Moment Coefficient of Skewness Calculator
Use this unified calculator to find the moment coefficient of skewness for both ungrouped (raw) data and grouped (frequency distribution) data. This is the most comprehensive skewness measure, using central moments.
Quick Start
Choose your data type, enter your values, and click Calculate:
| Moment 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): | |
| Mean of X values: | |
| First Central Moment (μ₁): | |
| Second Central Moment (μ₂): | |
| Third Central Moment (μ₃): | |
| Fourth Central Moment (μ₄): | |
| Coefficient of Skewness (β₁): | |
| Coefficient of Skewness (γ₁): | |
Understanding Moment Coefficient of Skewness
The moment coefficient of skewness is the most theoretically rigorous skewness measure because:
- ✅ Uses all data points and their deviations (complete information)
- ✅ Based on central moments (mathematical foundation)
- ✅ Provides two related measures (β₁ and γ₁)
- ✅ Most sensitive to distribution shape
- ✅ Used in advanced statistical theory
Two Forms of the Coefficient
This calculator provides both:
- β₁ (Beta): The second form (algebraic) - always ≥ 0
- γ₁ (Gamma): The standardized form (practical) - can be negative or positive
Formulas
Central Moments
$$\mu_r = \frac{1}{N}\sum_{i=1}^{N}(x_i - \overline{x})^r$$
Where:
- $\mu_1$ = First central moment (always = 0)
- $\mu_2$ = Second central moment (= variance)
- $\mu_3$ = Third central moment (measures skewness)
- $\mu_4$ = Fourth central moment (relates to kurtosis)
Moment Coefficient of Skewness (β₁ Form)
$$\beta_1 = \frac{\mu_3^2}{\mu_2^3}$$
Properties:
- Always ≥ 0
- β₁ = 0 for symmetric distribution
- β₁ > 0 for any skewed distribution
- Doesn’t distinguish left vs. right skew
Moment Coefficient of Skewness (γ₁ Form)
$$\gamma_1 = \sqrt{\beta_1} = \frac{\mu_3}{\mu_2^{3/2}}$$
Or equivalently (accounting for sign):
$$\gamma_1 = \frac{\mu_3}{(\mu_2)^{3/2}}$$
Properties:
- Can be negative or positive
- γ₁ = 0 for symmetric distribution
- γ₁ > 0 for right-skewed
- γ₁ < 0 for left-skewed
- Ranges approximately from -2 to +2
Interpretation
γ₁ Interpretation Scale
| γ₁ Value | Distribution | Interpretation |
|---|---|---|
| γ₁ < -1 | Highly left-skewed | Strong left tail |
| -1 ≤ γ₁ < -0.5 | Moderately left-skewed | Notable left tail |
| -0.5 ≤ γ₁ < 0 | Slightly left-skewed | Mild left asymmetry |
| γ₁ = 0 | Perfectly symmetric | No skewness |
| 0 < γ₁ ≤ 0.5 | Slightly right-skewed | Mild right asymmetry |
| 0.5 < γ₁ ≤ 1 | Moderately right-skewed | Notable right tail |
| γ₁ > 1 | Highly right-skewed | Strong right tail |
How to Use This Calculator
For Ungrouped (Raw) Data
Step 1: Select “Ungrouped (Raw Data)”
Step 2: Enter your data values separated by commas
Step 3: Click “Calculate”
Results will show:
- Mean and central moments (μ₁ through μ₄)
- Both skewness coefficients (β₁ and γ₁)
For Grouped (Frequency Distribution) Data
Step 1: Select “Grouped (Frequency Distribution)”
Step 2: Choose distribution type (Discrete or Continuous)
Step 3: Enter classes and frequencies
Step 4: Click “Calculate”
Results will show:
- All calculated values
Worked Examples
Example 1: Ungrouped Data - Symmetric Distribution
Data: Values: 2, 4, 6, 8, 10, 12, 14
Find the moment coefficient of skewness.
Solution:
Step 1: Calculate Mean
$$\overline{x} = \frac{2+4+6+8+10+12+14}{7} = 8$$
Step 2: Calculate Central Moments
Deviations: -6, -4, -2, 0, 2, 4, 6
μ₁ = 0 (always) μ₂ = (36+16+4+0+4+16+36)/7 = 112/7 = 16 μ₃ = (-216-64-8+0+8+64+216)/7 = 0/7 = 0 μ₄ = (1296+256+16+0+16+256+1296)/7 = 3136/7 = 448
Step 3: Calculate Skewness Coefficients
$$\beta_1 = \frac{0^2}{16^3} = 0$$
$$\gamma_1 = \frac{0}{16^{3/2}} = 0$$
Answer: β₁ = 0, γ₁ = 0
Interpretation: Perfect symmetry; both coefficients are zero.
Example 2: Ungrouped Data - Right-Skewed Distribution
Data: Values: 1, 2, 3, 4, 5, 15
Find the moment coefficient of skewness.
Solution:
Step 1: Calculate Mean
$$\overline{x} = \frac{1+2+3+4+5+15}{6} = 5$$
Step 2: Calculate Central Moments
Deviations: -4, -3, -2, -1, 0, 10
μ₁ = 0 μ₂ = (16+9+4+1+0+100)/6 = 130/6 = 21.67 μ₃ = (-64-27-8-1+0+1000)/6 = 900/6 = 150 μ₄ = (256+81+16+1+0+10000)/6 = 10354/6 = 1725.67
Step 3: Calculate Skewness Coefficients
$$\beta_1 = \frac{150^2}{21.67^3} = \frac{22500}{10154.8} = 2.215$$
$$\gamma_1 = \frac{150}{21.67^{1.5}} = \frac{150}{100.98} = 1.485$$
Answer: β₁ ≈ 2.215, γ₁ ≈ 1.485
Interpretation: Highly right-skewed distribution; extreme positive skewness due to the outlier value 15.
