Ch 3 - Central Tendency - student

Chapter 3: Central Tendency

Review of Frequency Distribution Tables

• Benefits of Frequency Distribution Tables:

  • They summarize data, providing a clear view of the distribution of values.

  • Show the number of occurrences of each value or class interval in a dataset.

• Grouped Frequency Distribution Table:

  • Similar benefits as frequency tables but simplify data into class intervals to make interpretation easier.

• Example Measurement:

  • Measuring daily TV watching time with apparent limits for highest class intervals:

    • 0-59 minutes

    • 60-119 minutes

    • 120-179 minutes

  • Variable: Minutes of TV watched (Continuous)

  • Scale of Measurement: Ratio.

Graphing Frequency Distributions

• To graph frequency distribution:

  • Select an appropriate graph (likely a histogram for continuous data).

• Interval Width of 60:

  • Considered good for this dataset as it balances detail and readability, accommodating group statistics effectively.

Introduction to Central Tendency

• Definition:

  • A single score representing the most typical or average value in a dataset.

  • It defines the center of a distribution.

• Type of Statistics: Descriptive statistics.

Problems with Central Tendency

• Selection of Values:

  • No standardized method for calculation leads to variations in central tendency scores.

  • Different methods yield different results, especially with skewed distributions.

Measuring Central Tendency

• Three Methods:

  1. Mean: The average value, calculated as the sum of all scores divided by the number of scores.

  2. Median: The middle value in a sorted list of scores (50% below, 50% above).

  3. Mode: The most frequently occurring score or category in the distribution.

• Average Definition:

  • Often refers to the mean but encompasses mean, median, and mode.

The Mean

• Definition:

  • Arithmetic average, symbolized as:

    • μ (population mean) = ΣX / N

    • M (sample mean) = ΣX / n

• Characteristics:

  • Represents the balance point of a distribution.

  • Example: Equal distribution of items among individuals (e.g., burritos = 6 each).

• Calculation Example:

  Data: 1, 4, 11, 15, ... Sum all data points and divide by count (n).

• Change Factors:

  • Modifying scores affects ΣX but not n.

  • Mean trends toward the direction of changed scores but not proportionally.

• Adding or Removing Scores:

  • Both actions alter ΣX and n, shifting mean toward or away from the added or removed score.

• Weighted Means:

  • Calculation for populations divided into groups:

  • Combined mean = (Sum of all scores) / (Total size)

  • Important in assessing combined scores across different demographics.

• Usage and Limitations of the Mean:

  • Preferred for representing central tendency

  • Not useful when dealing with extreme scores or nominal/ordinal scales.

The Median

• Definition:

  • Midpoint score in a ranked distribution.

  • Divides the data into two equal halves.

• Calculation Process:

  • Order data from lowest to highest.

  • For odd n, find the middle score; for even n, average middle two scores.

• When to Use the Median:

  • Effective when data is skewed or has outliers.

The Mode

• Definition:

  • Most frequently occurring score in a distribution.

  • Can be calculated through a frequency table.

• Change Implications:

  • Adjusting scores may modify the mode based on frequency shifts.

Impact of Skewed Distributions

• Mean vs. Median vs. Mode:

  • Skews affect the placement of mean, median, and mode differently, revealing insights about the data centrality.

• Specific Effects:

  • Mean: Strongly affected by extreme values.

  • Median: Mildly impacted.

  • Mode: Unaffected by skew.

Reporting Central Tendency in Research:

• Use standard notation for means (M).

• Median and mode lack standardized symbols; common practice includes displaying all together in graphics.

Statistical Integrity Considerations:

• Skews can lead to misleading representations of averages in reported data.

Practice Problems:

• Engage in deriving the mean, median, and mode from provided datasets for practical understanding.