Which way the data leans — in all three conventions, with the noise shown.
Which way it leans — in all three conventions
Skewness over 10 values
1.098042
Right-skewed — a longer tail on the high side. Standard error 0.6870, so this is 1.598 standard errors from zero.
g₁ — moment ratio
1.098042
most textbooks
G₁ — adjusted
1.302118
Excel SKEW, SAS, SPSS
b₁ — Minitab
0.937525
Standard error
0.687043
under normality
Three formulas, three answers: 1.0980, 1.3021 and 0.9375. The adjusted version is larger than the moment ratio by a factor of √(n(n − 1))/(n − 2) = 1.18585 at n = 10 — 18.6% higher. That factor is 1.49 at n = 5, 1.19 at n = 10, 1.05 at n = 30 and 1.0015 at n = 1000, so the conventions diverge precisely where small samples make the statistic unreliable anyway. If a spreadsheet and a textbook disagree about your skewness, this is why: Excel’s SKEWreturns G₁, and most courses define g₁.
This skewness is within the range a perfectly normal sample of 10 produces by chance. The standard error is 0.6870, so anything up to about ±1.3741 is unremarkable here. Fixed rules like “above 0.5 means skewed” ignore sample size entirely, and at small n they flag noise: at n = 10 the 95% band is roughly ±1.4, and at n = 30 still ±0.85. Judge against the band, not the number.
Mean
11.100000
Median
8.000000
Mean − median
3.100000
the intuitive skew signal
Pearson's second
1.003558
3(mean − median)/s
The mean sits 3.10000 above the median. That gap is the version of skewness you can actually see, and it is where the moment ratio’s sign comes from. It is not a reliable rule in general — distributions exist with a positive moment skewness and a mean below the median — but on ordinary unimodal data the two agree, and the gap is far easier to explain to a reader than a third moment is. for a six-value counterexample: g₁ is +0.361, unambiguously right-skewed, while the mean (9.833) sits below the median (10).
What this tool shows
“The” skewness names three different formulas. On ten values they read 1.098, 1.302 and 0.938, and Excel returns the middle one while most courses teach the first. This shows all three, plus the standard error — because at n = 30 a perfectly normal sample swings ±0.85 by chance.
g₁, the moment ratio most textbooks define
G₁, what Excel’s SKEW, SAS and SPSS return
b₁, Minitab’s variant
The standard error, so the number can be judged
Pearson’s second coefficient, a different quantity
Why “mean above median” is not a reliable rule
Three conventions Standard error shown Counterexample built in Exact moment ratios
[7, 3, 11, 11, 18, 9] is right-skewed with its mean below its median.
Updated 8 September 2026 · Works in any browser, no installation
Skewness measures asymmetry. Positive means a longer tail to the right, negative to the left, zero means symmetric. What complicates it is that three formulas share the name, and that the statistic is considerably noisier than the thresholds it gets judged against.
At a glance
Formula shown
g₁ = m₃/m₂^1.5, where mₖ is the kth central moment divided by n. The sample-adjusted version is G₁ = g₁·√(n(n−1))/(n−2), which is what Excel’s SKEW returns; Minitab uses b₁ = g₁·((n−1)/n)^1.5. Under normality the standard error of g₁ is √(6n(n−1)/((n−2)(n+1)(n+3))).
Scenario support
Deciding whether to log-transform; checking a normality assumption before a t-test; describing income, duration or price data; interpreting why a mean and a median differ; reconciling a spreadsheet with a textbook.
Educational estimate
Planning support from the values you enter — not professional advice.
Three formulas, one name
If a spreadsheet and a statistics course give you different skewness values for the same data, neither is broken. They are computing different things that share a name.
g₁ is the moment ratio, m₃/m₂1.5— the definition most textbooks give and the one that follows directly from the theoretical quantity.
G₁ is the sample-adjusted version, g₁ multiplied by √(n(n − 1))/(n − 2). It corrects a bias: g₁ systematically underestimates a population’s skewness in small samples. Excel’s SKEW, SAS and SPSS all return this one.
b₁ is Minitab’s, g₁ times ((n − 1)/n)1.5, which goes the other way and is smaller than g₁.