Example 3: Grouped Data (Discrete) - Test Scores
Problem: Score distribution for 40 students. Calculate moment coefficient of skewness.
| Score | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|
| Frequency | 2 | 5 | 18 | 12 | 3 |
Solution:
Step 1: Calculate Mean
$$\overline{x} = \frac{2(2)+3(5)+4(18)+5(12)+6(3)}{40} = \frac{167}{40} = 4.175$$
Step 2: Calculate Central Moments
Deviations: -2.175, -1.175, -0.175, 0.825, 1.825
μ₂ = [2(4.73)+5(1.38)+18(0.03)+12(0.68)+3(3.33)]/40 = 33.75/40 = 0.844
μ₃ = [2(-10.28)+5(-1.62)+18(-0.005)+12(0.56)+3(6.07)]/40 = 0.84/40 = 0.021
μ₄ = [2(22.37)+5(1.91)+18(0.0001)+12(0.46)+3(11.07)]/40 = 101.7/40 = 2.54
Step 3: Calculate Skewness Coefficients
$$\beta_1 = \frac{0.021^2}{0.844^3} = \frac{0.00044}{0.601} = 0.00073$$
$$\gamma_1 = \frac{0.021}{0.844^{1.5}} = \frac{0.021}{0.775} = 0.027$$
Answer: β₁ ≈ 0.00073, γ₁ ≈ 0.027
Interpretation: Nearly perfect symmetry; distribution is well-balanced with very slight positive skewness.
Example 4: Grouped Data (Continuous) - Age Distribution
Problem: Age distribution of 50 people. Calculate moment coefficient of skewness.
| Age Group | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 |
|---|---|---|---|---|---|
| Frequency | 3 | 8 | 18 | 14 | 7 |
Solution:
Step 1: Use Midpoints and Calculate Mean
| Midpoint | Frequency | f×x |
|---|---|---|
| 25 | 3 | 75 |
| 35 | 8 | 280 |
| 45 | 18 | 810 |
| 55 | 14 | 770 |
| 65 | 7 | 455 |
| Total | 50 | 2,390 |
$$\overline{x} = 2390/50 = 47.8$$
Step 2: Calculate Central Moments
Deviations from 47.8: -22.8, -12.8, -2.8, 7.2, 17.2
μ₂ = [3(519.8)+8(163.8)+18(7.84)+14(51.84)+7(295.8)]/50 = 4544.8/50 = 90.9
μ₃ = [3(-11851)+8(-2097)+18(-21.95)+14(373.25)+7(5088.45)]/50 = 1064/50 = 21.28
μ₄ = [3(270408)+8(26842)+18(61.47)+14(2689.0)+7(87520)]/50 = 677480/50 = 13549.6
Step 3: Calculate Skewness Coefficients
$$\beta_1 = \frac{21.28^2}{90.9^3} = \frac{452.8}{749968} = 0.000603$$
$$\gamma_1 = \frac{21.28}{90.9^{1.5}} = \frac{21.28}{866.4} = 0.0245$$
Answer: β₁ ≈ 0.000603, γ₁ ≈ 0.0245
Interpretation: Nearly symmetric distribution; very mild positive skewness slightly favoring higher ages.
Comparison: All Skewness Measures
| Measure | Formula | Range | Sensitivity | When to Use |
|---|---|---|---|---|
| Moment | μ₃/(σ³) | -2 to +2 | Very high | Complete analysis |
| Pearson’s | 3(Mean-Median)/SD | -3 to +3 | All data | Standard measure |
| Bowley’s | (Q₃+Q₁-2Q₂)/(Q₃-Q₁) | -1 to +1 | Middle 50% | Robust |
| Kelly’s | (D₉+D₁-2D₅)/(D₉-D₁) | -1 to +1 | Outer 80% | Tail focus |
When to Use Moment Coefficient
✅ Use when:
- Theoretical precision is needed
- Publishing advanced statistical analysis
- Complete moment information is needed
- Interested in relationship with kurtosis
❌ Avoid when:
- Simple, quick assessment needed (use Pearson’s)
- Robustness to outliers critical (use Bowley’s)
- Practical business decision required (use Pearson’s)
Understanding Central Moments
What Each Moment Tells You
| Moment | Meaning | Interpretation |
|---|---|---|
| μ₁ | First central moment | Always 0 (by definition) |
| μ₂ | Second central moment | Variance (spread) |
| μ₃ | Third central moment | Related to skewness |
| μ₄ | Fourth central moment | Related to kurtosis (peakedness) |
Common Mistakes to Avoid
❌ WRONG: Confusing β₁ and γ₁ ✓ RIGHT: β₁ is always ≥ 0; γ₁ can be negative
❌ WRONG: Using population formula for sample ✓ RIGHT: Use n-1 in variance calculation for samples
❌ WRONG: Comparing moment coefficient directly to Pearson’s ✓ RIGHT: Remember different ranges: moment (-2 to +2) vs Pearson’s (-3 to +3)
❌ WRONG: Ignoring that μ₁ is always 0 ✓ RIGHT: μ₁ = 0 by mathematical definition
Key Differences: Ungrouped vs. Grouped Data
| Aspect | Ungrouped | Grouped |
|---|---|---|
| Calculation | Direct from raw values | Using class midpoints |
| Precision | Exact moments | Approximate moments |
| Computational | More complex | Simplified with frequencies |
| Information | Complete | Some loss from grouping |
Related Calculators & Tutorials
Explore other skewness measures:
Related kurtosis measure:
Related statistics:
Learn More: