4TH-QUARTER-NOTES

Measure of Central Tendency (Ungrouped Data)

Overview

• Measure of Central Tendency: Indicates the tendency of scores to cluster around a central point.

Types of Measures

• Mean:

  • The average of a set of data.

  • Formula: 𝑥̅ = ∑𝑥 / N

    • 𝑥̅ = mean

    • ∑𝑥 = sum of scores

    • N = total number of scores

• Median:

  • The middle value in an ordered list of data, with half the values below and half above it.

  • To find the median, arrange scores from lowest to highest and identify the middle score.

• Mode:

  • The most frequently occurring score(s).

  • Bimodal: Two modes exist.

  • Multimodal: Three or more modes.

Example 1

• Nine students' scores: 83, 68, 62, 80, 70, 95, 83, 75, 75

  • Mean: 𝑥̅ = (83+68+62+80+70+95+83+75+75) / 9 = 691 / 9 = 76.78

  • Median: Arranged Scores: 62, 68, 70, 75, 75, 80, 83, 83, 95

    • The 5th score = 75

  • Mode: The most frequent: 75 and 83 (both appear twice).

Example 2

• Ten smokers' estimates: 12, 10, 10, 10, 15, 14, 15, 8, 9, 4

  • Mean: 𝑥̅ = (12+10+10+10+15+14+15+8+9+4) / 10 = 107 / 10 = 10.7

  • Median: Arranged Scores: 4, 8, 9, 10, 10, 10, 12, 14, 15, 15

    • (10 + 10) / 2 = 10

  • Mode: The most frequent score: 10 (appears thrice).

Frequency Distribution

Overview

1. Define Frequency Distribution.

2. Organize data in a frequency distribution table.

Types of Frequency Distribution Tables

• Quantitative Frequency Distribution:

  • Data tabulated based on numerical classes or intervals.

  • Example:

    • Class Interval | Frequency (f)

      • 98-99 | 9

      • 96-97 | 9

      • 94-95 | 13

      • ...

• Qualitative Frequency Distribution:

  • Data tabulated based on descriptions.

  • Example:

    • Courses | No. of Students

      • BSED Mathematics | 30

      • BSED Biological Science | 28

      • ...

Class Definitions

• Class Limits:

  • Lowest and highest values in each class.

• Lower Class Limit: Lowest value in a class.

• Upper Class Limit: Highest value in a class.

• Class Boundary: Midway between the upper limit of one class and the lower limit of the next class.

Constructing Frequency Distribution Table

1. Determine the highest and lowest values.

2. Calculate the range.

3. Decide on the number of classes (ideally between 5 and 20; use Sturges' formula: K = 1 + 3.3 log n).

4. Determine the size of class intervals (cs = R/K).

5. Construct class intervals and tally frequencies.

6. Calculate the midpoint and less than cumulative frequency.

Measure of Central Tendency (Grouped Data - Mean)

Overview

1. Illustrate measure of central tendency for grouped data.

2. Compute for mean.

Mean Formula

• 𝑥̅ = ∑𝑓𝑥 / N

  • Where:

    • 𝑓 = frequency

    • 𝑥 = class mark

    • N = total frequency

Example Calculation

• For employees' ages:

  • Sample class intervals and frequencies arranged into a table (e.g., ages 59-63).

  • Compute class marks.

  • Sum all frequencies multiplied by class marks to get ∑𝑓𝑥.

Measure of Central Tendency (Grouped Data - Median)

Overview

1. Compute for median in grouped data.

Median Formula

• Median = Lm + (𝑁/2 - 𝐶𝑓𝑏) / 𝑓𝑞1 × i

  • Where:

    • Lm = lower boundary of median class

    • 𝐶𝑓𝑏 = cumulative frequency below the median class

    • 𝑓𝑞1 = frequency of median class

    • i = class interval

    • N = total frequency

Example Calculation

• Determine frequency median by identifying the median class using cumulative frequency.

Measure of Central Tendency (Grouped Data - Mode)

Overview

1. Compute for mode in grouped data.

Mode Formula

• Mode = Lbmo + (𝑫1 / (𝑫1 + 𝑫2) × i

  • Where variables represent:

    • Lbmo = lower boundary of modal class (class with highest frequency)

    • D1 = difference between highest frequency and the frequency above

    • D2 = difference between highest frequency and the frequency below

    • i = class interval.