Skewness: Concept, Symmetric & Asymmetric Distributions

1. What Is Skewness?

The word Skewness means “lack of symmetry” or “asymmetry”.

While Measures of Central Tendency indicate the center, and Measures of Dispersion indicate the spread, Measures of Skewness indicate the direction and degree of tilt (asymmetry) of the distribution.

Prof. Simpson and Kafka’s Definition:
“Measures of skewness tell us the direction and the extent of asymmetry in a series, and indicate whether observations are clustered more on one side of the average than on the other.”

2. Distinction Between Dispersion and Skewness

Basis	Dispersion (Spread)	Skewness (Tilt / Asymmetry)
What it Measures	The magnitude or amount of variation around the average.	The direction and departure from a symmetrical shape.
Focus	How widely scattered the data is.	Whether data is stretched towards the right (high) or left (low).
Algebraic Sign	Dispersion values are always positive (cannot be negative).	Skewness can be zero, positive (`+`), or negative (`-`).
Graphic Visual	A wide, flat curve vs. a narrow, tall curve.	A balanced bell curve vs. a lopsided, one-tailed curve.

3. The 3 Fundamental Shapes of Distributions (Activity 2 Visuals)

The Three Shapes of Data Distributions

1. Symmetrical (Normal)

Mean = Median = Mode
Perfect mirror-image bell shape. Zero skewness (Sk = 0). Equidistant quartiles: (Q3 - M) = (M - Q1).

2. Positively Skewed (Right-Tailed)

Mean > Median > Mode
Tail stretches out to the right (higher values). Extreme high outliers pull the Mean to the right. (Q3 - M) > (M - Q1).

3. Negatively Skewed (Left-Tailed)

Mean < Median < Mode
Tail stretches out to the left (lower values). Extreme low outliers pull the Mean to the left. (Q3 - M) < (M - Q1).

In-Depth Breakdown of the 3 Shapes:

A. Perfectly Symmetrical Distribution

The curve has a central peak and drops symmetrically on both sides like a classic bell.
If folded down the vertical center line, the left half perfectly overlaps the right half.
Mathematical Condition:
Mean = Median = Mode
Quartile Symmetry: (Q3 - Median) = (Median - Q1)

Symmetrical Bell Curve:
            /\
           /  \
          /    \
        /        \
-------/----------\-------
       Mean=Median=Mode

B. Positively Skewed Distribution (Right-Skewed)

Frequencies rise rapidly on the left, reach a peak quickly, and then tail off slowly towards high values on the right.
Mathematical Condition:
Mean > Median > Mode
Quartile Condition: (Q3 - Median) > (Median - Q1)
Everyday Business Example:
Household Income in India! Most citizens earn modest incomes (clustered on the left, near the Mode), but a handful of billionaires pull the Arithmetic Mean far to the right.

Positively Skewed (Right-Tailed):
       /\
      /  \
     /    \
    /      \________
---/----------------\------
  Mode  Median  Mean   --> Tail extends to right

C. Negatively Skewed Distribution (Left-Skewed)

The tail of the curve stretches out towards the left (low values), while the bulk of observations cluster on the right.
Mathematical Condition:
Mean < Median < Mode
Quartile Condition: (Q3 - Median) < (Median - Q1)
Everyday Example:
Age at Retirement or Age at Death: In developed societies, most citizens retire or pass away between ages 60 and 85 (Mode on the right), with a long thin tail extending to rare early retirements on the left.

Negatively Skewed (Left-Tailed):
            /\
           /  \
   _______/    \
  /             \
-/---------------\--------
Mean  Median  Mode    <-- Tail extends to left

4. Tests of Skewness: How to Detect Asymmetry

A distribution is confirmed to be skewed (asymmetrical) if ANY of the following 4 conditions are observed:

The central measures are not identical: Mean ≠ Median ≠ Mode.
The quartiles are not equidistant from the median: (Q3 - Median) ≠ (Median - Q1).
The sum of positive deviations from median does not equal negative deviations: Σ(X - M) ≠ 0.
The frequencies on either side of the mode are unequal when plotted on a graph.

Skewness: Concept, Symmetric & Asymmetric Distributions | BBA Notes