Skewness Calculator
Asymmetry, computed two ways: the raw moment coefficient g₁ beside its small-sample-corrected form G₁ — the third life of the divisor doctrine — with the tail direction named and no magic bar for “significant” asymmetry.
Skewness Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Moment-based only — no rank/quantile skewness alternatives
- Small samples: g₁ is biased; G₁ corrects but stays jumpy below n = 20
In short: Data 2, 4, 4, 4, 5, 5, 7, 9: the raw skewness is g₁ = 0.656250 and the bias-corrected G₁ = 0.818488 — right-skewed: the mass crowds left of the mean while a long right tail (that 9) drags the average toward it. The page prints BOTH coefficients because they are not the same tool: g₁ is what the moment arithmetic gives, G₁ = √(n(n−1))/(n−2)·g₁ unbiases it for small samples — at n = 8 they differ by 0.162238, the same lesson the variance page teaches with n vs n − 1, third life. A constant list is refused: shape needs spread.
Formula
g₁ = m₃/m₂^1.5 · G₁ = √(n(n−1))/(n−2) · g₁
m₂ and m₃ are the second and third central moments of YOUR list. The adjusted form is the Excel-SKEW convention — printed beside the raw moment version so the small-sample correction is visible instead of silent.
Worked Example
- Paste 3 to 500 values.
- Read the adjusted G₁ (the headline) beside the raw g₁ — the gap is small-sample bias, priced.
- The direction card names which tail is long.
- Sanity-check against the mean–median order the summary cards print.
Defaults: g₁ = 0.656250, G₁ = 0.818488, right-skewed (the 9’s tail). Constant lists and n < 3 refused — shape needs spread and a third moment.
Strengths & Limits Of This Model
Where this engine is strong
- Raw and adjusted coefficients printed together — the correction priced, not hidden
- Direction named with the mean–median order shown on your own data
Where it stops
- No significance test for skewness (no standard error printed)
- Says nothing about modality — two humps can hide behind g₁ = 0
Practical Use Cases
Data QC
is this variable roughly symmetric?
Methods
report G₁ beside the normality claim
Teaching
moments beyond mean and variance
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Skewness Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why two skewness numbers?
The same n vs n − 1 story, third life. g₁ is the raw moment ratio — what the arithmetic gives; G₁ multiplies by √(n(n−1))/(n−2) to remove its small-sample bias, and is what Excel’s SKEW prints. At n = 8 the gap here is 0.162238 — real money in small samples, vanishing as n grows. This page prints both so the correction is never silent.
What does positive skewness mean?
A long RIGHT tail: most values crowd at the low end while a few large ones drag the mean above the median — incomes and waiting times live there. Negative mirrors it. Zero means moment-symmetric, which is weaker than it sounds: the correlation page’s x² card is perfectly asymmetric by eye yet has g₁ = 0 by symmetry.
Is 0.656250 “very” skewed?
The page declines to grade it — there is no universal bar, and inventing one (“above 1 is bad”) would be a convention wearing a lab coat. What is honest: the direction card, the mean–median gap on your own data, and comparison against a reference variable you already trust.
Why does skewness need at least 3 values?
The third central moment needs at least three deviations to be estimable at all, and G₁’s correction divides by n − 2 — at n = 3 the adjusted form is already wild. The refusal names the arithmetic; the advice is that skewness stabilizes only well past n = 20.
My data is a list of residuals — does that change anything?
No — the arithmetic never asks what the numbers mean. Residual skewness is one of the standard regression sanity checks (the regression page’s residual card lists them), and this page computes it on any list you paste.
Why did my constant list get refused?
Shape needs spread: a constant list has m₂ = 0 and g₁ = 0/0 — not a symmetric distribution, no distribution at all. The page refuses rather than printing a decorated 0, the same discipline the CV page applies at mean 0.
How does skewness relate to the normal distribution page?
The normal curve is the skew-0, kurtosis-0 reference: every normality argument is measured against it. This page prices ONE of those coordinates for your data; the kurtosis page prices the other; the normal page prices probabilities once the shape is granted.
Should I transform my data because of skewness?
Only for a REASON: many models lean on symmetry, and a log or square-root transform may buy it — at the cost of interpreting a new scale. The transformation question is downstream; this page prices the asymmetry honestly so the decision is made on a number, not a squint at a chart.