On the tool’s default ten values they read 1.098, 1.302 and 0.938. The largest is 39% above the smallest. All three are defensible and all three are labelled “skewness”.
The gap depends entirely on n. The adjustment factor is 1.49 at n = 5, 1.19 at n = 10, 1.05 at n = 30, 1.015 at n = 100 and 1.0015 at n = 1000. So the conventions diverge most sharply exactly where the statistic is least trustworthy in the first place — which is a reason to report n alongside, and to name the convention when the number matters.
In R, moments::skewness gives g₁ while e1071::skewness(type = 2) gives G₁. Two packages in one language, two answers.
It is noisier than the thresholds assume
“Skewness above 0.5 means the data is skewed” and “between −1 and 1 is acceptable” circulate as rules. Both ignore sample size, and sample size is most of the story.
Under normality, the standard error of g₁ is about √(6/n). At n = 30 that is 0.427, so a 95% band runs to roughly ±0.85. A perfectly normal sample of thirty produces a skewness of 0.7 quite regularly. Flagging it as skewed is flagging noise.
At n = 10 the band is about ±1.4. Almost any skewness you can compute from ten values is consistent with a symmetric population. The statistic is nearly uninformative there, which is worth knowing before basing a transformation decision on it.
At n = 1000 the band is ±0.15, and now a skewness of 0.3 is unambiguous. The same number means different things at different sample sizes, and that is what a fixed threshold cannot express.
The tool reports how many standard errors your skewness sits from zero, which is the comparison that actually answers the question. Roughly: above 2 is real, below 2 is not distinguishable from a symmetric population.
The corollary cuts the other way too. With a very large sample almost any skewness becomes “significant”, including one far too small to affect anything. At n = 100,000 a skewness of 0.02 clears two standard errors and means nothing in practice. This is the same significance-versus-magnitude problem that afflicts every large-sample test.
The mean-median rule is not a theorem
“If the mean is above the median, the data is right-skewed” is taught almost universally and is not true.
Here is a six-value counterexample: 7, 3, 11, 11, 18, 9. The moment skewness g₁ is +0.361, unambiguously right-skewed. The mean is 9.833 and the median is 10. The mean sits below the median while the skewness is positive. You can check every step of that by hand, and the tool loads it as a preset.
The rule fails because the two quantities measure different things. The moment skewness is driven by cubed deviations, so a few far-out values dominate it. The mean-median gap responds to the bulk of the distribution. Usually they point the same way; there is no theorem saying they must, and discrete or multimodal data breaks the correspondence regularly.
This matters more than it sounds. The rule is often used in reverse — comparing a mean and a median to infer the skew — and that inference can be wrong. If the direction of the asymmetry matters, compute the skewness rather than deducing it.
The gap is still the better thing to report. “The mean income is 20% above the median” tells a reader something concrete about the distribution; “the skewness is 1.098” tells almost nobody anything. Use the moment measure to decide, and the gap to communicate. The mean, median and mode calculator shows that gap directly.
Pearson’s second coefficient, 3(mean − median)/s, is the formalised version of the gap. It is a legitimate skewness measure with its own name and it is not on the same scale as g₁, so quoting one against a threshold meant for the other is another way to get the wrong answer.
What to do about it
Finding skew is only useful if it changes something. Four things it can change.
Report the median instead of the mean. Often the whole fix. For right-skewed data the median is the typical value and the mean answers a different question — the total divided by the count, which matters for a budget and misleads as a description.
Consider a log transform, if the process is multiplicative. Strongly right-skewed positive data is frequently lognormal, and taking logs makes it symmetric and makes ordinary methods valid. Remember to interpret carefully afterwards: exponentiating the mean of the logs gives the geometric mean, which equals the median rather than the arithmetic mean.
Do not panic about a t-test. Skewness affects the central limit theorem’s convergence rate, and with a moderate sample a two-sided test is fairly robust. The problem is sharper for one-sided tests and for the tails: on an exponential population at n = 30, a nominal one-sided 5% test really runs at 5.9%.
