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Kelly’s Coefficient of Skewness – Formula, Calculation & Interpretation

Kelly’s Coefficient of Skewness is an extension of Bowley’s measure, but instead of quartiles, it uses deciles. Because deciles divide data into 10 equal parts, Kelly’s measure captures skewness more sensitively — especially when the distribution is irregular.

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Meaning of Kelly’s Skewness

Kelly’s Skewness compares the distances between:

• D1 (1st decile) → lower 10% point
• Median → 50% point
• D9 (9th decile) → upper 90% point

If the distribution is symmetric:

D9 – Median = Median – D1

When these two distances differ, the distribution becomes skewed.

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Formula & Components

Kelly’s measure is calculated as:

Kelly’s Skewness = (D9 + D1 - 2*M) / (D9 - D1)

Where:

• D1 = first decile
• M = median
• D9 = ninth decile

Range: -1 to +1

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Steps to Calculate

1. Arrange the data or prepare cumulative frequencies for grouped data.

2. Find positions of D1, Median, D9.

   • D1 at N/10
   • Median at N/2
   • D9 at 9N/10

3. For grouped data, use the decile formula:

   Dk = L + [(Position – CF_before) / f] * h
   

4. Substitute D1, Median, D9 into Kelly’s formula.

5. Interpret sign and magnitude.

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Solved Example (Ungrouped Data)

Data: 4, 6, 8, 10, 11, 13, 15, 18, 20, 25

Step 1: Position of deciles

• N = 10
• D1 at N/10 = 1st value → 4
• Median at N/2 = between 5th & 6th → (11 + 13)/2 = 12
• D9 at 9N/10 = 9th value → 20

Step 2: Apply formula

Sk = (20 + 4 - 2*12) / (20 - 4)
   = (24 - 24) / 16
   = 0

Interpretation: Perfect symmetry.

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Solved Example (Grouped Data)

Class Interval	Frequency
0–10	6
10–20	8
20–30	12
30–40	10
40–50	4

Total N = 40

Step 1: Cumulative Frequency

Class	f	CF
0–10	6	6
10–20	8	14
20–30	12	26
30–40	10	36
40–50	4	40

Step 2: Locate D1, Median, D9

• D1 at N/10 = 40/10 = 4 → Lies in 0–10
• Median at N/2 = 20 → Lies in 20–30
• D9 at 9N/10 = 36 → Lies in 30–40

Step 3: Compute deciles using formula

D1 (0–10):

• L = 0, h = 10, CF_before = 0, f = 6

D1 = 0 + (4/6)*10 = 6.67

Median (20–30):

• L = 20, CF_before = 14, f = 12

M = 20 + ((20 - 14)/12)*10 = 25

D9 (30–40):

• L = 30, CF_before = 26, f = 10

D9 = 30 + ((36 - 26)/10)*10 = 40

Step 4: Apply Kelly’s Formula

Sk = (40 + 6.67 - 2*25) / (40 - 6.67)
   = (46.67 - 50) / 33.33
   = -3.33 / 33.33
   = -0.10

Interpretation: Slight negative skewness.

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Interpretation of Results

• Positive Kelly’s Sk → Right tail longer (positive skew)
• Negative Kelly’s Sk → Left tail longer (negative skew)
• Values near 0 indicate approximate symmetry
• Values closer to +1 or -1 indicate strong skewness

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Important Notes

• Kelly’s measure is more sensitive than Bowley’s because deciles split data more finely.
• Not affected heavily by extreme values.
• Best used for uneven or highly skewed distributions.
• Lies between -1 and +1.

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Quiz Time! 🎯