Kelly’s Coefficient of Skewness Calculator
Use this unified calculator to find Kelly’s coefficient of skewness for both ungrouped (raw) data and grouped (frequency distribution) data. Measure distribution asymmetry using deciles (10th and 90th percentiles).
Quick Start
Choose your data type, enter your values, and click Calculate:
| Kelly's Coefficient of Skewness Calculator | |
|---|---|
| Data Type | Ungrouped (Raw Data) Grouped (Frequency Distribution) |
| Enter the X Values (Separated by comma,) | |
| Type of Frequency Distribution | DiscreteContinuous |
| Enter the Classes for X (Separated by comma,) | |
| Enter the frequencies (f) (Separated by comma,) | |
| Results | |
| Number of Observations (N): | |
| Ascending order of X values: | |
| First Decile (D₁): | |
| Median (D₅): | |
| Ninth Decile (D₉): | |
| Kelly's Coefficient of Skewness: | |
Understanding Kelly’s Coefficient of Skewness
Kelly’s coefficient is a decile-based measure of skewness that:
- Uses extreme deciles (D₁ and D₉) plus the median (D₅)
- Ranges from -1 to +1
- Focuses on the tail behavior of the distribution
- More sensitive to extreme values than Bowley’s
- Useful for understanding data behavior in the extreme percentiles
Key Difference from Bowley’s
- Bowley’s: Uses Q₁, Q₂, Q₃ (focuses on middle 50% of data)
- Kelly’s: Uses D₁, D₅, D₉ (focuses on tail behavior from 10th to 90th percentile)
Interpretation Scale
| Kelly’s Coefficient | Distribution Shape | Interpretation |
|---|---|---|
| 0 | Perfectly symmetric | No skewness; symmetric tails |
| Between -0.5 and 0.5 | Approximately symmetric | Mild skewness; relatively balanced |
| < -0.5 | Negatively skewed (left) | Left tail extends further |
| > 0.5 | Positively skewed (right) | Right tail extends further |
How to Use This Calculator
For Ungrouped (Raw) Data
Step 1: Select “Ungrouped (Raw Data)” as your data type
Step 2: Enter your data values separated by commas (e.g., 10, 15, 20, 25, 30)
Step 3: Click “Calculate”
Results will show:
- Number of observations (N)
- Sorted values in ascending order
- First decile (D₁ = 10th percentile)
- Median (D₅ = 50th percentile)
- Ninth decile (D₉ = 90th percentile)
- Kelly’s coefficient of skewness
For Grouped (Frequency Distribution) Data
Step 1: Select “Grouped (Frequency Distribution)” as your data type
Step 2: Choose frequency distribution type:
- Discrete: For individual values
- Continuous: For class intervals
Step 3: Enter class values or intervals and frequencies
Step 4: Click “Calculate”
Results will show:
- All five calculated values
Formula
Kelly’s Coefficient of Skewness
$$S_k = \frac{D_9 + D_1 - 2 \times D_5}{D_9 - D_1}$$
Where:
- $D_1$ = First decile (10th percentile)
- $D_5$ = Fifth decile = Median (50th percentile)
- $D_9$ = Ninth decile (90th percentile)
Understanding the Formula
- Numerator: $(D_9 + D_1 - 2D_5)$ measures deviation of median from the midpoint of D₁ and D₉
- Denominator: $(D_9 - D_1)$ is the range excluding first 10% and last 10%, normalizing to -1 to +1 scale
Why Kelly’s Coefficient?
Advantages:
- ✅ Focuses on 80% of data (D₁ to D₉), ignoring extreme outliers
- ✅ Better captures tail behavior than Bowley’s
- ✅ More stable for very skewed distributions
- ✅ Uses deciles which are common in practice
- ✅ Ranges from -1 to +1 for easy interpretation
Disadvantages:
- ❌ Ignores extreme values (which might be important)
- ❌ Less well-known than Pearson’s or moment coefficients
- ❌ Still ignores some distribution information
Worked Examples
Example 1: Ungrouped Data - Symmetric Distribution
Data: Values: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100
Find Kelly’s coefficient of skewness.
