Skewness Overview

Skewness measures the asymmetry of a distribution. It indicates whether data is symmetrically distributed or skewed toward one side.

Interpretation:

  • Skewness = 0: Perfectly symmetric (bell-shaped)
  • Skewness > 0: Positively skewed (right-skewed, tail extends right)
  • Skewness < 0: Negatively skewed (left-skewed, tail extends left)

Pearson’s Coefficient of Skewness

Pearson’s coefficient uses the relationship between mean, median, and standard deviation.

Formula

$$S_K = \frac{3(\text{Mean} - \text{Median})}{s}$$

where:

  • Mean = $\bar{x}$
  • Median = $M$
  • s = Standard deviation

Pearson’s Skewness for Ungrouped Data

Example:

Age (in years) of 6 students: 22, 25, 24, 23, 24, 20

Calculate Pearson’s skewness.

Solution:

$x_i$ $x_i^2$
22 484
25 625
24 576
23 529
24 576
20 400
138 3190

Mean: $$\bar{x} = \frac{138}{6} = 23 \text{ years}$$

Median: Arranged: 20, 22, 23, 24, 24, 25 $$M = \frac{23 + 24}{2} = 23.5 \text{ years}$$

Variance: $$s^2 = \frac{1}{5}\left(3190 - \frac{138^2}{6}\right) = \frac{1}{5}(3190 - 3174) = 3.2$$

Standard Deviation: $$s = \sqrt{3.2} = 1.789$$

Pearson’s Skewness: $$S_K = \frac{3(23 - 23.5)}{1.789} = \frac{-1.5}{1.789} = -0.839$$

Interpretation: Negatively skewed (slightly left-skewed)

Pearson’s Skewness for Grouped Data

Example:

Class f Midpoint
0-10 5 5
10-20 12 15
20-30 18 25
30-40 10 35
40-50 5 45

Calculate Pearson’s skewness (μ = 25.5, M ≈ 23.3, s ≈ 12.1)

$$S_K = \frac{3(25.5 - 23.3)}{12.1} = 0.54$$

Interpretation: Positively skewed

Bowley’s Coefficient of Skewness

Bowley’s coefficient uses quartiles and is based on the position of the median relative to Q₁ and Q₃.

Formula

$$S_{Bowley} = \frac{(Q_3 - M) - (M - Q_1)}{Q_3 - Q_1} = \frac{Q_3 + Q_1 - 2M}{Q_3 - Q_1}$$

Interpretation:

  • Falls between -1 and 1
  • 0 = symmetric
  • Positive value = right-skewed
  • Negative value = left-skewed

Bowley’s Skewness for Ungrouped Data

Example:

Hospital stay (days) for 15 patients: 5, 6, 8, 9, 9, 10, 10, 10, 10, 12, 12, 13, 13, 14, 15

Calculate Bowley’s skewness.

Solution:

  • Q₁ = 9
  • M = 10 (median, 8th value)
  • Q₃ = 13

$$S_{Bowley} = \frac{(13 - 10) - (10 - 9)}{13 - 9} = \frac{3 - 1}{4} = \frac{2}{4} = 0.5$$

Interpretation: Positively skewed

Bowley’s Skewness for Grouped Data

Using grouped data quartile formulas: $$S_{Bowley} = \frac{Q_3 + Q_1 - 2M}{Q_3 - Q_1}$$

Kelly’s Coefficient of Skewness

Kelly’s coefficient uses deciles (D₉ and D₁) instead of quartiles, making it more stable for large datasets.

Formula

$$S_{Kelly} = \frac{(D_9 - M) - (M - D_1)}{D_9 - D_1} = \frac{D_9 + D_1 - 2M}{D_9 - D_1}$$

Interpretation:

  • Similar to Bowley’s, ranges from -1 to 1
  • Uses outer deciles (10th and 90th percentiles)
  • More resistant to extreme values

Example: Kelly’s Skewness

For a distribution with:

  • D₁ = 15
  • M = 45
  • D₉ = 75

$$S_{Kelly} = \frac{(75 - 45) - (45 - 15)}{75 - 15} = \frac{30 - 30}{60} = 0$$

Interpretation: Symmetric distribution

Moment Coefficient of Skewness

The moment coefficient uses the third central moment, providing the most theoretically rigorous measure.

Formula

For ungrouped data: $$\gamma_1 = \frac{m_3}{s^3}$$

where:

  • $m_3 = \frac{\sum(x_i - \bar{x})^3}{n}$ (third central moment)
  • $s$ = standard deviation

For grouped data: $$\gamma_1 = \frac{\sum f_i(m_i - \bar{x})^3}{n \cdot s^3}$$

Moment Coefficient for Ungrouped Data

Example:

Test scores: 45, 52, 58, 62, 68

Calculate moment coefficient skewness.

Solution:

$x_i$ $(x_i - \bar{x})$ $(x_i - \bar{x})^2$ $(x_i - \bar{x})^3$
45 -12 144 -1728
52 -5 25 -125
58 1 1 1
62 5 25 125
68 11 121 1331

Mean: $\bar{x} = 57$

Third Central Moment: $$m_3 = \frac{-1728 - 125 + 1 + 125 + 1331}{5} = \frac{-396}{5} = -79.2$$

Variance: $s^2 = \frac{316}{5} = 63.2$, so $s = 7.95$

Moment Coefficient: $$\gamma_1 = \frac{-79.2}{(7.95)^3} = \frac{-79.2}{502.46} = -0.158$$

Moment Coefficient for Grouped Data

$$\gamma_1 = \frac{\sum f_i(m_i - \bar{x})^3}{n \cdot s^3}$$

Calculate using class midpoints, frequencies, and deviations from mean.

Comparison of Skewness Methods

Method Advantages Disadvantages
Pearson Simple, intuitive Only uses 2 measures (mean, median)
Bowley Based on quartiles, robust Less sensitive to extreme values
Kelly Uses deciles, more stable More complex to calculate
Moment Theoretically rigorous Affected by extreme outliers

Interpretation Guidelines

Skewness Value Distribution Type Characteristics
-1 to 0 Left-skewed Tail extends left, mean < median
0 Symmetric Balanced distribution
0 to 1 Right-skewed Tail extends right, mean > median
> 1 or < -1 Highly skewed Extreme asymmetry

Visual Representation

Left-Skewed        Symmetric         Right-Skewed
   (Negative)         (Zero)           (Positive)

      |  |                |              |  |
      |  |                |              |  |
   |  |  |             |  |  |         |  |  |
|  |  |  |          |  |  |  |      |  |  |  |
────────────────────────────────────────────────
Mean                Mean              Mean
Median              Median            Median

Tail ←              Symmetric         → Tail

Related Shape Measures:

Related Concepts:

Applications:

Applications

  • Income Distribution: Identifying wealth inequality
  • Quality Control: Detecting process imbalances
  • Data Transformation: Deciding normalization techniques
  • Statistical Testing: Checking assumptions
  • Pattern Recognition: Understanding data characteristics

References

  1. Anderson, D.R., Sweeney, D.J., & Williams, T.A. (2018). Statistics for Business and Economics (14th ed.). Cengage Learning. - Comprehensive treatment of skewness calculation methods and interpretation.

  2. NIST/SEMATECH. (2023). e-Handbook of Statistical Methods. Retrieved from https://www.itl.nist.gov/div898/handbook/ - Authoritative reference on statistical measures including moment-based skewness.