Measures of Central Tendency
Computing and Interpreting Measures of Central Tendency (Mean, Median, Mode)
Learning Standards
Content Standard
The learner demonstrates understanding of key concepts of measures of central tendency and variability
Performance Standard
The learner is able to conduct systematically a mini-research applying the concepts of mean, median, and mode
Learning Competency
Calculate and interpret the mean, median, and mode of ungrouped and grouped data
Code: M10SP-IIIa-1
Complete Lesson Plan
Learning Objectives
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Define mean, median, and mode
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Calculate the mean, median, and mode of ungrouped data
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Compute measures of central tendency for grouped data
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Interpret and apply measures of central tendency in real-life situations
Lesson Procedures
motivation
Show class test scores: 78, 85, 92, 78, 88, 90, 78. Ask: 'What score appears most often? What is the average? What score is in the middle? How can we describe this class's performance?'
presentation
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MEASURE 1: MEAN (Average)
Formula: Mean = (Sum of all values) / (Number of values)
Example (Ungrouped Data): Test scores: 85, 90, 78, 92, 88
- Sum = 85 + 90 + 78 + 92 + 88 = 433
- Number of values = 5
- Mean = 433 ÷ 5 = 86.6
Interpretation: The average test score is 86.6
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MEASURE 2: MEDIAN (Middle Value)
Steps:
- Arrange data in ascending order
- Find the middle value
- If odd number of values: middle value
- If even number of values: average of two middle values
Example 1 (Odd): Data: 12, 15, 18, 20, 25
- Already arranged
- Middle value (3rd position) = 18
- Median = 18
Example 2 (Even): Data: 10, 15, 18, 20, 25, 30
- Two middle values: 18 and 20
- Median = (18 + 20) ÷ 2 = 19
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MEASURE 3: MODE (Most Frequent)
Definition: Value that appears most often
Example 1 (Unimodal): Data: 5, 7, 8, 7, 9, 7, 10
- 7 appears 3 times (most frequent)
- Mode = 7
Example 2 (Bimodal): Data: 3, 5, 5, 7, 8, 8, 9
- 5 and 8 both appear twice
- Mode = 5 and 8 (bimodal)
Example 3 (No mode): Data: 2, 4, 6, 8, 10
- All appear once
- No mode
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GROUPED DATA
Mean for Grouped Data: Formula: Mean = Σ(f × x) / Σf (where f = frequency, x = class mark)
Example: | Score | Frequency (f) | Class Mark (x) | f × x | |-------|--------------|----------------|-------| | 71-75 | 3 | 73 | 219 | | 76-80 | 5 | 78 | 390 | | 81-85 | 7 | 83 | 581 | | 86-90 | 4 | 88 | 352 | | 91-95 | 1 | 93 | 93 |
Σf = 20, Σ(f×x) = 1635 Mean = 1635 ÷ 20 = 81.75
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When to Use Each Measure:
- Mean: Best for normally distributed data, affected by outliers
- Median: Best when there are outliers or skewed data
- Mode: Best for categorical data or identifying most common value
generalization
Questions:
- What are the three measures of central tendency?
- How do we calculate each one?
- When is each measure most useful?
- How do outliers affect each measure?
- Why are these measures important in real life?
guided practice
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Calculate together:
Dataset: 65, 70, 75, 70, 80, 85, 70, 90
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Find the Mean:
- Sum = ?
- Mean = ?
-
Find the Median:
- Arrange: 65, 70, 70, 70, 75, 80, 85, 90
- Even number (8 values)
- Median = ?
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Find the Mode:
- Most frequent = ?
-
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Discuss: Which measure best represents this data?
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Practice with real-life scenario: 'Monthly expenses: ₱5000, ₱5500, ₱4800, ₱5200, ₱20000'
- Calculate all three measures
- Which is most representative? Why?
independent practice
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Activity 1: Calculate mean, median, and mode:
- Data: 12, 15, 18, 15, 20, 22, 15, 25
- Data: 45, 50, 55, 60, 65, 70
- Data: 8, 10, 12, 14, 16, 18, 20
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Activity 2: Solve for grouped data:
| Class Interval | Frequency | |----------------|----------| | 10-19 | 4 | | 20-29 | 6 | | 30-39 | 8 | | 40-49 | 5 | | 50-59 | 2 |
Find the mean.
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Activity 3: Word Problems:
- Five students scored 78, 85, 82, 90, and 85 on a quiz. Find mean, median, and mode.
- Which measure is affected most by an outlier? Give an example.
- A company has salaries: ₱15000, ₱18000, ₱16000, ₱17000, ₱50000. Which measure best represents typical salary? Why?
preliminary activities
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Prayer and greetings
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Review: Basic statistics concepts, data organization
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Quick activity: Survey class on a topic (e.g., hours of study per day)
Assessment
answers
- Mean = 34.375, Median = 32.5, Mode = 25
- Mean ≈ 79.5
- Mean = 156.1 cm, Median = 155 cm, Mode = 152 cm
- Median not affected by extremely high/low incomes (outliers)
- (Accept any valid dataset with explanation)
evaluation
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Solve:
- Find mean, median, and mode: 25, 30, 25, 35, 40, 25, 45, 50
- Calculate the mean for this grouped data:
| Score | Frequency | |-------|----------| | 60-69 | 5 | | 70-79 | 8 | | 80-89 | 12 | | 90-99 | 5 |
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Problem: Ten students' heights (in cm): 150, 152, 155, 152, 158, 160, 152, 162, 165, 155. Find all three measures.
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Why might the median be better than the mean for representing income data?
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Create your own dataset (10 values) where mean ≠ median ≠ mode
Materials & Resources
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Sample data sets (printed)
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Scientific calculator
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Manila paper and markers
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Graph paper
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Real-life statistical data examples
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Practice worksheets
assignment
Homework:
- Collect data from 15 classmates (e.g., allowance, study hours) and calculate all measures
- Research: How are these measures used in weather forecasting?
- Complete textbook exercises on page ___
Remarks:
- Emphasize real-world applications
- Use current events data for relevance (COVID cases, sports statistics, etc.)
subject matter
Topic: Measures of Central Tendency
Key Concepts:
- Measures of Central Tendency describe the center or typical value of a data set
- Mean (average): Sum of all values divided by the number of values
- Median: Middle value when data is arranged in order
- Mode: Most frequently occurring value
- Used in statistics to summarize and analyze data
Materials:
- Data sets (printed or on board)
- Calculator
- Manila paper and markers
- Graphing materials
- Real-life data examples