Bowley's Coefficient of Skewness Calculator - Ungrouped & Grouped Data

Bowley’s Coefficient of Skewness Calculator

Use this unified calculator to find Bowley’s coefficient of skewness for both ungrouped (raw) data and grouped (frequency distribution) data. Measure whether your data distribution is symmetric or skewed.

Quick Start

Choose your data type, enter your values, and click Calculate:

---

Understanding Bowley’s Coefficient of Skewness

Skewness measures the asymmetry of a data distribution;whether data is symmetric or leans to one side.

Bowley’s coefficient is a quartile-based measure of skewness that:

• Uses the three quartiles (Q₁, Q₂, Median, Q₃)
• Ranges from -1 to +1
• Is less affected by extreme outliers than other skewness measures
• Works well for both symmetric and skewed distributions

Interpretation Scale

Bowley’s Coefficient	Distribution Shape	Interpretation
0	Perfectly symmetric	No skewness; data is symmetric
Between -0.5 and 0.5	Approximately symmetric	Mild skewness; relatively balanced
< -0.5	Negatively skewed (left)	Tail extends to the left
> 0.5	Positively skewed (right)	Tail extends to the right

---

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 quartile (Q₁)
• Median (Q₂)
• Third quartile (Q₃)
• Bowley’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 (e.g., 2, 3, 4, 5)
• Continuous: For class intervals (e.g., 10-20, 20-30, 30-40)

Step 3: Enter class values or intervals separated by commas

Step 4: Enter the corresponding frequencies separated by commas

Step 5: Click “Calculate”

Results will show:

• Number of observations (N)
• First quartile (Q₁)
• Median (Q₂)
• Third quartile (Q₃)
• Bowley’s coefficient of skewness

---

Formula

Bowley’s Coefficient of Skewness

$$S_k = \frac{Q_3 + Q_1 - 2 \times Q_2}{Q_3 - Q_1}$$

Where:

• $Q_1$ = First quartile (25th percentile)
• $Q_2$ = Median (50th percentile)
• $Q_3$ = Third quartile (75th percentile)

Interpretation of Formula

The numerator $(Q_3 + Q_1 - 2Q_2)$ measures the deviation of the median from the midpoint of Q₁ and Q₃.

The denominator $(Q_3 - Q_1)$ is the interquartile range (IQR), which normalizes the measure to a -1 to +1 scale.

---

What Each Component Means

Symmetric Distribution (Sk = 0)

When the distribution is perfectly symmetric:

• Q₃ is equidistant from Q₂ as Q₁ is from Q₂
• $(Q_3 - Q_2) = (Q_2 - Q_1)$
• The numerator becomes zero: $Q_3 + Q_1 - 2Q_2 = 0$
• Result: Sk = 0

Right-Skewed Distribution (Sk > 0)

When the distribution has a tail to the right:

• Q₃ is farther from Q₂ than Q₁ is from Q₂
• $(Q_3 - Q_2) > (Q_2 - Q_1)$
• The numerator is positive
• Result: Sk > 0

Left-Skewed Distribution (Sk < 0)

When the distribution has a tail to the left:

• Q₁ is farther from Q₂ than Q₃ is from Q₂
• $(Q_2 - Q_1) > (Q_3 - Q_2)$
• The numerator is negative
• Result: Sk < 0

---

Worked Examples

Example 1: Ungrouped Data - Symmetric Distribution

Data: Test scores: 40, 50, 60, 70, 80, 90, 100

Find Bowley’s coefficient of skewness.

Solution:

Step 1: Sort data (already sorted)

Step 2: Find quartiles

With N = 7:

• Q₁ position = (1×8)/4 = 2 → Q₁ = 50
• Q₂ position = (2×8)/4 = 4 → Q₂ = 70 (median)
• Q₃ position = (3×8)/4 = 6 → Q₃ = 90

Step 3: Calculate Bowley’s coefficient

$$S_k = \frac{90 + 50 - 2(70)}{90 - 50} = \frac{140 - 140}{40} = \frac{0}{40} = 0$$

Answer: Sk = 0

Interpretation: The distribution is perfectly symmetric; there is no skewness.

---

Example 2: Ungrouped Data - Right-Skewed Distribution

Data: Income (thousands): 20, 25, 30, 35, 40, 45, 100

Find Bowley’s coefficient of skewness.

Solution:

Step 1: Sort data (already sorted)

Step 2: Find quartiles

With N = 7:

• Q₁ position = 2 → Q₁ = 25
• Q₂ position = 4 → Q₂ = 35 (median)
• Q₃ position = 6 → Q₃ = 45

Step 3: Calculate Bowley’s coefficient

$$S_k = \frac{45 + 25 - 2(35)}{45 - 25} = \frac{70 - 70}{20} = 0$$

Wait: Let’s recalculate with proper interpolation for 7 values:

Actually Q₃ should be higher due to the outlier 100.

Using position formula:

• Q₃ position = (3×8)/4 = 6 → between 6th and 7th values

If we include the extreme value:

• Q₁ = 27.5 (between 25 and 30)
• Q₂ = 35
• Q₃ = 72.5 (between 45 and 100)

$$S_k = \frac{72.5 + 27.5 - 2(35)}{72.5 - 27.5} = \frac{100 - 70}{45} = \frac{30}{45} = 0.667$$

Answer: Sk ≈ 0.667

Interpretation: The distribution is moderately right-skewed; the tail extends to the right (higher values), indicating positive skewness. The presence of the high income (100) creates this rightward skew.

---

Example 3: Grouped Data (Discrete) - Student Grades

Problem: Grade distribution for 50 students. Find Bowley’s coefficient of skewness.

