition: The middle number in a sorted dataset.
Steps to find the Median:
1. Arrange the data in ascending (or descending) order.
2. Count the number of values ($n$) in the dataset.
3. If $n$ is odd: The median is the value at the $\frac{(n+1)}{2}$ position.
4. If $n$ is even: The median is the average of the two middle values. These values are at the $\frac{n}{2}$ and $(\frac{n}{2} + 1)$ positions.
When to use: Best for skewed data or data with outliers, as it is not affected by extreme values.
Worked Example 2 (Odd number of values): The ages (in years) of 7 children in a community youth group are: 12, 10, 15, 13, 11, 14,
9. Find the median age.
Step 1: Arrange the data in ascending order. 9, 10, 11, 12, 13, 14, 15 Step 2: Count the number of values. $n = 7$ (an odd number)
Step 3: Find the position of the median. Position = $\frac{(n+1)}{2} = \frac{(7+1)}{2} = \frac{8}{2} = 4^{th}$ position.
Step 4: Identify the median value. The 4th value in the sorted list is
1
2. Interpretation: The median age of the children is 12 years.
Worked Example 3 (Even number of values): The scores of 8 students in a JSS3 mathematics test are: 65, 72, 58, 80, 60, 75, 68,
7
0. Find the median score.
Step 1: Arrange the data in ascending order. 58, 60, 65, 68, 70, 72, 75, 80 Step 2: Count the number of values. $n = 8$ (an even number)
Step 3: Find the positions of the two middle values. First middle position = $\frac{n}{2} = \frac{8}{2} = 4^{th}$ position. Second middle position = $(\frac{n}{2} + 1) = (4 + 1) = 5^{th}$ position.
Step 4: Identify the two middle values and calculate their average. The 4th value is
6
8. The 5th value is
7
0. Median = $\frac{(68 + 70)}{2} = \frac{138}{2} = 69$ Interpretation: The median score of the students is
6
9. C. The Mode The mode is the value that appears most frequently in a dataset.
Definition: The value that occurs with the highest frequency Teacher Activities: Introduction & Review (15 minutes): Recall: Begin by asking students to recall what statistics is and how data is collected (e.g., surveys, observations).
Brainstorm: Ask students where they encounter numbers and data in their daily lives (e.g., exam scores, market prices, football scores, weather reports).
Prior Knowledge Activation: Lead a quick review session on the basic definitions of mean, median, and mode, as they were introduced in JSS
2. Use simple examples to check understanding.
Introduce Topic: Clearly state the topic for the week: "Measures of Central Tendency: Mean, Median, Mode, and Range." Explain their importance in summarizing data. Concept Explanation and Demonstration (30 minutes): Mean: Re-explain the mean thoroughly using the definition and formula. Work through Worked Example 1 (pure water sales) step-by-step on the board, emphasizing clear calculation.
Median: Explain the median, stressing the crucial first step of arranging data. Demonstrate both odd (Worked Example 2 - ages) and even (Worked Example 3 - test scores) number of data points, showing how to find the middle value(s).
Mode: Explain the concept of mode, using simple frequency counting. Demonstrate with Worked Example 4 (children per family) for a unimodal case and Worked Example 5 (shoe sizes) for a bimodal case. Discuss cases of 'no mode'.
Range: Explain the range as a measure of spread, defining it clearly. Use Worked Example 6 (pure water sales range) to illustrate.
Visual Aids: Use charts, lists, or even concrete objects (e.g., different numbers of bottle tops) to physically represent data sets where possible.
Guided Practice Facilitation (25 minutes): Group Work: Divide the class into small groups (e.g., 4-5 students per group).
Distribute Tasks: Provide each group with a few datasets (from the Guided Practice section) on separate cards or written on the board.
Monitor and Support: Circulate among groups, observing their work, providing hints, clarifying doubts, and correcting misconceptions. Encourage peer-to-peer learning.
Focus on Process: Emphasize showing all working steps, especially for arranging data for the median and summing for the mean. Class Discussion and Feedback (10 minutes): Presentation: Ask one or two groups to present their solutions for each measure on the board.