Solution:
Step 1: Arrange data (already sorted)
Step 2: Find deciles
With N = 10:
-
D₁ position = (1×11)/10 = 1.1 → Between 1st and 2nd values
- D₁ = 10 + 0.1(20-10) = 11
-
D₅ position = (5×11)/10 = 5.5 → Between 5th and 6th values
- D₅ = 50 + 0.5(60-50) = 55
-
D₉ position = (9×11)/10 = 9.9 → Between 9th and 10th values
- D₉ = 90 + 0.9(100-90) = 99
Step 3: Calculate Kelly’s coefficient
$$S_k = \frac{99 + 11 - 2(55)}{99 - 11} = \frac{110 - 110}{88} = 0$$
Answer: Sk = 0
Interpretation: Distribution is perfectly symmetric; the tails are equally balanced.
Example 2: Ungrouped Data - Right-Skewed Distribution
Data: Income (thousands): 20, 22, 24, 26, 28, 30, 32, 35, 40, 100
Find Kelly’s coefficient of skewness.
Solution:
Step 1: Data is sorted
Step 2: Find deciles
With N = 10:
-
D₁ = 20 + 0.1(22-20) = 20.2
-
D₅ = 28 + 0.5(30-28) = 29
-
D₉ = 40 + 0.9(100-40) = 94
Step 3: Calculate Kelly’s coefficient
$$S_k = \frac{94 + 20.2 - 2(29)}{94 - 20.2} = \frac{114.2 - 58}{73.8} = \frac{56.2}{73.8} = 0.761$$
Answer: Sk ≈ 0.761
Interpretation: Strong positive skewness; the right tail (higher values) extends significantly, especially the extreme value of 100. Distribution is considerably right-skewed.
Example 3: Grouped Data (Discrete) - Income Distribution
Problem: Income distribution for 100 employees (in $1000s). Find Kelly’s coefficient of skewness.
| Income | 20 | 30 | 40 | 50 | 60 | 70 | 80 |
|---|---|---|---|---|---|---|---|
| Frequency | 5 | 10 | 15 | 30 | 20 | 12 | 8 |
Solution:
Step 1: Calculate cumulative frequencies
| Income | Frequency | Cumulative |
|---|---|---|
| 20 | 5 | 5 |
| 30 | 10 | 15 |
| 40 | 15 | 30 |
| 50 | 30 | 60 |
| 60 | 20 | 80 |
| 70 | 12 | 92 |
| 80 | 8 | 100 |
Step 2: Find D₁ (10th percentile)
Position = (1×100)/10 = 10
- Cumulative ≥ 10 is 15 → D₁ = 30
Step 3: Find D₅ (Median)
Position = (5×100)/10 = 50
- Cumulative ≥ 50 is 60 → D₅ = 50
Step 4: Find D₉ (90th percentile)
Position = (9×100)/10 = 90
- Cumulative ≥ 90 is 92 → D₉ = 70
Step 5: Calculate Kelly’s coefficient
$$S_k = \frac{70 + 30 - 2(50)}{70 - 30} = \frac{100 - 100}{40} = 0$$
Answer: Sk = 0
Interpretation: Despite the distribution shape, Kelly’s coefficient indicates symmetric tail behavior around the median. The 90% central range is symmetric.
