What is Skewness? The Intuition

Imagine you're looking at a class's test scores. Most students score around 70%, a few score higher, and a few score lower. If you plot this data, you usually get a bell-shaped curve — symmetric, with the hump in the middle.

But real data is rarely that neat. Sometimes the hump is pushed to one side, with a long "tail" stretching out on the other side. That asymmetry — that lopsidedness — is what we call skewness.

Think of a seesaw. A symmetric distribution balances perfectly at its centre. A skewed distribution is like a seesaw with more weight on one side — the balance point (the mean) gets pulled toward the heavier side, while the tail stretches away on the lighter side.

The Two Flavours of Skewness

Positive skew (right-skewed): The tail is longer on the right. Most data is bunched up on the left, with a few unusually large values pulling the mean to the right. Example: household income — most people earn modest amounts, but a few billionaires create a long right tail.

Negative skew (left-skewed): The tail is longer on the left. Most data is on the right, with a few unusually small values. Example: age at death in a developed country — most people die old (right side), but infant mortality creates a left tail.

The Precise Definition

Skewness is a numerical measure of asymmetry. The most common formula (the moment-based one) is:

where $\bar{x}$ is the mean, $s$ is the standard deviation, and $n$ is the number of data points.

Why the cube? Because cubing preserves the sign — positive deviations stay positive, negative ones stay negative — and amplifies large deviations. So if there's a long tail on the right, those large positive cubed terms dominate, giving a positive skewness value. A symmetric distribution gives zero (the positive and negative cubes cancel out).

What the Number Tells You

• Skewness = 0: Perfectly symmetric (like a normal distribution).
• Skewness > 0: Positive skew (right tail longer).
• Skewness < 0: Negative skew (left tail longer). …