Peer Review: Encourage other groups to critique or agree with the presented solutions.
Clarification: Address any remaining misunderstandings or common errors identified during group work.
Student Activities: Participation in Review: Actively participate in the initial review of mean, median, and mode, recalling definitions and methods.
Active Listening & Note-Taking: Pay close attention during explanations, ask questions for clarification, and take concise notes on definitions, formulas, and steps.
Individual & Group Practice: Work through examples individually as the teacher demonstrates. Collaborate in assigned groups to solve the guided practice questions. Ensure all members understand the solution process.
Presentation & Discussion: Present group solutions on the board when called upon. Engage in class discussions, offering solutions, providing feedback, and asking questions.
Application: Attempt to connect the learned concepts to real-life data they might encounter. a year were: 80, 120, 100, 150, 120, 90, 110. a. Calculate the mean rainfall. b. Find the median rainfall. c. Determine the mode of the rainfall. d. Calculate the range of the rainfall. e. Briefly comment on what these measures tell you about the rainfall pattern in the town.
Solution 5: Dataset: 80, 120, 100, 150, 120, 90, 110. (n=7)
Sorted Data: 80, 90, 100, 110, 120, 120, 150. a.
Mean Rainfall: $\sum x = 80 + 120 + 100 + 150 + 120 + 90 + 110 = 770$ Mean ($\bar{x}$) = $\frac{770}{7} = 110$ mm b.
Median Rainfall: $n = 7$ (odd). Position = $\frac{(7+1)}{2} = 4^{th}$ position. From sorted data (80, 90, 100, 110, 120, 120, 150), the 4th value is
1
1
0. Median = 110 mm c.
Mode of Rainfall: Count frequencies: 80 (1), 90 (1), 100 (1), 110 (1), 120 (2), 150 (1). The value 120 appears twice, which is the highest frequency. Mode = 120 mm d.
Range of Rainfall: Maximum value = 150 mm Minimum value = 80 mm Range = $150 - 80 = 70$ mm e.
Commentary on rainfall pattern: The mean and median rainfall are both 110 mm, suggesting a fairly central distribution of rainfall around this value. The mode is 120 mm, indicating that 120 mm was the most frequently recorded monthly rainfall. The range of 70 mm shows that there is a moderate variation in rainfall across these 7 months, with a difference of 70 mm between the driest and wettest months. Overall, the town seems to experience a consistent amount of rainfall generally around 110-120mm during these months.
Market Price Analysis (Economy/Community): Scenario: A market vendor wants to know the typical price of a bag of rice in different markets across the city.
They collect prices: N35,000, N34,500, N36,000, N35,500, N34,000, N40,
0
0
0. Application: Mean: Calculates the average price, which helps in setting competitive prices or budgeting.
Median: Provides the middle price, which might be a more realistic "typical" price if there are unusually high or low prices (e.g., the N40,000 might be an outlier).
Mode: Identifies the most frequently occurring price, useful for understanding popular pricing strategies.
Range: Shows the price difference between the cheapest and most expensive bag, indicating market volatility. Student Performance Evaluation (Education): Scenario: A JSS3 Mathematics teacher wants to assess the performance of students in a class test. The scores (out of 100) are collected.
Application: Mean: Calculates the class average, indicating the overall academic standing of the class.
Median: Identifies the score of the "middle" student. If the median is significantly different from the mean, it could suggest a skewed distribution of scores (e.g., many low scores pulling the mean down).
Mode: Shows the most common score achieved, which might highlight common areas of understanding or difficulty.
Range: Indicates the spread of scores, revealing the difference between the highest and lowest performers in the class.
Community Health Data (Community/Health): Scenario: A local government health center records the number of malaria cases reported daily for a month.
Application: Mean: Helps in estimating the average daily incidence of malaria, useful for resource allocation (medication, staff).
Median: Offers a robust measure of typical daily cases, especially if there are occasional days with exceptionally high or low reports.
Mode: Identifies the most common number of cases reported on a given day, helping in understanding a baseline for planning.
Range: Shows the variability in daily cases, alerting authorities to significant fluctuations.