Skewness Calculator

What Is Skewness?

Skewness measures the asymmetry of a distribution. A symmetric distribution has skewness of zero—the left and right sides are mirror images. Asymmetric distributions have non-zero skewness:

• Positive skewness (right-skewed): Long tail extends to the right. Mode < Median < Mean.
• Negative skewness (left-skewed): Long tail extends to the left. Mean < Median < Mode.

Think of the tail pointing the direction of the skew.

Visualizing Skewness

Symmetric (Skewness ≈ 0): Heights, IQ scores, measurement errors

Right-skewed (Positive): Income, house prices, insurance claims—most values are modest, but a few are very large.

Left-skewed (Negative): Age at death in developed countries, test scores on an easy exam—most values are high, but some are low.

The Skewness Formula

Sample skewness (Fisher's formula):

$g_1 = \frac{n}{(n-1)(n-2)} \sum_{i=1}^{n} \left(\frac{x_i - \bar{x}}{s}\right)^3$

Where:

• $n$ is sample size
• $\bar{x}$ is sample mean
• $s$ is sample standard deviation
• The sum runs over all observations

The cubing preserves the sign of deviations, making values above the mean contribute positively and values below contribute negatively.

Interpreting Skewness Values

These are guidelines—context matters. A skewness of 0.3 might be trivial for large samples but notable for small ones.

Why Skewness Matters

Statistical Method Selection

Many statistical techniques assume symmetric (often normal) distributions:

• t-tests and ANOVA assume normal distributions
• Linear regression assumes normally distributed residuals

Highly skewed data may require:

• Transformations (log, square root)
• Non-parametric alternatives
• Generalized linear models

Understanding Your Data

Skewness reveals data characteristics:

• Income data is typically right-skewed (skewness > 0)
• Reaction times are usually right-skewed
• Percentage data near boundaries often shows skewness

Mean vs. Median Decisions

For symmetric data, mean and median are similar. For skewed data:

• Right-skewed: Mean > Median (mean pulled toward large values)
• Left-skewed: Mean < Median (mean pulled toward small values)

When reporting central tendency for skewed data, the median often better represents the "typical" value.

Applications

Finance

Stock returns show slight negative skewness—small gains are more common, but occasional large losses create left-skew. This asymmetry challenges portfolio models assuming symmetry.

Insurance

Claim amounts are strongly right-skewed. Most claims are small, but rare large claims drive total costs. Actuaries must account for this when pricing policies.

Healthcare

Length of hospital stays is right-skewed. Most patients stay briefly, but some require extended care. Planning based on the mean overestimates typical stays.

Quality Control

Measurement errors are expected to be symmetric. Skewness indicates systematic bias in the measurement process.

Economics

Income and wealth distributions are famously right-skewed. Bill Gates entering a room dramatically changes the mean income but barely affects the median.

Historical Development

The formal study of distribution shape dates to the 19th century.

Karl Pearson developed the first skewness coefficient around 1895:

$\text{Pearson's skewness} = \frac{3(\text{Mean} - \text{Median})}{\text{Standard Deviation}}$

This simpler formula captures the mean-median relationship but is less commonly used today.

Ronald Fisher refined skewness measures in the 1920s, introducing the formula based on the third moment that is now standard.

The moment-based approach connects skewness to a broader framework where:

• First moment: Mean
• Second moment: Variance
• Third moment: Skewness
• Fourth moment: Kurtosis

Different Skewness Formulas

Pearson's First Coefficient

$\frac{\text{Mean} - \text{Mode}}{\text{Standard Deviation}}$

Requires a clear unimodal distribution.

Pearson's Second Coefficient

$\frac{3(\text{Mean} - \text{Median})}{\text{Standard Deviation}}$

Uses the relationship: $\text{Mean} - \text{Mode} \approx 3 \times (\text{Mean} - \text{Median})$

Moment Coefficient (Standard)

Based on the third standardized moment—what this calculator computes.

Quantile-Based Skewness

$\frac{Q_3 + Q_1 - 2 \times \text{Median}}{Q_3 - Q_1}$

Uses quartiles, making it robust to outliers.

Skewness and Transformations

Common transformations to reduce skewness:

Log transformation is most common for right-skewed positive data. It's particularly effective for data spanning multiple orders of magnitude.

After transformation, check that skewness is closer to zero.

Testing for Significant Skewness

For samples from a symmetric population, expected skewness is 0 but samples will show some skewness by chance.

Standard error of skewness:

$SE_{skew} \approx \sqrt{\frac{6}{n}}$

A z-test for skewness:

$z = \frac{\text{Skewness}}{SE_{skew}}$

If |z| > 2, skewness is statistically significant (rough rule).

Example: n = 100, skewness = 0.5

$SE_{skew} = \sqrt{6/100} = 0.245$

$z = 0.5 / 0.245 = 2.04$

Marginally significant—the population might genuinely be skewed.

Skewness and Sample Size

For small samples, skewness estimates are unstable. A single extreme value can dramatically affect skewness.

As sample size increases:

• Skewness estimates become more reliable
• Standard error decreases (SE ∝ 1/√n)
• Confidence intervals narrow

For samples under 20, interpret skewness cautiously.

Related Concepts

Kurtosis

While skewness measures asymmetry, kurtosis measures tail heaviness. Together, they describe distribution shape beyond mean and standard deviation.

Symmetric Distributions

Normal, uniform, logistic, and Cauchy distributions are symmetric (skewness = 0). However, they differ dramatically in kurtosis.

Bimodal Distributions

Bimodal data can have low skewness if symmetric, but high skewness if peaks are unequal. Skewness doesn't detect multimodality.

Related Calculations

After calculating skewness, explore:

• Kurtosis Calculator — Measure tail heaviness
• Mean Calculator — Affected by skewness
• Median Calculator — Resistant to skewness
• Standard Deviation Calculator — Spread measure
• Five-Number Summary Calculator — Distribution overview

Summary

Skewness quantifies distribution asymmetry, distinguishing left-tailed from right-tailed distributions. Understanding skewness guides the choice between mean and median for describing center, influences which statistical methods are appropriate, and reveals underlying data characteristics. Whether analyzing income distributions, insurance claims, or test scores, skewness provides essential information about distribution shape that measures of center and spread alone cannot capture.