MATH154-4 Quantitative Methods: Skewness and Kurtosis

Course Outcome 1: Measures of Shapes

• Course name: MATH154-4 Quantitative Methods
• Department: Mathematics, Mapua University

Objectives

• Compute and interpret the coefficient of skewness and kurtosis
• Differentiate between the coefficient of skewness and kurtosis
• Identify three types of skewness
• Identify three types of kurtosis

Skewness

• Definition: The degree of asymmetry of a distribution around its mean. It measures how the data deviates from being symmetrical.
• Interpretation: Symmetric, positively skewed, or negatively skewed.

Discussion on Skewness

• Symmetric Distribution: A distribution that can be folded along a vertical axis to reveal mirroring sides.
• Skewed Distribution: A distribution that lacks symmetry with respect to a vertical axis.

Types of Skewness

• Visual representations (Figures (a), (b), and (c)) depict the three types.
  • Figure (a) is skewed to the right (positive skewness) with a longer tail on the right side.
  • Figure (b) is symmetrical.
  • Figure (c) is skewed to the left (negative skewness) with a longer tail on the left side.

Skewness of Data

• Positive Skewness: Mode < Median < Mean
• Symmetric: Mode = Median = Mean
• Negative Skewness: Mode > Median > Mean

Formula - Ungrouped Data (Skewness)

• Sk = Σ(x - x̄)³/ns³
  • Sk = coefficient of skewness
  • x = score/observation
  • x̄ = mean
  • s = standard deviation
  • n = number of observations

Formula - Grouped Data (Skewness)

• Sk = Σf(x – x̄)³/ns³
  • Sk = coefficient of skewness
  • x = class mark
  • x̄ = mean
  • s = standard deviation
  • f = frequency
  • n = total number of frequency

Interpretation of Skewness Values

• Sk < 0: Negatively skewed ("skewed to the left")
• Sk = 0: Symmetric
• Sk > 0: Positively skewed ("skewed to the right")
• |Sk| < 1/2: Approximately Symmetric
• 1/2 ≤ |Sk| ≤ 1: Moderately skewed
• |Sk| > 1: Highly skewed

Kurtosis

• Definition: A measure of the degree to which a unimodal distribution is peaked. It describes the state or quality of flatness or peakedness of the curve describing a frequency distribution.

Types of Kurtosis

• Leptokurtic: High peak, heavy tails
• Mesokurtic: Normal distribution (average)
• Platykurtic: Flat peak, thin tails

Formula - Ungrouped Data (Kurtosis)

• K = Σ(x - x̄)⁴/ns⁴
  • K = coefficient of Kurtosis
  • x = score/observation
  • x̄ = mean
  • s = standard deviation
  • n = number of observations

Formula - Grouped Data (Kurtosis)

• K = Σf(x - x̄)⁴/ns⁴
  • K = coefficient of Kurtosis
  • x = class mark
  • x̄ = mean
  • s = standard deviation
  • f = frequency
  • n = total number of frequency

Interpretation of Kurtosis Values

• K < 3: Platykurtic
• K = 3: Mesokurtic
• K > 3: Leptokurtic

Example

• Data on dresses made by a factory in 10 days (52, 57, 55, 63, 50, 52, 60, 58, 54, 56)
• Calculate skewness and kurtosis, then interpret the results (Solution steps provided in the screenshots).

Summary (Skewness)

• Skewness measures asymmetry in a distribution.
• Positive skewness: mode < median < mean
• Symmetric: mode = median = mean
• Negative skewness: mode > median > mean

Summary (Kurtosis)

• Kurtosis measures the peakedness of a distribution.
• Platykurtic: K < 3
• Mesokurtic: K = 3
• Leptokurtic: K > 3

References

• (List of references from provided text)

This quiz covers the concepts of skewness and kurtosis in statistics, focusing on computation and interpretation. Learn about the different types of skewness and how they affect data distribution. Engage with visual representations to enhance your understanding of these statistical measures.