Unit III Β· Lecture 21 of 40 Β· B.Sc. Data Science, 1st Year, Semester II
Skewness is the measure of the asymmetry of a probability distribution or dataset about its mean. It tells us whether the data is balanced equally on both sides of the centre or leaning more to one side.
Imagine a seesaw balanced at the mean. In a symmetric distribution, the seesaw is perfectly level β equal weight on both sides. In a right-skewed distribution, the right tail is longer (heavier), pulling the seesaw down on the right β like a long tail dragging behind. In a left-skewed distribution, the left tail is longer.
1. Symmetric (Skewness = 0):
Mean = Median = Mode. Left and right tails are mirror images.
2. Positively Skewed (Right-skewed):
Long tail on the right. Mean > Median > Mode.
3. Negatively Skewed (Left-skewed):
Long tail on the left. Mean < Median < Mode.
Right-skewed: Income distribution (most people earn moderate,ε°ζ° earn very high), house prices, insurance claims.
Left-skewed: Exam scores in an easy test (most score high, few score very low), age at retirement.
Symmetric: Heights of adults, IQ scores, measurement errors.
Click each card to see the shape, the relationship between mean, median, and mode, and a real-world example:
Skewness = 0
Skewness > 0
Skewness < 0
Karl Pearson's coefficient of skewness (SK) is based on the difference between the mean and the mode, expressed in terms of the standard deviation. It measures the extent and direction of departure from symmetry.
SK = (Mean β Mode) / Ο
Where:
β’ Mean = arithmetic average (xΜ)
β’ Mode = most frequent value (Mo)
β’ Ο = standard deviation
Range: β3 β€ SK β€ +3 (theoretically, though values beyond Β±1 are rare in practice)
SK = 3(Mean β Median) / Ο
This version uses the empirical relationship: Mean β Mode β 3(Mean β Median).
It is used when the mode is ill-defined (e.g., bimodal data or data with no clear peak).
For a dataset: Mean = 50, Mode = 45, Standard Deviation = 10. Find Karl Pearson's coefficient of skewness.
Given: Mean (xΜ) = 50, Mode (Mo) = 45, Ο = 10
Formula: SK = (Mean β Mode) / Ο
Calculation:
SK = (50 β 45) / 10 = 5 / 10 = +0.5
The distribution is moderately positively skewed. The mean is greater than the mode, indicating a longer right tail. Most values cluster on the left, with some high values pulling the mean to the right.
The marks of 100 students have: Mean = 65, Median = 68, SD = 12. Find the coefficient of skewness and interpret.
Given: Mean = 65, Median = 68, Ο = 12
Formula: SK = 3(Mean β Median) / Ο
Calculation:
SK = 3(65 β 68) / 12 = 3(β3) / 12 = β9 / 12 = β0.75
The distribution is moderately negatively skewed (left-skewed). The mean (65) is less than the median (68), indicating a longer left tail. Most students scored above average, but a few very low scores pulled the mean down.
Bowley's coefficient of skewness (SQ) is based on quartiles. It measures skewness by comparing the distances of the first and third quartiles from the median. It is a positional measure β it does not use the mean or standard deviation.
SQ = (Qβ + Qβ β 2Qβ) / (Qβ β Qβ)
Where:
β’ Qβ = First quartile (25th percentile)
β’ Qβ = Second quartile = Median (50th percentile)
β’ Qβ = Third quartile (75th percentile)
Range: β1 β€ SQ β€ +1
Alternative form:
SQ = [(Qβ β Qβ) β (Qβ β Qβ)] / (Qβ β Qβ)
In a symmetric distribution, Qβ is exactly halfway between Qβ and Qβ, so (Qβ β Qβ) = (Qβ β Qβ), and SQ = 0. If Qβ is farther from Qβ than Qβ is, the right half is "stretched" β positive skew. If Qβ is farther, the left half is stretched β negative skew.
For a distribution: Qβ = 30, Qβ (Median) = 40, Qβ = 65. Find Bowley's coefficient of skewness.
