Skewness Formula

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

The Formula

skewness=n(n−1)(n−2)∑(xi−xˉs)3\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 γ1\gamma_1 or g1g_1. Positive values (γ1>0\gamma_1 > 0) indicate a right tail; negative values (γ1<0\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 g1=n(n−1)(n−2)∑i=1n(xi−xˉs)3g_1 = \frac{n}{(n-1)(n-2)} \sum_{i=1}^{n} \left(\frac{x_i - \bar{x}}{s}\right)^3. For a population, γ1=E[(X−μσ)3]\gamma_1 = E\left[\left(\frac{X - \mu}{\sigma}\right)^3\right]. Skewness is zero for symmetric distributions.

Worked Examples

Example 1

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A dataset has mean xˉ=12\bar{x}=12, median =10=10, and mode =8=8. Describe the skew and justify using the order of these three measures.

Answer

right (positive) skew\text{right (positive) skew}

First step

1
Order: mode << median << mean (8<10<128<10<12).

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Example 2

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A dataset has mean 2525, median 3030, mode 3535. Describe the skew using the order of mean, median, and mode.

Example 3

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Data: {1,1,2,2,3,3,4,4,5,5}\{1,1,2,2,3,3,4,4,5,5\}. Compute the skew direction (no need to plug into the formula).

Common Mistakes

  • Confusing the tail direction with where most data lies - The safer move is to ask "Am I interpreting the whole distribution or a value position inside it, rather than just computing a single summary?" and then state the data source, denominator, or variable before interpreting the result.
  • Assuming all distributions are symmetric - The safer move is to ask "Am I interpreting the whole distribution or a value position inside it, rather than just computing a single summary?" and then state the data source, denominator, or variable before interpreting the result.
  • Forgetting that outliers heavily influence skewness - The safer move is to ask "Am I interpreting the whole distribution or a value position inside it, rather than just computing a single summary?" and then state the data source, denominator, or variable before interpreting the result.
  • Choosing skewness from a keyword alone - Keywords like shape, percentile, quartile are only clues; the data structure must match the concept.

Why This Formula Matters

Skewness helps students read data as a whole pattern instead of a pile of disconnected values. That habit matters because many statistical decisions depend on where a value sits in context, how symmetric the pattern is, and whether a simple summary would hide important structure.

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 γ1\gamma_1 or g1g_1. Positive values (γ1>0\gamma_1 > 0) indicate a right tail; negative values (γ1<0\gamma_1 < 0) indicate a left tail; zero means symmetric.

Why is the Skewness formula important in Statistics?

Skewness helps students read data as a whole pattern instead of a pile of disconnected values. That habit matters because many statistical decisions depend on where a value sits in context, how symmetric the pattern is, and whether a simple summary would hide important structure.

What do students get wrong about Skewness?

Students often know a procedure related to skewness but skip the recognition step: Am I interpreting the whole distribution or a value position inside it, rather than just computing a single summary? That leads to a calculation or graph that looks reasonable but answers a different question.

What should I learn before the Skewness formula?

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

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