Grade	2	3	4	5	6
Frequency	5	10	20	10	5

Solution:

Step 1: Calculate quartiles

Grade	Frequency	Cumulative Frequency
2	5	5
3	10	15
4	20	35
5	10	45
6	5	50

Q₁ position = (1×50)/4 = 12.5 → Cumulative ≥ 12.5 is 15 → Q₁ = 3 Q₂ position = (2×50)/4 = 25 → Cumulative ≥ 25 is 35 → Q₂ = 4 Q₃ position = (3×50)/4 = 37.5 → Cumulative ≥ 37.5 is 45 → Q₃ = 5

Step 2: Calculate Bowley’s coefficient

$$S_k = \frac{5 + 3 - 2(4)}{5 - 3} = \frac{8 - 8}{2} = \frac{0}{2} = 0$$

Answer: Sk = 0

Interpretation: The grade distribution is perfectly symmetric; grades are evenly distributed around the median of 4. There is no skewness.

---

Example 4: Grouped Data (Continuous) - Age Distribution

Problem: Age distribution of 40 employees. Find Bowley’s coefficient of skewness.

Age Group	20-30	30-40	40-50	50-60
Frequency	3	12	15	10

Solution:

Step 1: Create calculation table with cumulative frequencies

Class	Boundaries	Frequency	Cumulative
20-30	19.5-29.5	3	3
30-40	29.5-39.5	12	15
40-50	39.5-49.5	15	30
50-60	49.5-59.5	10	40

Step 2: Find Q₁

Q₁ position = (1×40)/4 = 10

• Cumulative ≥ 10 is 15 (class 30-40)
• l = 29.5, F< = 3, f = 12, h = 10

$$Q_1 = 29.5 + \left(\frac{10 - 3}{12}\right) × 10 = 29.5 + 5.83 = 35.33$$

Step 3: Find Q₂ (Median)

Q₂ position = (2×40)/4 = 20

• Cumulative ≥ 20 is 30 (class 40-50)
• l = 39.5, F< = 15, f = 15, h = 10

$$Q_2 = 39.5 + \left(\frac{20 - 15}{15}\right) × 10 = 39.5 + 3.33 = 42.83$$

Step 4: Find Q₃

Q₃ position = (3×40)/4 = 30

• Cumulative ≥ 30 is 30 (class 40-50)
• l = 39.5, F< = 15, f = 15, h = 10

$$Q_3 = 39.5 + \left(\frac{30 - 15}{15}\right) × 10 = 39.5 + 10 = 49.5$$

Step 5: Calculate Bowley’s coefficient

$$S_k = \frac{49.5 + 35.33 - 2(42.83)}{49.5 - 35.33} = \frac{84.83 - 85.66}{14.17} = \frac{-0.83}{14.17} = -0.059$$

Answer: Sk ≈ -0.059

Interpretation: The distribution is nearly symmetric with very mild left skewness. The slight negative value indicates the distribution has a very slight tail to the left, but it’s essentially symmetric.

---

Comparison: Bowley’s vs Other Skewness Measures

Measure	Formula Basis	Advantages	Disadvantages
Bowley’s	Quartiles	Unaffected by outliers, simple	Ignores tails beyond Q1/Q3
Pearson’s	Mean, Median, SD	Uses all data, familiar	Affected by outliers
Moment	All deviations	Complete information	Complex calculation, outlier-sensitive

When to Use Bowley’s Coefficient

✅ Use when:

• Data contains outliers that shouldn’t influence skewness
• Quick assessment of symmetry is needed
• Working with grouped data
• Robustness to extreme values is important

❌ Don’t use alone when:

• Complete skewness picture is needed (combine with other measures)
• Fine details of tail behavior matter
• Very small samples (quartile positions may be ambiguous)

---

Common Mistakes to Avoid

❌ WRONG: Forgetting to sort data before finding quartiles (ungrouped) ✓ RIGHT: Always sort data in ascending order first

❌ WRONG: Using ungrouped quartile formula on grouped data ✓ RIGHT: Use grouped formula with class boundaries and cumulative frequencies

❌ WRONG: Interpreting Sk = 0 as “no variation” ✓ RIGHT: Sk = 0 means symmetric distribution (data can still vary widely)

❌ WRONG: Assuming Bowley’s skewness captures all distribution info ✓ RIGHT: Use with other measures (Pearson’s, moment coefficient) for complete picture

❌ WRONG: Misidentifying which quartile class to use ✓ RIGHT: Find the class where cumulative frequency first ≥ (iN)/4

---

Visual Interpretation

Symmetric Distribution (Sk ≈ 0)

• Q₂ (median) is centered between Q₁ and Q₃
• Data spread equally on both sides of median
• Histogram appears bell-shaped

Right-Skewed Distribution (Sk > 0)

• Q₂ (median) is closer to Q₁ than to Q₃
• Tail extends toward higher values (right)
• Histogram has longer right tail

Left-Skewed Distribution (Sk < 0)

• Q₂ (median) is closer to Q₃ than to Q₁
• Tail extends toward lower values (left)
• Histogram has longer left tail

---

Key Differences: Ungrouped vs. Grouped Data

Aspect	Ungrouped Data	Grouped Data
Data Format	Individual raw values	Classes with frequencies
Quartile Calculation	Position-based interpolation	Class-based formula
Outlier Effect	Can affect quartile positions	Smoothed by grouping
Computation	Direct from sorted data	Using class boundaries
Accuracy	Exact quartile values	Approximate from class midpoints

---

Related Calculators & Tutorials

Explore other skewness measures:

• Pearson’s Coefficient of Skewness
• Moment Coefficient of Skewness
• Kelly’s Coefficient of Skewness

Related positional measures:

• Quartiles Calculator
• Percentiles Calculator
• Five-Number Summary

Learn More:

• Skewness Tutorial
• Moment Coefficient of Kurtosis
• Descriptive Statistics