Study Notes on Measures of Skewness

Measures of Skewness

Overview of Unit Five

• Focus on measures of skewness.
• End goals for students:
  • Identify and compute measures of skewness in data.
  • Integrate understanding of all descriptive measures, including:
  • Mean
  • Median
  • Mode
  • Standard Deviation

Definition of Skewness

• Skewness refers to the distribution of data and indicates where most observations lie.
• Positive Skewness: Distribution stretches to the right.
• Negative Skewness: Distribution stretches to the left.

Common Shapes of Distributions

1. Symmetrical Shape

   • Data is evenly distributed.
   • Relationships between measures:
     • Mean = Median = Mode.

2. Positive Skewness (Skewed to the Right)

   • Characterized by a long tail extending to the right.
   • Relationships among measures:
     • Mean > Median > Mode
   • Important Note:
     • The difference between mean and median should always be positive.

3. Negative Skewness (Skewed to the Left)

   • Characterized by a long tail extending to the left.
   • Relationships among measures:
     • Mean < Median < Mode
   • Important Note:
     • The difference between mean and median should always be negative.

Equations for Calculating Skewness

• Pearson's Skewness (SKP)
  • Formula: $SKP = \frac{3(\text{Mean} - \text{Median})}{\text{Standard Deviation}}$
  • Interpretation:
  • If Mean > Median, the data is positively skewed.
  • If Mean < Median, the data is negatively skewed.
• Alternative Pearson's Skewness Equation
  • Formula: $SKP = \frac{\text{Mean} - \text{Mode}}{\text{Standard Deviation}}$
  • Interpretation:
  • If Mean > Mode, the data is positively skewed.
  • If Mean < Mode, the data is negatively skewed.

Skewness Values and Their Interpretations

• Range of skewness values: (-∞, +∞)
• Excessive skewness:
  • Values greater than 1 or less than -1 indicate excessive skewness.
• Interpretation of skewness results:
  • Positive skewness: Data is skewed to the right.
  • Negative skewness: Data is skewed to the left.
  • Zero skewness: Data is symmetrically distributed (Mean = Median = Mode).

Bowley's Skewness (SKB)

• Formula for calculating skewness using quartiles:
  • $SKB = \frac{(Q<em>3 - Q</em>2) - (Q<em>2 - Q</em>1)}{IQR}$
  • Where:
  • $Q_3$ = Upper Quartile
  • $Q_2$ = Median
  • $Q_1$ = Lower Quartile
  • $IQR = Q<em>3 - Q</em>1$ (Interquartile Range)
• Interpretation of Bowley’s Skewness results:
  • Zero: Data is symmetrically distributed.
  • Positive: Data is positively skewed.
  • Negative: Data is negatively skewed.

Calculating Descriptive Statistics

• Two scenarios for calculating descriptive statistics:
  1. Ungrouped Data: Use specific equations for ungrouped data to find:
  • Mean
  • Median
  • Mode
  • Standard Deviation
  • Quartiles
  1. Grouped Data: Use specific equations for grouped data to find:
  • Mean
  • Median
  • Mode
  • Standard Deviation
  • Quartiles
• After calculating descriptive statistics, apply them to Pearson’s or Bowley’s equations for skewness.

Conclusion

• Exercise two provided for practice.
• Questions can be directed to:
  • Professor Mitilin
  • Mister Rabo

End of Unit Five Notes