1. Introduction to Skewness
Skewness measures the asymmetry of a data distribution around its mean. In an ideal normal distribution, the data is symmetrically distributed, meaning the mean, median, and mode are equal. However, in real-world scenarios, data is often skewed due to the presence of outliers, extreme values, or natural data trends.
Skewness helps in business analytics, finance, and quality control by indicating whether data is positively or negatively skewed, which affects forecasting, risk assessment, and decision-making.
2. Types of Skewness
There are three types of skewness:
(i) Symmetrical Distribution (Zero Skewness)
A distribution is symmetrical if it has no skewness.
- The mean, median, and mode are equal.
- The data is evenly distributed on both sides.
- The skewness value is zero.
- Example: Bell-shaped normal distribution (e.g., heights of people in a population).
- Mean, median, and mode are approximately equal, showing a balanced dataset.
(ii) Positive Skewness (Right Skewed Distribution)
A distribution is positively skewed if:
- The tail extends towards the right (higher values).
- Mean > Median > Mode.
- Skewness value is greater than zero.
- Example: Income distribution in a country, where most people earn lower wages but a few high earners pull the mean upwards.
- The tail extends to the right, and the mean is greater than the median.
(iii) Negative Skewness (Left Skewed Distribution)
A distribution is negatively skewed if:
- The tail extends towards the left (lower values).
- Mean < Median < Mode.
- Skewness value is less than zero.
- Example: Age at retirement, where most people retire at a certain age, but a few retire much earlier.
- The tail extends to the left, and the mean is less than the median.
3. Formula for Skewness
Skewness is mathematically calculated using different methods, but the most common formula is Pearson’s Coefficient of Skewness:Skewness=3(Xˉ−Median)s\text{Skewness} = \frac{3(\bar{X} – \text{Median})}{s}Skewness=s3(Xˉ−Median)
Where:
- Xˉ\bar{X}Xˉ = Mean
- Median = Middle value of the dataset
- sss = Standard Deviation
Another formula for sample skewness is:Skewness=∑(Xi−Xˉ)3ns3\text{Skewness} = \frac{\sum (X_i – \bar{X})^3}{n s^3}Skewness=ns3∑(Xi−Xˉ)3
Where:
- XiX_iXi = Individual data points
- sss = Standard deviation
- nnn = Total number of observations
4. Example Calculation
A company tracks daily sales (₹ ‘000s) for 7 days:
| Day | Sales (X) |
|---|---|
| Mon | 20 |
| Tue | 22 |
| Wed | 25 |
| Thu | 27 |
| Fri | 30 |
| Sat | 50 |
| Sun | 70 |
Step 1: Calculate the Mean
Xˉ=20+22+25+27+30+50+707=2447=34.86\bar{X} = \frac{20 + 22 + 25 + 27 + 30 + 50 + 70}{7} = \frac{244}{7} = 34.86Xˉ=720+22+25+27+30+50+70=7244=34.86
Step 2: Find the Median
- Ordered dataset: 20, 22, 25, 27, 30, 50, 70
- Median (middle value) = 27
Step 3: Calculate Standard Deviation (s)
Using the standard deviation formula:s=∑(Xi−Xˉ)2ns = \sqrt{\frac{\sum (X_i – \bar{X})^2}{n}}s=n∑(Xi−Xˉ)2
The calculated standard deviation is 17.52.
Step 4: Compute Skewness Using Pearson’s Formula
Skewness=3(34.86−27)17.52=3(7.86)17.52=23.5817.52=1.35\text{Skewness} = \frac{3(34.86 – 27)}{17.52} = \frac{3(7.86)}{17.52} = \frac{23.58}{17.52} = 1.35Skewness=17.523(34.86−27)=17.523(7.86)=17.5223.58=1.35
Since the skewness value is positive (1.35), this dataset is positively skewed (right-skewed).
5. Interpretation of Skewness Values
| Skewness Value | Interpretation |
|---|---|
| 0 | Symmetrical (Normal Distribution) |
| Between -0.5 and 0.5 | Approximately symmetrical |
| Between -1 and -0.5 | Moderately negatively skewed |
| Less than -1 | Highly negatively skewed |
| Between 0.5 and 1 | Moderately positively skewed |
| Greater than 1 | Highly positively skewed |
A higher skewness value (>1 or <-1) indicates a highly asymmetrical distribution, whereas a skewness value near zero suggests a normal distribution.
6. Applications of Skewness in Business Statistics
(i) Financial Market Analysis
- Stock returns often exhibit skewness.
- Positive skewness means more small losses and rare large gains.
- Negative skewness indicates frequent small gains but rare large losses.
(ii) Sales Forecasting and Demand Analysis
- Businesses analyze sales skewness to identify patterns in customer purchases.
- Right skewness may indicate few very high sales days (e.g., festival season spikes).
(iii) Income Distribution
- Salary distributions are typically positively skewed, where most employees earn an average salary, but a few earn exceptionally high amounts.
(iv) Customer Behavior in Marketing
- Purchase frequency and spending habits show skewness, helping marketers segment customers into high-spending and low-spending groups.
(v) Quality Control in Manufacturing
- A skewed distribution in product weights or defect rates can indicate an issue in manufacturing processes.
7. Advantages of Skewness Analysis
✔ Identifies Data Asymmetry – Helps detect extreme values that impact decision-making.
✔ Aids in Risk Management – Used in finance and investments to assess risk.
✔ Improves Forecasting – Helps in sales predictions, inventory planning, and business strategy.
✔ Enhances Market Research – Assists in customer segmentation and pricing strategies.
8. Limitations of Skewness Analysis
❌ Does Not Measure Spread – Skewness only shows direction of asymmetry, not how much data varies.
❌ Sensitive to Outliers – One extreme value can distort skewness calculations.
❌ Difficult Interpretation for Small Data – Skewness calculations may not be meaningful with limited data points.
9. Conclusion
Skewness is an essential measure of data distribution asymmetry, helping businesses and analysts understand whether data is right-skewed, left-skewed, or symmetrical. It is widely applied in finance, marketing, risk assessment, and quality control to detect patterns and make data-driven decisions.