# Skewness & Kurtosis Calculator

Free skewness and kurtosis calculator. Assess the symmetry and tail heaviness of your data distribution using Pearson's skewness and excess kurtosis.

## What this calculates

Calculate skewness and kurtosis to understand the shape of your data distribution. Skewness measures symmetry; kurtosis measures tail heaviness relative to a normal distribution.

## Inputs

- **Mean (x̄)** — The arithmetic mean of the data.
- **Median** — The median of the data.
- **Mode** — The mode of the data (for Pearson's skewness).
- **Standard Deviation (s)** — min 0.0001 — The standard deviation of the data.
- **Sample Size (n)** — min 4 — Number of observations.
- **Σ(xᵢ - x̄)³ (optional)** — Sum of cubed deviations from the mean. Required for exact skewness.
- **Σ(xᵢ - x̄)⁴ (optional)** — Sum of fourth-power deviations. Required for exact kurtosis.

## Outputs

- **Pearson's First Skewness (mode)** — (Mean - Mode) / Std Dev.
- **Pearson's Second Skewness (median)** — 3(Mean - Median) / Std Dev.
- **Sample Skewness (moment-based)** — Fisher's skewness coefficient (if cubed deviations provided).
- **Skewness Interpretation** — formatted as text — What the skewness value indicates.
- **Excess Kurtosis** — Kurtosis relative to normal distribution (normal = 0).
- **Kurtosis Interpretation** — formatted as text — What the kurtosis value indicates.

## Details

Skewness measures the asymmetry of a distribution:

- Skewness = 0: Symmetric (like normal distribution)

- Skewness > 0: Right-skewed (tail extends right, mean > median)

- Skewness < 0: Left-skewed (tail extends left, mean < median)

Pearson's Approximations

- First: (Mean - Mode) / SD

- Second: 3(Mean - Median) / SD

Kurtosis measures tail heaviness:

- Excess kurtosis = 0: Mesokurtic (normal distribution)

- Excess kurtosis > 0: Leptokurtic (heavy tails, more outliers)

- Excess kurtosis < 0: Platykurtic (light tails, fewer outliers)

## Frequently Asked Questions

**Q: What is the difference between kurtosis and excess kurtosis?**

A: Kurtosis (raw) for a normal distribution is 3. Excess kurtosis subtracts 3, so a normal distribution has excess kurtosis of 0. Most statistical software reports excess kurtosis. This calculator reports excess kurtosis so that 0 = normal.

**Q: Why are there different formulas for skewness?**

A: Pearson's first coefficient uses the mode (which can be unreliable), Pearson's second uses the median (more robust), and the moment-based (Fisher's) coefficient uses actual deviations from the mean (most precise but requires raw data or summed deviations). All measure the same concept.

**Q: How much skewness is acceptable for assuming normality?**

A: A common rule of thumb: |skewness| < 2 and |excess kurtosis| < 7 are often considered acceptable for most parametric tests. For strict normality, |skewness| < 0.5 is preferred. With large sample sizes (n > 30), moderate skewness is less problematic due to the Central Limit Theorem.

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Source: https://vastcalc.com/calculators/statistics/skewness-kurtosis
Category: Statistics
Last updated: 2026-04-21