Skewness and Kurtosis Calculator

Paste your data and get mean, variance, standard deviation, skewness, kurtosis, and excess kurtosis, plus a concise interpretation of your distribution’s shape.

Built for stats, data science, and quality engineering

Uses central-moment definitions consistent with common statistical references, and reports both population and sample variance to bridge textbook and software results.

Author: CalcDomain Stats Team

Reviewed by: Applied statistician

Last updated: 2025

Educational use only. For regulated reporting, clinical trials, or safety-critical analyses, follow your organisation’s statistical standards and validate all results independently.

Interactive skewness and kurtosis calculator

Separate values with commas, spaces, tabs, or line breaks. Non-numeric entries will trigger an error.

Variance convention

Skewness and kurtosis are always computed from central moments with denominator n; this toggle affects only the reported variance and standard deviation line.

Numeric formatting

Results may switch to scientific notation for very large or small values.

Summary statistics

n (count)
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Mean
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Variance
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Std. deviation
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Shape measures

Skewness
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Kurtosis
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Excess kurtosis
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Normal distribution reference: skewness = 0, kurtosis = 3, excess kurtosis = 0.

Plain-English interpretation

Results and a qualitative interpretation of skewness and kurtosis will appear here.

Definitions of skewness and kurtosis

Let \(x_1, x_2, \dots, x_n\) be your data and let \(\bar{x}\) be the sample mean. The central moments of order 2, 3, and 4 are:

\[ m_2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2,\quad m_3 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^3,\quad m_4 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^4. \]

In this calculator, skewness and kurtosis are defined as:

\[ \text{Skewness} = \gamma_1 = \frac{m_3}{m_2^{3/2}},\qquad \text{Kurtosis} = \gamma_2 = \frac{m_4}{m_2^{2}},\qquad \text{Excess kurtosis} = \gamma_2 - 3. \]
  • \(\gamma_1 = 0\) for a perfectly symmetric distribution.
  • \(\gamma_2 = 3\) (excess kurtosis 0) for a normal distribution.
  • \(\gamma_2 > 3\) (excess > 0) means heavier tails than normal (“leptokurtic”).
  • \(\gamma_2 < 3\) (excess < 0) means lighter tails than normal (“platykurtic”).

Population vs sample variance and standard deviation

The central moment \(m_2\) uses denominator \(n\), which corresponds to population variance. For the sample variance often used in statistical inference, the denominator is \(n-1\):

\[ s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2,\qquad s = \sqrt{s^2}. \]

The calculator reports both conventions: it always uses central moments with denominator \(n\) for skewness and kurtosis, and lets you choose which convention to display for the variance and standard deviation line.

Interpreting skewness

A common rule of thumb for skewness \(\gamma_1\) is:

  • \(|\gamma_1| < 0.5\): roughly symmetric.
  • \(0.5 \le |\gamma_1| < 1.0\): moderately skewed.
  • \(|\gamma_1| \ge 1.0\): strongly skewed.

The sign tells you the direction:

  • \(\gamma_1 > 0\): right-skewed (longer right tail; a few large values).
  • \(\gamma_1 < 0\): left-skewed (longer left tail; a few small values).

Interpreting kurtosis

Kurtosis focuses on tail behaviour relative to the normal distribution:

  • \(\gamma_2 \approx 3\) (excess ≈ 0): similar tails to normal (“mesokurtic”).
  • \(\gamma_2 > 3\) (excess > 0): heavier tails and more outliers (“leptokurtic”).
  • \(\gamma_2 < 3\) (excess < 0): lighter tails, fewer outliers (“platykurtic”).

Remember that sample estimates of kurtosis can be noisy, especially for small \(n\). Always look at skewness, kurtosis, and visual diagnostics (histogram, boxplot, Q-Q plot) together.

Worked example

Consider the small data set \(x = \{2, 3, 4, 8, 9\}\).

