Measures of Central Tendency of Ungrouped Data

Quiz 1.1: Integer Arithmetic (06 - 15 - 26)

Answer the following arithmetic operations (P.S: 14):

• 1. $4 - 5$
• 2. $-4 - 5$
• 3. $-4 - (-5)$
• 4. $4 - (-5)$
• 5. $5 - 4$
• 6. $-5 - 4$
• 7. $-5 - (-4)$
• 8. $5 - (-4)$
• 9. $4(5)$
• 10. $-4(5)$
• 11. $-4(-5)$
• 12. $4(-5)$
• 13. $6 \div 2$
• 14. $-6 \div -2$
• 15. $-6 \div (2)$
• 16. $6 \div (-2)$

Understanding Ungrouped and Grouped Data

• Ungrouped Data: This is raw data collected in its original, actual form. It is simply a list of individual values.

  • Example 1: Shoe sizes of Grade 8 students in MCSNHS.
  • Example 2: The age of Grade 8 students in MCSNHS.
  • Example 3: The number of absences of students in a specific term.

• Grouped Data: This is raw information that has been organized, categorized, and bundled into distinct intervals or "classes" rather than being maintained as a list of individual numbers.

Theoretical Overview of Measures of Central Tendency

• Definition: A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set. These are sometimes referred to as measures of central location.
• Core Purpose: It is used to describe a set of data where the measures cluster or concentrate at a specific point.
• Guided Questioning for Conceptual Understanding:
  • 1. What is the middle value of the indicated numbers in the number line on the left?
  • 2. What is the middle value of the indicated numbers in the number line on the right?
  • 3. How were you able to find the answer for each given number line?

The Mean (Arithmetic Mean)

• Definition: The mean is the sum of the data values divided by the total number of values. It is considered the arithmetic mean and the "balance point" of a distribution.
• Impact of Outliers: It is greatly affected by extreme scores. An extreme value, or "outlier," is a number significantly greater or significantly less than most of the other numbers in a distribution.
• Best Use Case: It is best used for symmetrical, normal distributions that do not contain heavy outliers.
• Formula for Ungrouped Data:
  • $\bar{x} = \frac{\sum x}{N}$
  • where $\bar{x} =\text{mean}$
  • $\sum x =\text{the sum of all values}$
  • $N =\text{number of values}$
  • Note: The symbol $\sum$ (sigma) is a Greek letter used to indicate summation.
• Mean Examples:
  • Problem 1: Number of books owned by 10 students: $3, 5, 4, 2, 6, 4, 3, 5, 4, 3$
  • Problem 2: Number of siblings of 9 students: $1, 2, 3, 2, 1, 4, 2, 3, 1$

The Median

• Definition: The middle value or central value of an ordered distribution. It is known as the "halfway marker" because it splits the data perfectly into two equal halves (50% above, 50% below).
• Characteristics: The data set must be arranged in either increasing or decreasing order. It is the best measure used for cases with extreme values because it is less affected by them.
• Best Use Case: Skewed numerical data or datasets containing strong outliers.
• Rules for Calculation:
  • 1. Arrange the data set in increasing order.
  • 2. If the data set contains an odd number of values, use the formula for the nth term to find the position: $median = \frac{n+1}{2}th$
  • 3. Count from left to right (or vice versa) to find that specific position's value.
  • 4. If the data set contains an even number of values, the median is the average of the two middle values.
• Median Examples:
  • Problem 1: Number of stickers on notebooks of 9 students: $2, 5, 3, 4, 1, 6, 3, 2, 4$
  • Problem 2: Number of hours of sleep of 8 students: $1, 2, 3, 2, 1, 4, 2, 3$

The Mode

• Definition: The most frequent value in a set of data. It represents "peak popularity" within the dataset.
• Multiple Modes: If two or more measures appear an equal number of times (and that number is the highest frequency), each value is considered a mode.
• No Mode: If every measure in the dataset appears the same number of times, the set has no mode.
• Best Use Case: Categorical (nominal) data or for identifying the most popular choice.

Types of Mode

• Unimodal Mode: A collection of data with exactly one mode.
  • Example: Set $B = \{20, 14, 16, 17, 14, 18, 14, 19\}$. The mode is $14$.
• Bimodal Mode: A set of data containing two modes. This means two separate values share the highest frequency.
  • Example: Set $B = \{8, 12, 12, 14, 15, 19, 17, 19\}$. The modes are $12$ and $19$.
• Trimodal Mode: A collection of data containing three modes.
  • Example: Set $B = \{2, 2, 2, 3, 7, 7, 5, 6, 5, 4, 7, 5, 8\}$. The modes are $2$, $7$, and $5$.
• Multimodal Mode: A set of data including four or more modes.
  • Example: Set $B = \{101, 82, 82, 95, 95, 100, 90, 90, 101, 96\}$. The modes are $82, 95, 90, 101$.

