Skewness Formula

The Formula

\text{skewness} = \frac{n}{(n-1)(n-2)} \sum\left(\frac{x_i - \bar{x}}{s}\right)^3

When to use: A right-skewed distribution has a long tail to the right (a few very large values); left-skewed has a long tail to the left.

Quick Example

Income distribution is right-skewed: most earn moderate incomes, but a few earn millions, pulling the mean up.

Notation

Skewness is denoted \gamma_1 or g_1. Positive values (\gamma_1 > 0) indicate a right tail; negative values (\gamma_1 < 0) indicate a left tail; zero means symmetric.

What This Formula Means

A measure of how asymmetric a probability distribution is around its mean — positive skew tails right, negative skew tails left.

A right-skewed distribution has a long tail to the right (a few very large values); left-skewed has a long tail to the left.

Formal View

The sample skewness is g_1 = \frac{n}{(n-1)(n-2)} \sum_{i=1}^{n} \left(\frac{x_i - \bar{x}}{s}\right)^3. For a population, \gamma_1 = E\left[\left(\frac{X - \mu}{\sigma}\right)^3\right]. Skewness is zero for symmetric distributions.

Common Mistakes

  • Confusing the tail direction with where most data lies
  • Assuming all distributions are symmetric
  • Forgetting that outliers heavily influence skewness

Why This Formula Matters

Skewness tells you whether the mean or median is a better measure of center and whether standard statistical methods (which often assume symmetry) are appropriate for your data.

Frequently Asked Questions

What is the Skewness formula?

A measure of how asymmetric a probability distribution is around its mean — positive skew tails right, negative skew tails left.

How do you use the Skewness formula?

A right-skewed distribution has a long tail to the right (a few very large values); left-skewed has a long tail to the left.

What do the symbols mean in the Skewness formula?

Skewness is denoted \gamma_1 or g_1. Positive values (\gamma_1 > 0) indicate a right tail; negative values (\gamma_1 < 0) indicate a left tail; zero means symmetric.

Why is the Skewness formula important in Statistics?

Skewness tells you whether the mean or median is a better measure of center and whether standard statistical methods (which often assume symmetry) are appropriate for your data.

What do students get wrong about Skewness?

Positive skewness means the tail extends to the right, not that most values are large.

What should I learn before the Skewness formula?

Before studying the Skewness formula, you should understand: distribution shape.

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