Example 4: Grouped Data (Continuous) - Age Distribution
Problem: Age distribution of 80 customers. Find Kelly’s coefficient of skewness.
| Age Group | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 |
|---|---|---|---|---|---|
| Frequency | 5 | 15 | 30 | 20 | 10 |
Solution:
Step 1: Create cumulative frequency table
| Class | Boundaries | Frequency | Cumulative |
|---|---|---|---|
| 20-30 | 19.5-29.5 | 5 | 5 |
| 30-40 | 29.5-39.5 | 15 | 20 |
| 40-50 | 39.5-49.5 | 30 | 50 |
| 50-60 | 49.5-59.5 | 20 | 70 |
| 60-70 | 59.5-69.5 | 10 | 80 |
Step 2: Find D₁
Position = (1×80)/10 = 8
- Cumulative ≥ 8 is 20 (class 30-40)
- l = 29.5, F< = 5, f = 15, h = 10
$$D_1 = 29.5 + \left(\frac{8-5}{15}\right) \times 10 = 29.5 + 2 = 31.5$$
Step 3: Find D₅ (Median)
Position = (5×80)/10 = 40
- Cumulative ≥ 40 is 50 (class 40-50)
- l = 39.5, F< = 20, f = 30, h = 10
$$D_5 = 39.5 + \left(\frac{40-20}{30}\right) \times 10 = 39.5 + 6.67 = 46.17$$
Step 4: Find D₉
Position = (9×80)/10 = 72
- Cumulative ≥ 72 is 80 (class 60-70)
- l = 59.5, F< = 70, f = 10, h = 10
$$D_9 = 59.5 + \left(\frac{72-70}{10}\right) \times 10 = 59.5 + 2 = 61.5$$
Step 5: Calculate Kelly’s coefficient
$$S_k = \frac{61.5 + 31.5 - 2(46.17)}{61.5 - 31.5} = \frac{93 - 92.34}{30} = \frac{0.66}{30} = 0.022$$
Answer: Sk ≈ 0.022
Interpretation: Distribution is nearly perfectly symmetric; Kelly’s coefficient shows almost no skewness in the tail regions.
Comparison: Kelly’s vs Other Skewness Measures
| Measure | Formula Basis | Range | Focus | When to Use |
|---|---|---|---|---|
| Kelly’s | Deciles (D₁, D₅, D₉) | -1 to +1 | Outer 80% | Tail behavior |
| Bowley’s | Quartiles (Q₁, Q₂, Q₃) | -1 to +1 | Middle 50% | Central tendency |
| Pearson’s | Mean, Median, SD | -3 to +3 | All data | Standard measure |
| Moment | All deviations | -2 to +2 | Complete | Detailed analysis |
Key Differences: Ungrouped vs. Grouped Data
| Aspect | Ungrouped Data | Grouped Data |
|---|---|---|
| Data Format | Individual raw values | Classes with frequencies |
| Decile Calculation | Position-based interpolation | Class-based formula |
| Outlier Effect | Directly included | Smoothed by grouping |
| Extreme Values | Can dominate deciles | Less impact |
| Accuracy | Exact from raw data | Approximate from classes |
Common Mistakes to Avoid
❌ WRONG: Confusing Kelly’s with Bowley’s coefficient ✓ RIGHT: Kelly’s uses D₁, D₅, D₉; Bowley’s uses Q₁, Q₂, Q₃
❌ WRONG: Forgetting to sort data before finding deciles (ungrouped) ✓ RIGHT: Always sort in ascending order first
❌ WRONG: Using Kelly’s when Bowley’s is more appropriate (and vice versa) ✓ RIGHT: Choose based on focus: tails (Kelly’s) vs. center (Bowley’s)
❌ WRONG: Assuming Sk = 0 means no variation ✓ RIGHT: Sk = 0 means symmetric tail distribution; variation still exists
Visual Interpretation
Symmetric Distribution (Sk ≈ 0)
- D₉ - D₅ ≈ D₅ - D₁
- Outer 80% is evenly distributed
Right-Skewed (Sk > 0)
- D₉ - D₅ > D₅ - D₁
- Right tail (90th percentile region) extends further
Left-Skewed (Sk < 0)
- D₅ - D₁ > D₉ - D₅
- Left tail (10th percentile region) extends further
Related Calculators & Tutorials
Explore other skewness measures:
Related percentile measures:
Learn More: