Skewness

This Skewness Calculator helps you measure the asymmetry of your data distribution. It calculates the skewness coefficient, standard error, and statistical significance, helping you understand whether your data leans more towards one side. It also provides histogram with density estimation to visualize the distribution of your data.

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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.

Formulas

Pearson's First Coefficient of Skewness (Moment Coefficient):

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}}

Based on the third standardized moment of the distribution. More sensitive to extreme values.

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

SK_2 = \frac{3(\bar{x} - \text{median})}{s}

Based on the relationship between mean, median, and standard deviation. More robust to outliers.

Where:

$n$ = sample size
$x_i$ = individual values
$\bar{x}$ = sample mean
$s$ = sample standard deviation
$\text{median}$ = middle value when data is ordered

Note: While both coefficients measure skewness, they can give slightly different results. The First Coefficient is more commonly used in statistical software but can be more sensitive to outliers. The Second Coefficient is simpler to interpret and more robust to extreme values.

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