Measures of Central Tendency for Ungrouped Data (Grade 8 Review)

Channel & Lesson Context

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• Current video: Grade 8 (Revised K-12 Curriculum) – Measures of Central Tendency for Ungrouped Data.
• Learning Competencies:
  • Determine measures of central tendency of ungrouped data.
  • Draw conclusions from statistical data using those measures.

Quick Grade-7 Recall: Statistics & Data

• Statistics = branch of math concerned with collecting, organizing, presenting, analyzing, interpreting data.
  • Mnemonic COPIE:
  • C – Collecting
  • O – Organizing
  • P – Presenting
  • A – Analyzing (implicit in definition)
  • I – Interpreting
• Data = facts/numbers/observations used for analysis & decision-making.
  • Examples: exam scores, attendance records, weights, heights, etc.

Types of Data (Grade-7 Review)

1. Qualitative (Categorical)
   • Describes qualities/characteristics; cannot be measured numerically.
   • Sub-types:
     • Nominal – labels w/ no inherent order (e.g., civil status).
     • Ordinal – categories with order (e.g., satisfaction level: satisfied, neutral, unsatisfied).
2. Quantitative (Numerical)
   • Represents quantities; can be measured/ counted.
   • Sub-types:
     • Discrete – whole-number counts (e.g., number of books).
     • Continuous – measured values that may include decimals (e.g., temperature).

10-Item Identification Exercise (answers)

1. General average – Quantitative
2. Civil status – Qualitative
3. Annual income – Quantitative
4. Years in school – Quantitative
5. Educational attainment – Qualitative
6. Skin color – Qualitative
7. Age – Quantitative
8. Number of children – Quantitative
9. Weight – Quantitative
10. Social class – Qualitative

Ungrouped vs. Grouped Data

• Ungrouped data: raw list of individual values; no class intervals or frequency groupings.
  • Example 1 (General averages): 90, 85, 88, 79, 93
  • Example 2 (Weights): 38.3 kg, 47.4 kg, 31.9 kg, 52.16 kg
  • Example 3 (Skin color): fair, light, medium, dark, very dark

Measures of Central Tendency – Overview

• Single value that represents “center,” “typical value,” or “central location” of a data set.
• Three common measures for ungrouped data:
  1. Mean – arithmetic average.
  2. Median – middle value once data are ordered.
  3. Mode – most frequently occurring value(s).

1. Mean (Average)

• Definition: sum of data values divided by number of values.
• Symbol: \bar{x} (“x-bar”).
• Formula: \bar{x} = \dfrac{\sum x}{n}
  • \sum x = summation of all individual values.
  • n = total number of values.
• Verbal shortcut: “Add them all, divide by how many.”

Examples

1. Data: 4, 5, 9, 10, 12
   \bar{x} = \dfrac{4+5+9+10+12}{5} = \dfrac{40}{5} = 8
2. Quarterly grades (88, 90, 92, 91):
   \bar{x} = \dfrac{361}{4} = 90.25

2. Median (MDN or MED; symbol \tilde{x})

• Definition: middle value when data are ordered from least→greatest (or vice-versa).
• Rules:
  • Odd n → median is the single middle value.
  • Even n → median is the mean of the two middle values.

Step-by-Step

1. Arrange data.
2. Identify middle position(s).
3. Compute if necessary.

Examples

1. Data: 9, 4, 12, 10, 5 → Ordered: 4, 5, 9, 10, 12
   Middle value = 9 → \tilde{x}=9
2. Weights: 38.3, 47.4, 31.9, 52.16 → Ordered: 31.9, 38.3, 47.4, 52.16
   Two middles: 38.3 & 47.4
   \tilde{x} = \dfrac{38.3 + 47.4}{2} = 42.85

3. Mode

• Definition: value(s) that appear most frequently.
• Dataset may be:
  • Unimodal – one mode.
  • Bimodal – two modes.
  • Multimodal – three or more modes.
  • No mode – all values equally frequent.

Examples

1. 3, 1, 4, 3, 4, 0, 4 → Mode = 4 (appears 3×)
2. 1, 0, 5, 14, 14, 7, 2, 3, 7 → Modes = 14 & 7 (bimodal)
3. 2, 1, 5, 13, 8, 6 → No repetition → No mode

Combined Worked Problems

Exercise 1 – Shoe Sizes (Basketball Team)

Data: 7, 9, 11, 8, 8, 8, 7, 8, 9, 10, 8 (11 values)

• \bar{x} = \dfrac{93}{11} \approx 8.45 (rounded to 2 decimals)
• Ordered: 7, 7, 8, 8, 8, 8, 8, 9, 9, 10, 11 → Median = 8 (6th value)
• Mode = 8 (occurs 5×)

Exercise 2 – Quiz Scores (12 students)

Data: 9, 5, 3, 7, 9, 6, 3, 6, 7, 5, 8, 7

• Mean: \bar{x}=\dfrac{92}{12}=7.66\overline{6}\Rightarrow 7.67
• Ordered: 3, 3, 5, 5, 6, 6, 7, 7, 7, 8, 9, 9
  Middle positions = 6th & 7th → \tilde{x}=\dfrac{6+7}{2}=7.5
• Frequencies: 3(2×), 5(2×), 6(2×), 7(3×), 8(1×), 9(2×)
  Modes = 3, 5, 6, 7, 9 → multimodal

Exercise 3 – Skin Tone Survey (Nominal Data)

• Responses (12 students): dark, dark, very dark, medium, light, light, dark, very dark, medium, fair, dark, medium
• Appropriate measure: Mode (qualitative nominal data).
• Mode = dark (appears 4×).

Connections & Real-World Relevance

• Central tendency condenses large raw datasets (exam grades, body weights, demographics) into a single understandable value for quicker decision-making.
• In schools: average grade determines honors; median helpful when outliers skew mean; mode useful in inventory (most common shoe size to stock).
• Survey research: choosing the right measure depends on data type (mean/median require numeric; mode works for categorical).

Ethical & Philosophical Notes

• Skin-tone trivia: Populations near the equator (e.g., Philippines, Indonesia, Brazil, many African nations) tend to have darker skin due to increased melanin for UV protection.
• Takeaway: Skin color is an adaptive, biological trait—not a measure of personal worth. Respect diversity; avoid prejudice.

Numerical & Symbolic Summary

• \bar{x} = \dfrac{\sum x}{n}
• Median (odd n): \tilde{x}=x{(\frac{n+1}{2})} Median (even n): \tilde{x}=\dfrac{x{(\frac{n}{2})}+x_{(\frac{n}{2}+1)}}{2}
• Mode: identify value(s) with max frequency f_{max}.

Key Takeaways

• Identify your data type first; it dictates which measure(s) are meaningful.
• Mean sensitive to outliers; median robust; mode flexible for categorical data.
• Always order data before finding the median; always count frequencies for the mode.
• Round means to appropriate decimal places; state multiple modes explicitly.

Practice Activity (Self-Study)

• Video ends with additional problem set (pause & solve). Share answers in comment section for feedback.

Further Resources

• Review Grade-7 playlist for deeper coverage of data types & basic statistics.
• Editable PowerPoints available for teachers (see description).
• Follow Math eSiP on FB for updates.