Given: Qβ = 30, Qβ = 40, Qβ = 65
Formula: SQ = (Qβ + Qβ β 2Qβ) / (Qβ β Qβ)
Calculation:
SQ = (65 + 30 β 2Γ40) / (65 β 30)
SQ = (95 β 80) / 35
SQ = 15 / 35 = +0.4286
Verification using alternative form:
Qβ β Qβ = 65 β 40 = 25
Qβ β Qβ = 40 β 30 = 10
SQ = (25 β 10) / (25 + 10) = 15 / 35 = +0.4286 β
The distribution is moderately positively skewed. The right half (Qβ to Qβ = 25 units) is wider than the left half (Qβ to Qβ = 10 units), indicating a longer right tail.
Kelly's coefficient of skewness (SKel) uses percentiles (specifically the 10th and 90th percentiles) instead of quartiles. It is a more extreme positional measure that captures skewness in the tails.
SKel = (Pββ + Pββ β 2Median) / (Pββ β Pββ)
Where:
β’ Pββ = 10th percentile (Dβ or Decile 1)
β’ Pββ = 90th percentile (Dβ or Decile 9)
β’ Median = 50th percentile
Range: β1 β€ SKel β€ +1
Kelly's measure is similar to Bowley's but uses deciles instead of quartiles, making it sensitive to a wider portion of the distribution.
The moment-based coefficient of skewness uses the third central moment (ΞΌβ) relative to the second central moment (ΞΌβ). It is the most mathematically rigorous measure of skewness and is widely used in theoretical statistics.
Beta coefficient:
Ξ²β = ΞΌβΒ² / ΞΌβΒ³
Gamma coefficient (signed skewness):
Ξ³β = βΞ²β = ΞΌβ / ΞΌβ^(3/2)
β’ Ξ²β is always β₯ 0 (squared term)
β’ Ξ³β retains the sign of ΞΌβ
β’ Ξ³β = 0 β Symmetric
β’ Ξ³β > 0 β Right-skewed
β’ Ξ³β < 0 β Left-skewed
The third power preserves the sign of deviations:
β’ Positive deviations (xα΅’ > xΜ) contribute positive cubes.
β’ Negative deviations (xα΅’ < xΜ) contribute negative cubes.
In a symmetric distribution, these cancel out β ΞΌβ = 0.
If the right tail is longer β more large positive cubes β ΞΌβ > 0.
If the left tail is longer β more large negative cubes β ΞΌβ < 0.
Dividing by ΞΌβ^(3/2) standardises the measure, making it unit-free.
For a distribution, the central moments are: ΞΌβ = 16, ΞΌβ = β48. Find Ξ²β and Ξ³β and interpret.
Given: ΞΌβ = 16, ΞΌβ = β48
Ξ²β = ΞΌβΒ² / ΞΌβΒ³
Ξ²β = (β48)Β² / (16)Β³ = 2304 / 4096 = 0.5625
Ξ³β = ΞΌβ / ΞΌβ^(3/2)
Ξ³β = β48 / (16)^(3/2) = β48 / 64 = β0.75
Ξ²β = 0.5625 | Ξ³β = β0.75
Ξ³β is negative, indicating the distribution is negatively skewed (left-skewed). The left tail is longer than the right tail. The magnitude |Ξ³β| = 0.75 indicates moderate skewness.
For a frequency distribution, the first four raw moments about the origin are: ΞΌβ' = 5, ΞΌβ' = 30, ΞΌβ' = 200, ΞΌβ' = 1500. Find all measures of skewness.