  • \(n = 5\)
  • \(\bar{x} = (2+3+4+8+9)/5 = 26/5 = 5.2\)

The central moments are:

\[ m_2 = \frac{1}{5}\sum (x_i - 5.2)^2 \approx 7.76,\quad m_3 \approx 7.488,\quad m_4 \approx 132.55. \]

From these we obtain:

\[ \gamma_1 = \frac{m_3}{m_2^{3/2}} \approx 0.35 \quad (\text{slightly right-skewed}),\\[4pt] \gamma_2 = \frac{m_4}{m_2^{2}} \approx 2.20,\quad \text{excess} \approx -0.80 \quad (\text{lighter tails than normal}). \]

Skewness and kurtosis: FAQs

Audit: Complete
Formula (LaTeX) + variables + units
This section shows the formulas used by the calculator engine, plus variable definitions and units.
Formula (extracted LaTeX)
\[m_2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2,\quad m_3 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^3,\quad m_4 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^4.\]
m_2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2,\quad m_3 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^3,\quad m_4 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^4.
Formula (extracted LaTeX)
\[\text{Skewness} = \gamma_1 = \frac{m_3}{m_2^{3/2}},\qquad \text{Kurtosis} = \gamma_2 = \frac{m_4}{m_2^{2}},\qquad \text{Excess kurtosis} = \gamma_2 - 3.\]
\text{Skewness} = \gamma_1 = \frac{m_3}{m_2^{3/2}},\qquad \text{Kurtosis} = \gamma_2 = \frac{m_4}{m_2^{2}},\qquad \text{Excess kurtosis} = \gamma_2 - 3.
Formula (extracted LaTeX)
\[s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2,\qquad s = \sqrt{s^2}.\]
s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2,\qquad s = \sqrt{s^2}.
Formula (extracted LaTeX)
\[m_2 = \frac{1}{5}\sum (x_i - 5.2)^2 \approx 7.76,\quad m_3 \approx 7.488,\quad m_4 \approx 132.55.\]
m_2 = \frac{1}{5}\sum (x_i - 5.2)^2 \approx 7.76,\quad m_3 \approx 7.488,\quad m_4 \approx 132.55.
Formula (extracted LaTeX)
\[\gamma_1 = \frac{m_3}{m_2^{3/2}} \approx 0.35 \quad (\text{slightly right-skewed}),\\[4pt] \gamma_2 = \frac{m_4}{m_2^{2}} \approx 2.20,\quad \text{excess} \approx -0.80 \quad (\text{lighter tails than normal}).\]
\gamma_1 = \frac{m_3}{m_2^{3/2}} \approx 0.35 \quad (\text{slightly right-skewed}),\\[4pt] \gamma_2 = \frac{m_4}{m_2^{2}} \approx 2.20,\quad \text{excess} \approx -0.80 \quad (\text{lighter tails than normal}).
Formula (extracted LaTeX)
\[','\\]
','\
Formula (extracted text)
\[ m_2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2,\quad m_3 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^3,\quad m_4 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^4. \]
Formula (extracted text)
\[ \text{Skewness} = \gamma_1 = \frac{m_3}{m_2^{3/2}},\qquad \text{Kurtosis} = \gamma_2 = \frac{m_4}{m_2^{2}},\qquad \text{Excess kurtosis} = \gamma_2 - 3. \]
Formula (extracted text)
\[ s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2,\qquad s = \sqrt{s^2}. \]
Formula (extracted text)
\[ m_2 = \frac{1}{5}\sum (x_i - 5.2)^2 \approx 7.76,\quad m_3 \approx 7.488,\quad m_4 \approx 132.55. \]
Variables and units
  • No variables provided in audit spec.
Sources (authoritative):
Changelog
Version: 0.1.0-draft
Last code update: 2026-01-19
0.1.0-draft · 2026-01-19
  • Initial audit spec draft generated from HTML extraction (review required).
  • Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
  • Confirm sources are authoritative and relevant to the calculator methodology.
Verified by Ugo Candido on 2026-01-19
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