Quiz 1.2: General Concepts and Calculation (06 - 16 - 26)

Answer the following (P.S: 8):

• 1. It is also known as halfway marker. It splits your data perfectly into two equal halves. (Answer: Median)
• 2. It is best use if data sets contains strong outliers. (Answer: Median)
• 3. The most frequent value in a set of data. (Answer: Mode)
• 4. It is best use if it is symmetrical, normal distributions without heavy outliers. (Answer: Mean)
• 5. A number that is significantly greater or significantly less than most of the other numbers in a distribution is called? (Answer: Outlier or Extreme Value)
• Scenario: Number of hours of sleep of students: $6, 5, 3, 2, 1, 4, 2, 3, 5, 6$
  • 6. What is the mean?
  • 7. What is the median?
  • 8. What is the mode?
  • 9. What type of mode?

Practice 1.1: Applied Problems (06 - 16 - 26)

Find the mean, median, mode, and type of mode for each item. Show solutions.

• Item 1: The grades in Geometry of 10 students: $87, 84, 85, 85, 86, 90, 79, 82, 78, 76$.
• Item 2: Liza Soberano's first-quarter grades: Filipino (88), English (91), Math (90), Science (90), Araling Panlipunan (90), ESP (92), TLE (92), MAPEH (91).

Real-Life Applications and Selection of Appropriate Measures

Directions: Identify whether the Mean, Median, or Mode is most appropriate.

• 1. Most common online games played by Grade 8 students.
  • Appropriate Measure: Mode
• 2. Science test grades where all scored over 75, but 4 absent students are listed as 0.
  • Appropriate Measure: Median (to avoid the downward pull of the 0 outliers).
• 3. Determining the most common Electronic computer games played by youth today.
  • Appropriate Measure: Mode
• 4. Jose's performance across math exam scores: 80, 87, 68, 74, 81.
  • Appropriate Measure: Mean
• 5. Average monthly temperature in a city over a year.
  • Appropriate Measure: Mean
• 6. Exam scores with one extremely low score due to illness.
  • Appropriate Measure: Median
• 7. Favorite fruit among students in a class.
  • Appropriate Measure: Mode

Distribution Shapes and Central Tendency Relationships

Skewness describes a dataset that is unbalanced rather than perfectly symmetrical.

• Symmetrical Distribution (Normal Distribution):

  • Characterized by a perfectly symmetrical bell curve.
  • The mean, median, and mode are all perfectly identical and located at the center peak.
  • Relationship: $Mean = Median = Mode$
  • Real-Life Examples: Adult Human Height, Shoe Sizes.

• Negatively Skewed (Left-Skewed):

  • The long tail extends toward the lower values on the left.
  • Low-value outliers pull the mean down.
  • Relationship: $Mean < Median < Mode$
  • Real-Life Example: Age of Death (Lifespan), Age of Retirement at a corporation.

• Positively Skewed (Right-Skewed):

  • The long tail extends toward the higher values on the right.
  • High-value outliers pull the mean to the far right.
  • Relationship: $Mean > Median > Mode$
  • Real-Life Example: Student Bank Account Balances, Weekly Restaurant Wait Times.

Comparative Analysis of Distribution Shapes

• Example A (Standardized Math Test):
  • Given: Mean (75), Median (75), Mode (75).
  • Type: Normal Distribution.
• Example B (Weekly Restaurant Wait Times):
  • Given: Mean (38 min), Median (25 min), Mode (15 min).
  • Relationship: $38 > 25 > 15$
  • Type: Positively Skewed.
• Example C (Age of Retirement):
  • Given: Mean (61 y.o), Median (64 y.o), Mode (65 y.o).
  • Relationship: $61 < 64 < 65$
  • Type: Negatively Skewed.
• Example D (Heights of adult women):
  • Given: Mean (163 cm), Median (163.2 cm), Mode (163.5 cm).
  • Relationship: Very close, near center peak.
  • Type: Symmetrical / Normal.

Oral Assessment: Summary of Logic (06 - 16 - 26)

Answer the following (P.S: 4):

• 1. The long tail extends toward the higher numbers on the right. (Answer: Positively Skewed)
• 2. $Mean < Median < Mode$ (Answer: Negatively Skewed)
• 3. The long tail extends toward the lower numbers on the left. (Answer: Negatively Skewed)
• 4. The mean, median, and mode are all perfectly identical and located right at the center peak of the curve. (Answer: Normal Distribution)
• 5. Define the skew of: Mean (38), Median (25), Mode (15). (Answer: Positively Skewed)

Closing the Loop: Reflection on Learning

1. What are the key concepts of our lesson?
2. Which part of the lesson is the easiest for you? Why?
3. Which part of the lesson is the hardest for you? Why?
4. How are we as a class today?