Skewness

Distributions
definition

Also known as: skew, distribution skew

Grade 9-12

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

Definition

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

💡 Intuition

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.

🎯 Core Idea

Positive skew: tail on right, mean > median. Negative skew: tail on left, mean < median.

Example

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

Formula

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

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.

🌟 Why It 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.

💭 Hint When Stuck

To determine skewness, compare the mean and median. If the mean is greater than the median, the distribution is right-skewed (positive). If the mean is less than the median, it is left-skewed (negative). You can also look at a histogram: the direction of the longer tail tells you the skew direction.

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 Stuck Point

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

⚠️ Common Mistakes

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

Frequently Asked Questions

What is Skewness in Statistics?

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

What is the Skewness formula?

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

When do you use Skewness?

To determine skewness, compare the mean and median. If the mean is greater than the median, the distribution is right-skewed (positive). If the mean is less than the median, it is left-skewed (negative). You can also look at a histogram: the direction of the longer tail tells you the skew direction.

Prerequisites

distribution shape

Next Steps

mean vs median

How Skewness Connects to Other Ideas

To understand skewness, you should first be comfortable with distribution shape. Once you have a solid grasp of skewness, you can move on to mean vs median.

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