Skewness Calculator (G₁, g₁, Pearson & Galton skewness, kurtosis)

Understanding the skewness calculator — concept and principle

This calculator measures how lopsided a data set is. It gives the adjusted sample skewness G₁, the moment skewness g₁, Pearson's median skewness, Galton's quartile skewness, kurtosis and excess kurtosis, along with the mean, median and count. Use it for statistics homework, checking whether data are roughly normal, and describing income, reaction-time or other skewed data.

A few large values (11 and 15) pull the mean above the median: positive skew.

How it is calculated

Moment skewness: g₁ = [Σ(x − x̄)³ ÷ n] ÷ s³, with s computed using n.
Adjusted skewness: G₁ = √(n(n − 1)) ÷ (n − 2) × g₁, the version most software reports.
Pearson median skewness: 3(mean − median) ÷ s, with the sample SD (n − 1).
Galton skewness: (Q₁ + Q₃ − 2·median) ÷ (Q₃ − Q₁), quartiles by linear interpolation (as Excel QUARTILE.INC).
Kurtosis: [Σ(x − x̄)⁴ ÷ n] ÷ s⁴; excess kurtosis = kurtosis − 3.

The principle

Cubing the deviations keeps their sign and makes large deviations count much more than small ones. If a few values sit far above the mean, their positive cubes dominate and the skewness is positive (a long right tail). Dividing by s³ makes the measure unit-free. G₁ corrects the downward bias of g₁ in small samples. The median-based and quartile-based versions are less affected by single extreme values. Kurtosis uses fourth powers and describes how heavy the tails are; a normal distribution has kurtosis 3.

Worked example

Default data: 2, 3, 3, 4, 4, 4, 5, 5, 6, 8, 11, 15 (n = 12).

Mean = 70 ÷ 12 ≈ 5.83333; median = (4 + 5) ÷ 2 = 4.5.
Moment skewness g₁ ≈ 1.394.
Adjustment factor √(12 × 11) ÷ 10 ≈ 1.1489, so G₁ ≈ 1.602: a longer right tail.
Pearson: 3(5.83333 − 4.5) ÷ 3.786 ≈ 1.057. Quartiles 3.75, 4.5, 6.5 give Galton (3.75 + 6.5 − 9) ÷ 2.75 ≈ 0.4545.
Kurtosis ≈ 3.946, excess kurtosis ≈ 0.9463.

How to read the result

Where does your result fall?

Skewness near 0 means a roughly symmetric shape. A common rule of thumb treats values between about −0.5 and 0.5 as fairly symmetric and values beyond ±1 as strongly skewed, but these are guidelines, not tests. Positive skew often appears in incomes, house prices and waiting times. Positive excess kurtosis means heavier tails than a normal distribution.

Common mistakes

Comparing G₁ from one program with g₁ from another; check which formula is used.
Drawing conclusions from skewness with very small samples, where it is unstable.
Reading the sign backwards: positive skew means the tail points right, not the bulk.
Reporting kurtosis and excess kurtosis interchangeably.

Keep in mind: Skewness from a sample is itself uncertain, especially below about 30 values. A single outlier can change it a lot; look at a histogram before drawing conclusions.

Frequently asked questions

What does positive skewness mean?

The distribution has a longer tail on the right, so a few values are much larger than most.

Which skewness does Excel report?

Excel's SKEW function returns the adjusted version, G₁.

Is skewness 0 proof of a normal distribution?

No. Symmetric distributions that are not normal also have zero skewness.

Sources: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.11 Measures of Skewness and Kurtosis

Checked 9 Nov 2026. NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.11 Measures of Skewness and Kurtosis (Fisher–Pearson g₁ with s computed with N; adjusted G₁ = √(N(N − 1))/(N − 2)·g₁; kurtosis Σ(Y − Ȳ)⁴/N ÷ s⁴, excess kurtosis = kurtosis − 3); NIST Dataplot reference “SKEWNESS” (Galton skewness (Q₁ + Q₃ − 2Q₂)/(Q₃ − Q₁); Pearson 2 skewness 3(Ȳ − Ỹ)/s).

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