ΞΌβ = ΞΌβ' β (ΞΌβ')Β² = 30 β 25 = 5
ΞΌβ = ΞΌβ' β 3ΞΌβ'ΞΌβ' + 2(ΞΌβ')Β³ = 200 β 3(30)(5) + 2(125) = 200 β 450 + 250 = 0
ΞΌβ = ΞΌβ' β 4ΞΌβ'ΞΌβ' + 6ΞΌβ'(ΞΌβ')Β² β 3(ΞΌβ')β΄ = 1500 β 4(200)(5) + 6(30)(25) β 3(625)
= 1500 β 4000 + 4500 β 1875 = 125
Moment-based:
Ξ²β = ΞΌβΒ² / ΞΌβΒ³ = 0 / 125 = 0
Ξ³β = 0 / 5^(3/2) = 0
Since Ξ³β = 0, the distribution is perfectly symmetric.
Karl Pearson's (using empirical relation):
For symmetric distribution: Mean = Median = Mode = 5
SK = (Mean β Mode) / Ο = (5 β 5) / β5 = 0
Bowley's: For a symmetric distribution, Qβ is equidistant from Qβ and Qβ, so SQ = 0.
All three methods confirm the distribution is symmetric. This is expected since ΞΌβ = 0.
Enter your data to compute Karl Pearson's, Bowley's, and moment-based skewness:
| Feature | Karl Pearson's SK | Bowley's SQ | Moment-based Ξ³β |
|---|---|---|---|
| Based on | Mean, Mode/Median, SD | Qβ, Qβ, Qβ | ΞΌβ, ΞΌβ |
| Range | β3 to +3 | β1 to +1 | Theoretically unbounded |
| Uses all data? | Yes (via mean, SD) | No (only quartiles) | Yes (via moments) |
| Affected by outliers? | Yes (mean & SD are) | Less affected | Yes (ΞΌβ cubes deviations) |
| Open-end classes? | Cannot use (need mean, SD) | Can use (only quartiles) | Cannot use |
| Ease of computation | Easy | Easiest | Moderate |
| Mathematical rigour | Moderate | Low | High |
| Best for | Quick assessment with known mean & SD | Data given as quartiles or open-end | Theoretical analysis, advanced statistics |
Q1: A distribution with a longer right tail is called:
Q2: Karl Pearson's coefficient of skewness using the mode is:
Q3: If Mean = 40, Median = 45, Ο = 10, Karl Pearson's SK (using median) is:
Q4: Bowley's coefficient uses which statistical measures?
Q5: For a symmetric distribution, which of the following is true?
Q6: The moment-based skewness Ξ³β = ΞΌβ / ΞΌβ^(3/2) is negative when:
Q7: Which skewness measure is most robust to outliers?
Q8: If Mean = 60, Mode = 50, the estimated Median (using empirical relation) is:
1. Find Karl Pearson's coefficient of skewness if Mean = 100, Mode = 80, SD = 20. Interpret the result.
2. Given Qβ = 25, Median = 35, Qβ = 50. Find Bowley's coefficient of skewness.
3. The central moments of a distribution are ΞΌβ = 25, ΞΌβ = 75. Find Ξ²β and Ξ³β. Is the distribution skewed? In which direction?
4. For a grouped frequency distribution, the raw moments are: ΞΌβ' = 8, ΞΌβ' = 80, ΞΌβ' = 720. Find the central moments and the moment-based skewness.
5. The marks of students have Mean = 55, Median = 58, Mode = 62, SD = 15. Calculate Karl Pearson's SK using both formulas. Are the results consistent?
6. Explain why Bowley's coefficient is preferred over Karl Pearson's when data has open-end classes.
7. If every value in a dataset is multiplied by 3 and then 10 is added, what happens to the skewness? Verify mathematically.
8. BONUS: For a log-normal distribution, the mean is always greater than the median. What does this tell you about the skewness? Calculate Karl Pearson's SK for a log-normal distribution with mean = 100 and median = 90, SD = 30.
Questions? Let's discuss!
Lecture 22: Measures of Kurtosis
Lecture 23: Lorenz Curve & Measures of Inequality
Lecture 24: Review & Problem-Solving Session
Unit III Β· Lecture 21 Β· BDS113 Introduction to Statistics Β· B.Sc. Data Science, Semester II