Use a rank-based method and stop worrying. A Mann-Whitney test, a Spearman correlation or a bootstrap interval makes no symmetry assumption, so the skewness stops being a threat and becomes just a description.
What not to do is delete the tail. Trimming a distribution until it looks symmetric is not a transformation, it is discarding the observations that carry the most information about the behaviour you are trying to describe.
Transformations beyond the logarithm
The log transform is the default response to right skew, and it fails on two common shapes. Knowing the alternatives saves reaching for it where it cannot apply.
The log needs strictly positive data. A single zero breaks it, and negative values break it entirely. The habitual fix — adding 1 before logging — is not neutral: it changes the shape by an amount that depends on the units, so the same data in grams and kilograms transforms differently. If you use it, say so.
The square root is milder and handles zeros. It is the conventional choice for count data, where it also stabilises the variance of a Poisson process — a genuine second reason rather than just a weaker log.
Box-Cox chooses the exponent for you by fitting the power that best normalises the data, with the log as its limiting case. It still requires positive values. Yeo-Johnson is the variant that handles zeros and negatives, which makes it the right tool for profit, temperature change or any quantity that can go either way.
The reciprocal is the strongest of the common transforms and reverses the order of the data, so an interpretation stated on the transformed scale means the opposite of what it looks like. It is natural for rates — time per unit becomes units per time — and confusing anywhere else.
Consider not transforming at all. Every transform makes the coefficients harder to explain, and a rank-based method or a bootstrap interval keeps the data on its own scale while making no symmetry assumption. Transform when the process is genuinely multiplicative, not merely because a histogram leans.
Sources and methodology
References for the conventions and the counterexample.
Method. All three conventions are computed from the same central moments rather than by rescaling one another, so each is exact rather than derived. The standard error uses the exact normal-theory expression √(6n(n − 1)/((n − 2)(n + 1)(n + 3))) rather than the √(6/n) approximation, which overstates precision at small n where it matters most. The suite asserts the three conventions’ ratios against their algebraic definitions across a grid of n, that a symmetric dataset gives exactly zero under all three, that reversing a dataset negates the skewness exactly, and that [7, 3, 11, 11, 18, 9] has g₁ = +0.361 with a mean of 9.833 below its median of 10 — the counterexample that stops this page repeating the mean-median rule as a fact. That engine is verified on every change against 62 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
KurtosisExcess, raw and Excel KURT side by side, with the tail-weight reading made explicit — kurtosis measures tails, not peakedness, and "kurtosis 3" means two opposite things.
Mean, Median and ModeAll three centres marked on your own data, every mode rather than just the first, and the mean-median gap read as a direct measure of skew.
Standard DeviationSample and population standard deviation, plus variance, mean, median, quartiles, z-scores, outliers, and confidence intervals.
Lognormal DistributionTakes log-scale parameters or your data's own mean and SD and converts between them — because μ describes ln x, not x, and entering the wrong one is wrong by orders of magnitude with nothing to flag it.
Normal DistributionProbabilities under a normal curve in all four directions with the region shaded — and the empirical rule given exactly, because two standard deviations is 95.45% and the 95% everyone quotes sits at 1.96σ.
Central Limit TheoremComputes exact sampling distributions instead of simulating them, so you can see what a nominal 95% interval really covers — and that the two-sided figure hides a tail 36% too large.
An educational tool. Skewness is a noisy statistic at small sample sizes, and the three conventions in circulation differ most where n is smallest — report the sample size and the convention alongside any value.
Published a skewness calculator that shows all three conventions at once. On ten values they read 1.098, 1.302 and 0.938, and Excel's SKEW returns the middle one while most courses define the first — an 18.6% gap at n = 10 and 49% at n = 5.
Reports the standard error beside the value, because at n = 30 a perfectly normal sample swings plus or minus 0.85 by chance, which makes fixed thresholds like 'above 0.5 means skewed' a way of flagging noise.
Includes a six-value counterexample to the mean-median rule taught almost universally: 7, 3, 11, 11, 18, 9 has a moment skewness of +0.361 with a mean of 9.833 sitting below its median of 10.
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