Skewness

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Understanding Skewness

Definition

Skewness is a measure of asymmetry in a probability distribution. It quantifies how much a distribution deviates from perfect symmetry, where data is evenly distributed around the mean.

Formula

Sample Skewness (Pearson's Second Coefficient of Skewness):

g_1 = \frac{\frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^3}{(\frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^2)^{3/2}}

Where:

$n$ = sample size
$x_i$ = individual values
$\bar{x}$ = sample mean

Interpretation Guidelines

| g_1 |\leq 0.5

: Approximately symmetric

0.5 <| g_1 |< 1.0

: moderately skewed

| g_1 |\geq 1.0

: Highly skewed

Important Considerations

Sensitive to outliers due to the cubic term in numerator
Requires at least 3 observations to be calculated
Should be used alongside visualization for complete understanding

Visual Examples of Skewness

The following examples illustrate how skewness affects the shape of a distribution and the relationships between mean, median, and mode. Hover over the charts to explore the data points.

Approximately Symmetric Distribution

Skewness ≈ 0

Relationship: Mean ≈ Median ≈ Mode

The distribution is balanced around the mean, with similar tails on both sides.

Moderate Positive Skewness

0.5 < Skewness ≤ 1.0

Relationship: Mean > Median > Mode

The distribution has a moderate tail extending to the right.

High Positive Skewness

Skewness > 1.0

Relationship: Mean >> Median > Mode

The distribution has a long tail extending far to the right.

Moderate Negative Skewness

-1.0 ≤ Skewness < -0.5

Relationship: Mean < Median < Mode

The distribution has a moderate tail extending to the left.

High Negative Skewness

Skewness < -1.0

Relationship: Mode > Median >> Mean

The distribution has a long tail extending far to the left.

Key Takeaways

The mean is pulled in the direction of the skewness (tail)
The median is affected less by extreme values than the mean
The mode typically occurs at the peak of the distribution

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